calabi’s extremal k ahler metrics: an elementary introductionrodion/lib/p. gauduchon....

Calabi’s extremal Kahler metrics: An

elementary introduction

Paul Gauduchon

Contents

Introduction 7

Chapter 1. Elements of Kahler geometry 91.1. Kahler manifolds 91.2. Riemannian and symplectic duality 111.3. Inner product and exterior derivative 121.4. Lie derivative and the Cartan formula 131.5. Symplectic vector fields and Poisson brackets 141.6. Cauchy-Riemann operators and Chern connections 151.7. Chern connections of complex line bundles 201.8. Real vs complex viewpoint 221.9. The type of exterior forms 241.10. Riemannian and symplectic Hodge operators 251.11. Twisted exterior differential and twisted codifferential 271.12. J-invariant 2-forms, hermitian operators and Chern forms 321.13. The Hodge-Lepage decomposition 331.14. Kahler identities 361.15. Laplace and twisted Laplace operators 381.16. The Akizuki-Nakano identity 401.17. The ddc-lemma 411.18. Riemannian curvature and Bianchi identities 441.19. The Ricci form of a Kahler structure 451.20. The Calabi-Yau theorem 491.21. Kahler-Einstein metrics 511.22. Bochner identities and vanishing theorems 541.23. The Lichnerowicz fourth order operator 60

Chapter 2. Holomorphic vector fields on a compact Kahler manifold 652.1. The space of holomorphic vector fields 652.2. The space of Killing vector fields 682.3. Harmonic 1-forms and parallel vector fields 692.4. The ideal hred and the reduced automorphism group 712.5. The Calabi operator L+ and its conjugate 742.6. The space of hamiltonian Killing vector fields 76

Chapter 3. Calabi extremal Kahler metrics 793.1. The space of Kahler metrics within a fixed Kahler class 793.2. The Calabi functional 803.3. Semi-positivity of the second derivative at critical points 823.4. Holomorphic vector fields on a compact extremal Kahler manifold 843.5. Maximality of K(M, g) in H(M,J) 863.6. Kahler metrics with constant scalar curvature and Kahler-Einstein

metrics 87

3

4 CONTENTS

Chapter 4. More on the geometry of the space of Kahler metrics 914.1. The space of relative Kahler potentials 914.2. Aubin functionals 934.3. The Mabuchi metric 964.4. The Levi-Civita connection of the Mabuchi metric 974.5. Group action of H(M,J) on MΩ 984.6. The equation of geodesics 994.7. The Moser lift as a horizontal lift 1024.8. The geodesic equation as a Monge-Ampere equation 1044.9. The curvature of the Mabuchi metric 1064.10. The Mabuchi K-energy 1074.11. The Futaki-Mabuchi bilinear form 1094.12. The Futaki character 1114.13. The extremal vector field of a Kahler class 1134.14. Relative Mabuchi K-energy and relative Futaki character 1164.15. Uniqueness of extremal Kahler metrics 120

Chapter 5. The extremal Kahler cone 1255.1. LeBrun-Simanca openness theorem 1255.2. A few variational formulae 1265.3. The end of the proof of LeBrun-Simanca openness theorem 1315.4. Strongly extremal Kahler metrics 1325.5. The four-dimensional case: the Bach tensor 136

Chapter 6. The complex projective space 1436.1. The complex projective space as a complex manifold 1436.2. The Fubini-Study metric 1466.3. The complex projective space as a (co)adjoint orbit 1516.4. The blowing up process 1536.5. Blow-up of projective spaces and Hirzebruch surfaces 1576.6. Fano complex surfaces 160

Chapter 7. Kahler-Einstein metrics on Fano manifolds 1657.1. The Ricci potential of a Fano manifold 1657.2. Holomorphic vector fields and Poincare inequalities 1677.3. The modified Monge-Ampere equation 1717.4. The Tian constant 1777.5. The Ding functional 180

Chapter 8. Polarized Kahler manifolds 1858.1. Polarized Kahler manifolds 1858.2. The Bergman density of a polarized Kahler manifold 1908.3. The Kodaira embedding theorem 1938.4. Bergman metrics on polarized manifolds 1978.5. The Ricci form of a complex hypersurface 2028.6. The Donaldson gauge functional as a Kahler potential 2038.7. Balanced metrics 2078.8. S1-actions on polarized Kahler manifold 2098.9. The Futaki character and the Futaki-Mabuchi bilinear forms as

asymptotic invariants 216

Chapter 9. The scalar curvature as a momentum map 2219.1. The symplectic viewpoint: the space Jω0 221

CONTENTS 5

9.2. The space of symplectic almost complex structures as a Kahlermanifold 225

9.3. The Chern connection of an almost Kahler structure 2299.4. The hermitian scalar curvature 2339.5. The Mohsen formula 2369.6. Hamiltonian action of Ham(M,ω0) on ACω0

2399.7. A symplectic Futaki invariant 2429.8. The Mabuchi K-energy as a Kahler potential 2439.9. The relative case 246

Chapter 10. Extremal Kahler metrics on Hirzebruch-like ruled surfaces 24910.1. Complex ruled surfaces 25010.2. Hirzebruch-like ruled surfaces 25210.3. Admissible Kahler metrics 25710.4. Extremal admissible Kahler metrics 26510.5. Bach-flat Kahler metrics on Hirzebruch-like rules surfaces: the Page

metric 27010.6. Weakly selfdual Kahler metrics and hamiltonian 2-forms 27510.7. The Futaki character of admissible ruled surfaces 28210.8. The extremal vector field of a Hirzebruch-like ruled surface 28410.9. Extremal metrics on complex ruled surfaces 286

Appendix A. Bochner tensor and Apte formulae 291A.1. Curvature decomposition 291A.2. The Bochner tensor 293A.3. Apte formulae 295A.4. The four-dimensional case 297

Appendix B. The space of almost complex structures compatible with asymplectic form 299

B.1. The Cayley transformation 299B.2. Symplectic complex structures 300

Appendix. Bibliography 303

Introduction

sintroduction

7

CHAPTER 1

Elements of Kahler geometry

skaehler1.1. Kahler manifolds

ssdef

An almost hermitian manifold M = (M, g, J) is a manifold of (real) evendimension n = 2m, equipped with a riemannian metric g and a g-compatiblealmost complex structure J . By definition, J is a field of automorphisms ofthe tangent bundle TM such that J2 = −I, where I denotes the identity —TM can then be regarded as a complex vector bundle of rank m — and Jis said g-compatible if the identity g(JX, JY ) = g(X,Y ) holds for any twovector fields X,Y . The bilinear form ω defined by ω(X,Y ) := g(JX, Y ) isthen a (skew-symmetric) 2-form and is called the Kahler form of the almosthermitian structure.

The Levi-Civita connection D = Dg of a riemannian metric g is theunique connection on M , acting on vector fields and naturally extended toany kind of tensor fields, which is torsion-free and preserves the metric, i.e.is such that Dg = 0. It can be expressed by the following Koszul formula :

2g(DXZ, Y ) = X · g(Z, Y ) + Z · g(X,Y )− Y · g(X,Z)

+ g([X,Z], Y ) + g([Y,X], Z) + g(X, [Y, Z]),koszulkoszul (1.1.1)

for any vector fields X,Y, Z, where [·, ·] stands for the usual bracket of vectorfields.

A Kahler manifold is then defined as an almost hermitian manifold(M, g, J, ω) such that DJ = 0. An equivalent formulation is given by thefollowing proposition:

propbasic Proposition 1.1.1. An almost hermitian manifold (M, g, J, ω) is Kahlerif and only if J is integrable — so that (M,J) is a complex manifold — andω is closed — so that (M,ω) is a symplectic manifold.

Proof. By the celebrated Newlander-Nirenberg theorem [154], an al-most complex structure J is integrable if and only if the Nijenhuis tensorN = NJ defined by

NN (1.1.2) N(X,Y ) =1

4([JX, JY ]− J [JX, Y ]− J [X, JY ]− [X,Y ])

is identically zero. The rhs can then be expressed in terms of the covariantderivative DJ of J and we then get

NbisNbis (1.1.3) N(X,Y ) =1

4((DJXJ)Y − J(DXJ)Y − (DJY J)X + J(DY J)X) .

We readily infer that DJ = 0 implies N = 0. By its very definition, theKahler form ω is non-degenerate at each point of M . Moreover, its exteriorderivative dω is expressed in terms of its covariant derivative Dω by

symplecticsymplectic (1.1.4) dω(X,Y, Z) = (DXω)(Y,Z) + (DY ω)(Z,X) + (DZω)(X,Y ),

9

10 1. ELEMENTS OF KAHLER GEOMETRY

cf. (1.3.5). Since the Levi-Civita connection D preserves the metric g thetwo conditions DJ = 0 and Dω = 0 are equivalent and, by (1.1.4), implythat ω is closed, hence a symplectic 2-form. The converse is a consequenceof the identity

g((DXJ)Y,Z) =1

2(dω(X,Y, Z)− dω(X, JY, JZ))

+ 2g(JX,N(Y,Z)),KNKN (1.1.5)

which can be easily checked by using (1.1.3)-(1.1.4).

Remark 1.1.1. Formula (1.1.5) appears in [121], Vol. II; beware how-ever that Kobayashi-Nomizu use different conventions for N and for theexterior derivative; our own conventions will be made precise in the nextsections.

remalmostsymplectic Remark 1.1.2. An almost hermitian structure (M, g, J, ω) is called her-mitian if J is integrable, almost Kahler — or simply symplectic — if ω isclosed (hence symplectic). By the above proposition, a Kahler structurecan be alternatively defined as an almost hermitian structure which is bothhermitian and symplectic. The following criterion is often useful:

propintD Proposition 1.1.2. An almost hermitian structure (M, g, J) is hermit-ian if and only if the following identity holds:

DJDJ (1.1.6) DJXJ = J DXJ,

for each vector field X.

Proof. By (1.1.3), (1.1.6) readily implies N = 0, whereas the conversefollows from the reciprocal identity

1

2g((DXJ)Y, Z) + J(DJXJ)Y, Z) =

g(JX,N(Y,Z))− g(JY,N(Z,X))− g(JZ,N(X,Y )),

reciprocalreciprocal (1.1.7)

whose easy verification is left to the reader.

Another usefull integrability criterion for J is given by

lemmaintJ Lemma 1.1.1. An almost complex structure J is integrable if and onlyif the identity

LXJLXJ (1.1.8) LJXJ = JLXJ,holds for any vector field X.

Proof. Here and henceforth LX denotes the Lie derivative along X,acting on any sort of tensor fields, cf. Section 1.4. In the current case wehave that (LXJ)(Y ) = [X, JY ]− J [X,Y ], so that

LXJbisLXJbis (1.1.9) (LJXJ − JLXJ)(Y ) = 4N(X,Y ),

for any vector fields X,Y . The conclusion follows immediately.

Until Chapter 9 and unless otherwise specified, M = (M, g, J, ω) willdenote a connected Kahler manifold of (real) dimension n = 2m, with rie-mannian metric g, complex structure J and Kahler (symplectic) form ω.

1.2. RIEMANNIAN AND SYMPLECTIC DUALITY 11

1.2. Riemannian and symplectic dualityssdual

In general, the riemannian volume form, vg, of an oriented, n-dimensionalriemannian manifold (M, g) is determined by

(1.2.1) vg(e1, . . . , en) = 1,

for any (local) positively oriented orthonormal frame e1, . . . , en; with re-spect to any local, positively oriented, coordinate system x1, . . . , xn, we thenhave

volumeformgenvolumeformgen (1.2.2) vg = |∂/∂x1 ∧ . . . ∧ ∂/∂xn| dx1 ∧ . . . dxn,where |∂/∂x1 ∧ . . . ∧ ∂/∂xn| denotes the norm with respect to g of the n-multivector ∂/∂x1∧ . . .∧∂/∂xn, equal to the square root of the determinantof the matrix gij := g(∂/∂xi, ∂/∂xj), cf. Section 1.3.

In the current Kahler situation — more generally in the almost hermit-ian case — the riemannian volume form then coincides with the symplecticvolume form , i.e.

volumeformvolumeform (1.2.3) vg =ωm

m!

(easy verification).The riemannian duality is denoted by ], in the direction T ∗M → TM ,

and by [ in the direction TM → T ∗M : for any 1-form α and for any vectorfields X,Y , we then have α(Y ) = g(Y, α]) and X[(Y ) = g(Y,X). The samesymbols will be used for the induced isomorphisms from ΛpM = Λp(T ∗M)to Λp(TM) and vice versa.

The symplectic duality is similarly denoted by ]ω from T ∗M to TMand [ω from TM to T ∗M , with α(Y ) = ω(Y, α]ω) and X[ω(Y ) = ω(Y,X).Equivalently,

symplectic-dualsymplectic-dual (1.2.4) α = −ια]ωω, X[ω = −ιXω,for any 1-form α and any vector field X.

The action of J on T ∗M is defined in such a way that the isomorphisms] and [, as well as ]ω and [ω, are J-linear: for any 1-form we then set

J1formsJ1forms (1.2.5) (Jα)(X) = α(J−1X) = −α(JX).

This operator is then extended to p-forms byJpformsJpforms (1.2.6)

(Jψ)(X1, . . . , Xp) = ψ(J−1X1, . . . , J−1Xp) = (−1)pψ(JX1, . . . , JXp)

(Jf := f , if f is a 0-form, i.e. a function).As an operator acting on the space of p-forms, J then satisfies J2 =

(−1)p.The riemannian and the symplectic duality are then related to each other

by

(1.2.7) ψ]ω = Jψ],

for each p-form ψ. In particular, the symplectic gradient, gradω f , of a (real)function f , defined as the symplectic dual of df , is related to the (usual)riemannian gradient grad f by

(1.2.8) gradω f = Jgrad f.


The symplectic form ω allows one to define a Poisson bracket in thespace C∞(M,R) of smooth real functions on M . Recall that for any tworeal functions f1, f2 on any symplectic manifold (M,ω) of dimension n = 2m,the Poisson bracket f1, f2 = f1, f2ω is defined by

poisson1poisson1 (1.2.9) f1, f2ωm = mdf1 ∧ df2 ∧ ωm−1.

Equivalently,

f1, f2 = ω(gradωf1, gradωf2)

= gradωf1 · f2

= −gradωf2 · f1.

poisson2poisson2 (1.2.10)

Since ω is closed, we infer that

morphismmorphism (1.2.11) gradω(f1, f2) = [gradωf1, gradωf2],

from which it readily follows that the Poisson bracket satisfies the Jacobiidentity:

poisson-jacobipoisson-jacobi (1.2.12) f1, f2, f3+ f2, f3, f1+ f3, f1, f2 = 0.

The space C∞(M,R), equipped with the Poisson bracket, is then a Liealgebra. The identity (1.2.11) then means that the linear map

poissonpoisson (1.2.13) f 7→ gradωf

is a Lie algebra morphism from the space C∞(M,R), equipped with thePoisson bracket, to the space X (M) of vector fields, equipped with the usualbracket. Its range is the Lie algebra, hamω, of hamiltonian vector fields on(M,ω), cf. Section 1.5.

1.3. Inner product and exterior derivativessinner

The pointwise inner product of any kind of tensors is denoted by (·, ·).Except for p-forms, (·, ·) is the tensorial inner product, i.e. the inner productnaturally induced by g on the tensor algebra. For p-forms however we adoptthe following different, but usual convention:

unusualunusual (1.3.1) (α1 ∧ . . . ∧ αp, β1 ∧ . . . ∧ βp) = det((αi, βj)).

The same convention will be used for p-multivectors, i.e. for sections ofΛp(TM).

The inner product defined that way on the space of p-forms is equal to1p! times the tensorial inner product. In particular, for the Kahler form we

have |ω|2 = (ω, ω) = m, while the tensorial square norm of ω is equal to|g|2 = 2m. This choice is dictated by our convention for the exterior product, namely

(1.3.2) (α1 ∧ . . . ∧ αp)(X1, . . . , Xp) = det((αi(Xj)),

1.4. LIE DERIVATIVE 13

and for the exterior derivative

dψ(X0, X1, . . . , Xp) =

p∑j=0

(−1)jXj · ψ(X0, . . . , Xj , . . . , Xp)

+∑i<j

(−1)i+jψ([Xi, Xj ], X0, . . . , Xi, . . . , Xj , . . . , Xp),

extdifextdif (1.3.3)

where Xi means that the corresponding argument has been omitted. Theseconventions have become widely used and there is no real need to changethem (notice that these conventions amount to consider the exterior algebrabundle Λ•M as a quotient rather than a subbundle of the tensor algebrabundle).

For any local orthonormal frame e1, . . . , en, we also have

difframedifframe (1.3.4) dψ =n∑i=1

e∗i ∧Deiψ,

where e∗1, . . . .e∗n denotes the (algebraic) dual coframe, determined by e∗i (ej) =δij .. Equivalently,

dDdD (1.3.5) dψ(X0, . . . , Xp) =

p∑j=0

(−1)jDXjψ(X0, . . . , Xj , . . . , Xp),

i.e. dψ is a normalized anti-symmetrization of the covariant derivative Dψwith respect to the Levi-Civita connection (or any torsionless connection).

For all kinds of tensor, the global inner product is denoted by 〈·, ·〉:

(1.3.6) 〈·, ·〉 =

∫M

(·, ·) vg.

When complex tensors are involved, unless otherwise specified (·, ·) and 〈·, ·〉will denote the corresponding hermitian inner products, as defined in Section1.8.

1.4. Lie derivative and the Cartan formulasscartan

For any diffeomorphic map Φ from a manifold M to a manifold N , andfor any tensor field T — of any sort — on M , the direct image Φ · T isthe tensor field on N obtained by transporting the corresponding tensorfiber bundle over M to the tensor bundle of the same kind on N via thedifferential Φ∗ of Φ.

If X is a vector field on M , Φ ·X is then the vector field on N definedby (Φ ·X)y = Φ∗(XΦ−1(y)); if ψ is a p-form on M , Φ · ψ is the p-form on N

defined by (Φ · ψ)y(Y1, . . . , Yp) = ψΦ−1(y)(Φ−1∗ Y1, . . . ,Φ

−1∗ Yp); if A is a field

of endomorphisms of TM , Φ ·A is the field of endomorphisms of TN definedby (Φ ·A)y(Y ) = Φ∗(AΦ−1(y)(Φ

−1∗ Y )), etc...

These definitions hold in particular when M = N and Φ is a diffeo-morhism of M . We then get a group action of the group Diff(M) of diffeo-morphisms of M on the space of tensor fields.


The Lie derivative of T along a vector field X, denoted by LXT , is thendefined by

liederlieder (1.4.1) LXT = − d

dt |t=0(ΦX

t · T ),

where ΦXt denotes the (local) flow of X in some neighbourhood of the point

on consideration or, more generally, any curve ΦXt in Diff(M) such that

ΦX0 = I, the identity, and d

dt |t=0ΦXt = X.

The sign in (1.4.1) has been chosen in such a way that LXf = df(X)when f is a function, and we then have

(1.4.2) LXY = [X,Y ],

when Y is a vector field.By its very definition, the Lie derivative preserves each natural linear

operation on tensors.In particular LX is a derivation of the exterior algebra in the sense that

(1.4.3) LX(ψ ∧ ψ′) = LXψ ∧ ψ′ + ψ ∧ LXψ′,

for any p-form ψ and any p′-form ψ′, and commute with the exterior differ-ential. These facts, together with the above identity LXf = df(X) implythe following Cartan formula:

cartancartan (1.4.4) LXψ = ιXdψ + d(ιXψ),

for any p-form ψ (both sides are derivations of the exterior algebra, commuteto d and are equal to df(X) when ψ = f is a function; they then have thesame action on the whole exterior algebra).

In (1.4.4) and henceforth, ιXψ denotes the interior product of ψ by X,i.e., if p > 0, the (p− 1)-form defined by

(1.4.5) ιXψ(X1, . . . , Xp−1) = ψ(X,X1, . . . , Xp−1),

for any vector fields X1, . . . , Xp−1 (we agree that ιXf = 0 if f is a function).The Lie derivative is a Lie algebra action: for any two vector fields X,Y ,

and for any kind of vector field T , we have that

liecomliecom (1.4.6) L[X,Y ]T = LX(LY T )− LY (LXT ).

1.5. Symplectic vector fields and Poisson bracketssspoisson

If (M,ω) is a symplectic manifold, a vector field X is called symplecticif

(1.5.1) LXω = 0.

By the Cartan formula (1.4.4), this condition amounts to d(ιXω) = 0. A

vector field X is then symplectic if and only if its symplectic dual X[ω isclosed. It is called hamiltonian if X[ω if exact. We then have

(1.5.2) X = gradω hX ,

i.e.

(1.5.3) ιXω = −dhX ,

1.6. CAUCHY-RIEMANN OPERATORS AND CHERN CONNECTIONS 15

where hX is a real function and gradω is the symplectic gradient of X definedin Section 1.2: hX is called the momentum of X with respect to ω and isuniquely defined by X up to an additive constant.

By (1.4.6), the space of symplectic vector fields is a Lie algebra, denotedspω. By (1.2.11) the space of hamiltonian vector fields is a Lie algebraof spω, isomorphic to the space C∞(M,R)/R equipped with the Poissonbracket defined in Section 1.2; it will be denoted by hamω.

propbracket-ham Proposition 1.5.1. The Lie algebra hamω is an ideal of spω and con-tains its derived ideal:

derivedspderivedsp (1.5.4) [spω, spω] ⊂ hamω.

More precisely, for any two elements X,Y of spω, we have:

derivedsp-bisderivedsp-bis (1.5.5) [X,Y ] = gradω(ω(X,Y )).

Proof. It suffices to check (1.5.5). By the Cartan formula, for anyvector fields X,Y we have

d(ω(X,Y )) = −d(ιXιY ω)

= −LX(ιY ω) + ιX(d(ιY ω)

= −ι[X,Y ]ω − ιY LXω + ιXLY ω.(1.5.6)

When X,Y both belong to spω, this reduces to

(1.5.7) ι[X,Y ]ω = −d(ω(X,Y )),

which is the same as (1.5.5).

In the case that X = gradωhX and Y = gradωh

Y both belong to hamω,then ω(X,Y ) = hX , hY and (1.5.5) then reduces to (1.2.11).

1.6. Cauchy-Riemann operators and Chern connectionsssholochern

The material presented in this section and in the next Section 1.18 isof quite general character and, unless specifically stated, does not require Mbeing Kahler.

We start this section by recalling some general facts concerning the gen-eral theory of linear connections on vector bundles.

Let E be any real or complex vector bundle over some n-dimensionalmanifold M . A linear connection on E or, rather, the corresponding co-variant derivative, is a first-order linear differential operator ∇ acting onsections of E, with values in the tensor product T ∗M ⊗ E, which satisfiesthe following Leibniz identity :

(1.6.1) ∇(fs) = df ⊗ s+ f ∇s,for any section s of E and any (real or complex) function f (when E isa complex vector bundle, T ∗M ⊗ E is implicitly understood as the tensorproduct over C of the complexified cotangent bundle T ∗M ⊗ C with E).

Any linear connection∇ extends into an “exterior differential” d∇ actingon E-valued exterior forms, i.e. on sections of the tensor product E⊗Λ•M ,where Λ•M =

∑np=0 ΛpM , denotes the bundle of (real) exterior forms on M

(again, if E is a complex vector bundle, E ⊗ Λ•M is tacitly understood as


the tensor over C of E with the vector bundle Λ•M ⊗C of complex exteriorforms). This is done by setting

dnabladnabla (1.6.2) d∇(s⊗ φ) = ∇s ∧ φ+ s⊗ dφ,

for any decomposed section s⊗φ of E⊗Λ•M . Equivalently, for any E-valuedp-form ϕ, d∇ϕ is defined by

(d∇ϕ)(X0, X1, . . . , Xp) =

p∑i=0

(−1)i∇Xi(ϕ(X0, . . . , Xi, . . . , Xp))

+∑i<j

(−1)i+jϕ([Xi, Xj ], . . . , Xi, . . . , Xj , . . . , Xp),

diffnabladiffnabla (1.6.3)

for any vector fieldsX0, X1, . . . , Xp (compare with (1.3.3)). For any auxiliarytorsion-free linear connection D on M , denote by D∇ the induced covariantderivative acting on the space of E-valued exterior forms by D∇(s ⊗ φ) =∇s⊗ φ+ s⊗Dφ; then, the above formula can be rewritten as

dnablaDargdnablaDarg (1.6.4) (d∇Φ)X0,X1,...,Xp =

p∑k=0

(−1)k(D∇XkΦ)X0,...,Xk,...,Xp,

or, in a more concise way:

dnablaDdnablaD (1.6.5) d∇Φ =n∑i=1

e∗i ∧D∇eiΦ,

for any auxiliary frame e1, . . . , en of TM . Denote by End(E) the vectorbundle of R- or C-linerar endomorphisms of E, according to E being a realor a complex vector bundle. Then, the difference ∇′ −∇ of any two linearconnections ∇,∇′ is a End(E)-valued 1-form, whereas the curvature, R∇, of∇ is a End(E)-valued 2-form, determined by

curvature-defcurvature-def (1.6.6) R∇X,Y s = (∇[X,Y ] − [∇X ,∇Y ]) s,

for any vector fields X,Y on M and any section s of E. The curvature R∇

then appears as the obstruction for d∇ to be a cohomological operator as theusual exterior differential, via the following — easily checked — identity:

dddd (1.6.7) (d∇ d∇)ϕ = −R∇ ∧ ϕ,

for any E-valued p-form ϕ (in the rhs of (1.6.7) R∇ ∧ ϕ stands for the E-valued (p + 2)-form obtained by performing the formal exterior product ofR∇ and ϕ, with values in the tensor product End(E) ⊗ E, followed by theevaluation map End(E)⊗ E → E).

A linear connection ∇ is said to be integrable — or flat — if R∇ ≡ 0,equivalently if d∇ d∇ = 0.

To any linear connection ∇ on E is attached a horizontal distribution,H∇, defined on the total space of E as follows: for each ξ in E over x, H∇ξis defined by H∇ξ = s∗(TxM) ⊂ TξE for any local section s of E such that∇s = 0 at x, where s is viewed as a map from M to E and s∗ stands for its


differential; in other words, the pair (ξ,H∇ξ ) is the 1-jet at x of any sectionof E which is ∇-parallel at x. The tangent space TE then splits as

nablasplitnablasplit (1.6.8) TE = H∇ ⊕ T VE,

where T V stands for the vertical tangent space, whose fiber at ξ is T Vξ E :=

Tξ(Ex) ∼= Ex. For any section s of E, again viewed as a map from M to E,and for any vector X in TxM , the value of the covariant derivative ∇Xs atx is then given by

(1.6.9) (∇Xs)(x) = v∇(s∗(X)),

where v∇ : TE → T VE denotes the projection from TE to T VE along H∇;equivalently,

covderivativecovderivative (1.6.10) s∗(X) = X +∇Xs,

for any vector field X on M , if X denotes the horizontal lift of X on Ewith respect to H∇, i.e. the unique section of H∇ which projects to X.Alternatively, any section s of E can be regarded as a vertical vector fieldon the total space of E, constant on each fibre, and we then have

nablacovariantnablacovariant (1.6.11) ∇Xs = [X, s].

By using (1.6.10), one easily checks that the curvature R∇ can then bealternatively defined by

nablacurvaturenablacurvature (1.6.12) R∇X,Y ξ = v∇([X, Y ]ξ),

for any two vectors X,Y in TxM and any ξ in Ex (in the rhs of (1.6.12),

X, Y are the horizontal lifts of any extensions of X,Y and [X, Y ]ξ denotes

the value at ξ of their bracket on E). The vanishing of R∇ is thus equivalentto the integrability of H∇ in the sense of the Frobenius theorem. It followsthat a linear connection ∇ on E is integrable if and only if E can be locallytrivialized by ∇-parallel sections (for a more detailed discussion we refer thereader to [171] or to our notes [90]).

remcurvature Remark 1.6.1. The sign convention adopted in (1.6.6), hence also in(1.6.7) and (1.6.12), for the curvature of a linear connection suits with [28]and will be followed throughout these notes. The reader is warned howeverthat the opposite convention: R∇X,Y s = ([∇X ,∇Y ] −∇[X,Y ]) s is commonlyencountered in the more recent literature.

We now restrict our attention to the case when E is a complex vectorbundle of (complex) rank r defined over an almost complex manifold (M,J).

A Cauchy-Riemann operator ∂E on E is then defined as a first orderC-linear differential operator acting on sections of E with values in the(complex) tensor product E ⊗ Λ0,1M , satisfying the following Leibniz-likeidentity

CRCR (1.6.13) ∂(fs) = s⊗ ∂f + f ∂Es,

for any section s of E and any (complex-valued) function f on M . Here, ∂ =12(d− idc) denotes the usual Cauchy-Riemann operator acting on functions(note however that in the general almost complex context, we don’t have∂ ∂ = 0, as d, dc don’t commute, cf. Section 1.11).


Any C-linear connection ∇ on E can be written as

(1.6.14) ∇ = ∇1,0 +∇0,1,

where the (1, 0)-part ∇1,0 and the (0, 1)-part ∇0,1 are defined by

(1.6.15) ∇1,0X s =

1

2(∇Xs− i∇JXs), ∇0,1

X s =1

2(∇Xs+ i∇JXs),

and ∇0,1 clearly satisfies (1.6.13), hence is a Cauchy-Riemann operator.Conversely, any Cauchy-Riemann operator can be realized as the (0, 1)-partof a C-linear connection in several ways. If E is equipped with a hermitianinner product, h, a C-linear connection ∇ is called hermitian if it preservesh, i.e, if

(1.6.16) X · h(s1, s2) = h(∇Xs1, s2) + h(s1,∇Xs2),

for any sections s1, s2 of E and any (real) vector field X. We then have

propchern Proposition 1.6.1. Let E be a complex vector bundle over an almostcomplex manifold (M,J). Then, for any given hermitian inner product hon E, any Cauchy-Riemann operator ∂E on E is the (0, 1)-part of a uniquehermitian connection ∇.

Proof. We first prove uniqueness. If two C-linear connections havethe same (0, 1)-part, their difference, say η, is of type (1, 0), as a EndE-valued 1-form, hence satisfies the identity ηJX = iηX ; if, moreover, bothconnections preserve h, then ηX is h-skew hermitian for any vector X: thesetwo properties are clearly incompatible, unless η is identically zero.

To construct the connection, we first observe that any Cauchy-Riemannoperator ∂E defined on E induces a Cauchy-Riemann operator ∂E

∗defined

on the (complex) dual E∗ by the following duality formula:

(1.6.17) 〈∂E∗σ, s〉 = ∂〈σ, s〉 − 〈σ, ∂Es〉,for any section σ of E∗ and any section s of E, where 〈·, ·〉 stands for theduality between E and E∗. Denote by τ the hermitian duality from E toE∗ defined by 〈ξ1, τ(ξ2)〉 = h(ξ1, ξ2). We then define a connection ∇ by

chernchern (1.6.18) ∇s = τ−1(∂E∗τ(s)) + ∂Es,

for any section s of E. By its very definition, ∇ is C-linear and preserves thehermitian inner product h, hence is a hermitian connection on E, whereaswe clearly have ∇0,1 = ∂E .

When (M,J) is a complex manifold and E is a holomorphic vector bun-dle over M , then E admits an (integrable) canonical Cauchy-Riemann oper-ator ∂E , defined below, and for any hermitian inner product h, the resultinghermitian connection defined by Proposition 1.6.1 is usually called the Chernconnection of the hermitian holomorphic vector bundle (E, h).

In the more general context of this section, ∇ will still be called theChern connection of the hermitian vector bundle (E, h) with respect to thechosen Cauchy-Riemann operator ∂E .

In the sequel of this section, we assume that J is integrable.

Then, any Cauchy-Riemann operator ∂E extends to an operator, stilldenoted by ∂E , acting on sections of E ⊗ Λ0,∗M , where Λ0,∗M denotes the


bundle of (complex-valued) forms of type (0, q) on M , cf. Section 1.9 below.The action of ∂E on sections of E ⊗ Λ0,∗M is determined by its action ondecomposed (local) section s⊗ φ of E ⊗ Λ0,∗M , defined by:

dbarpartialdbarpartial (1.6.19) ∂E(s⊗ φ) = ∂Es ∧ φ+ s⊗ ∂φ

(compare with (1.6.2)).A Cauchy-Riemann operator ∂E is said to be integrable if its “curvature”

vanishes, i.e. if

CRintegrableCRintegrable (1.6.20) ∂E ∂E = 0.

A holomorphic vector bundle is defined as a vector bundle E in thecategory of holomorphic manifolds. This means that E is a complex vectorbundle over some complex manifold (M,J), which, as a manifold, is itselfequipped with a complex structure in such a way that all fibers are complexsubmanifolds of E and the two operations wich make E into a complex vectorbundle, namely the addition and the product by a complex number, are bothholomorphic; moreover, E can be locally trivialized in the usual sense by(local) holomorphic sections, i.e. by local sections which are holomorphic as(local) maps from M to E.

Any holomorphic (complex) vector bundle E over a complex manifold(M,J) admits a canonical Cauchy-Riemann operator ∂E defined by

holocanholocan (1.6.21) ∂Es =r∑i=1

∂fi ⊗ si,

for any section s of E, locally written as s =∑m

i=1 fisi with respect to

a a holomorphic local frame si of E, where ∂ = 12(d − idc) denotes the

usual Cauchy-Riemann operator acting on fonctions. The Cauchy-Riemannoperator ∂E defined that way is independent of the chosen holomorphicframe si and is evidently integrable, as ∂ ∂ = 0. It is characterized by thefact that a (local) section of E is holomorphic — as a map from an open setof M to E — if and only if it belongs to the kernel of ∂E . The converse isthe celebrated Koszul-Malgrange theorem:

thmKM Theorem 1.6.1 (Koszul-Malgrange [124]). Let (M,J) be a complex man-ifold of real dimension n = 2m and E a complex vector bundle over M . Let∂E be a Cauchy-Riemann operator on E satisfying the integrability condition(1.6.20). Then, E admits a unique structure of holomorphic vector bundlewhose canonical Cauchy-Riemann operator is ∂E.

Proof. The proof relies on the above construction, which makes The-orem 1.6.1 a direct consequence of the Newlander-Nirenberg theorem alongthe following line of argument. We first chose an auxiliary hermitian innerproduct on E and we complete ∂E into a hermitian connection ∇ accordingto the above construction. The tangent space TE then splits as in (1.6.8)and we construct an almost-complex structure, say JE , en E by transport-ing the complex structure J of M on the horizontal, via the isomorphismsH∇ξ = TxM , and the complex structure of the fibres of V on V . We then

check that JE is independent of the chosen auxiliary hermitian inner productand that the Nijenhuis tensor of JE is essentially identified with the square


∂E ∂E which, up to sign, is the (0, 2)-part of the curvature of ∇: by theNewlander-Nirenberg theorem, JE is then induced by a complex structureof E. It is then easy to show that this complex structure actually makes Einto a holomorphic vector bundle over M .

In view of Theorem 1.6.1, an integrable Cauchy-Riemann structure ona complex vector bundle E over a complex manifold (M,J) is called a holo-morphic structure on E. We then have

propholochern Proposition 1.6.2. Let E be a complex vector bundle over a complexmanifold (M,J). For any fixed hermitian inner product h on E, the map∇ 7→ ∇0,1 establishes a natural one to one correspondence between the setof all holomorphic structures of E and the set of all hermitian connectionswhose curvature is J-invariant.

Proof. For any C-linear connection ∇, it follows from (1.6.7) that its

curvature can be expressed as R∇ = −d∇ d∇, whose (0, 2)-part is −d∇0,1 d∇

0,1. It follow that the Cauchy-Riemann operator ∇0,1 is integrable if and

only if the (0, 2)-part of R∇ is zero. If ∇ is hermitian, the (2, 0)- and (0, 2)-parts of R∇ are related to each other by the identity R∇Z1,Z2

= −(R∇Z1,Z2

)∗,

for any two complex vector fields Z1, Z2, where ∗ stands for the hermitianadjoint: then, ∇0,1 is integrable if and only if R∇ is reduced to its J-invariantpart, i.e. satisfies R∇JX,JY = R∇X,Y . We then conclude by using Proposition1.6.1.

1.7. Chern connections of complex line bundlessschernline

In this section, we apply the theory of Chern connections developped inSection 1.6 to the particular case when E is a complex line bundle, i.e. acomplex vector bundle of rank 1. As such, it will be generally denoted byL, instead of E, and the Chern connection of a hermitian holomorphic linebundle will be generally denoted by ∇. The curvature, R∇, of ∇ is then ofthe form R∇ = i ρ∇, where ρ∇ is a real 2-form, called the curvature form of∇, or of (L, h). We denote by L the holomorphic fiber bundle obtained by

removing the zero section, Σ0, of L: L can then be viewed as the associated(holomorphic) C∗-principal bundle; the restriction of π to L is denoted π.

propLhermhol Proposition 1.7.1. Let (L, h) be a holomorphic line bundle over a com-plex manifold (M,J). Denote by s any non-vanishing holomorphic section

of L, i.e. any local holomorphic section of L defined on some open subsetU of M . Then, on U , the Chern connection ∇ and the curvature form ρ∇

have the following expressions:

∇s = ∂ log |s|2h ⊗ s

=1

2(d log |s|2h + i dc log |s|2h)⊗ s

chernlocalchernlocal (1.7.1)

where |s|2h stands for h(s, s), and

rhonablalocalrhonablalocal (1.7.2) ρ∇ = −1

2ddc log |s|2h.

Moreover, we have that

curvature-normcurvature-norm (1.7.3) π∗ρ∇ = −ddc log r,

1.7. CHERN CONNECTIONS OF COMPLEX LINE BUNDLES 21

where r stands the norm function on L determined by h, here restricted toL.

Proof. For simplicity, we denote by (·, ·) the hermitian inner producth. Since ∇ preserves h, we have that (∇Xs, s) = X · |s|2h − (s,∇Xs) and,similarly, (∇JXs, s) = JX · |s|2h − (s,∇JXs) for any (real) vector field X.Since s is a holomorphic section and ∇0,1 is equal to the Cauchy-Riemannoperator which determines the holomorphic structure of L, we have that∇JXs = i∇X , so that the latter identity can be rewritten as (∇JXs, s) =JX · |s|2h + i (s,∇Xs). Now, L is a complex line bundle: we then have that∇Xs = θ(X) s, for some complex 1-form θ defined on U . By combining theabove two identies, we immediately infer that θ = 1

2(d log |s|2h+i dc log |s|2h) =

∂X log |s|2h. This proves (1.7.1). By the very definition of R∇ — see Remark

1.6.1 below — we have that R∇s = −dθ ⊗ s = −12dd

c log |s|2h ⊗ i s; this

proves (1.7.2). If s is regarded as a (holomorphic) map from U to L, (1.7.2)can alternatively be read as: ρ∇ = −ddc(s∗ log r) = s∗(−ddc log r). Now,

on L, the 2-form −ddc log r is basic, i.e. −ddc log r = π∗ψ, for some 2-form ψ defined on M : this is because, if T denotes the generator of thenatural S1-action on L, we have that dc log r(T ) = 1, d log r(T ) = 0, so thatιTdd

c log r = ιJTddc log r = 0, and LTddc log r = LJTddc log r = 0. Since

π s = Id|U , we infer ρ∇ = s∗(−ddc log r) = s∗ π∗ψ = ψ, on U , hence on

M , as ρ∇ and ψ are both globally defined on M , and U can be chosen aneighbourhood of any point of M ; we thus obtain (1.7.3).

As a direct corollary, we get

propcherncompare Proposition 1.7.2. Let L be a holomorphic line bundle over a complexmanifold (M,J). For any two hermitian inner product h, h′ = e−2φh on L,the corresponding Chern connections ∇, ∇′ and the corresponding curvatureforms ρ∇, ρ∇

′are related by

cherncomparecherncompare (1.7.4) ∇′ = ∇− dφ− idcφ,

and

rhocherncomparerhocherncompare (1.7.5) ρ∇′

= ρ∇ + ddcφ.

Proof. The difference ∇′−∇ is naturally viewed as a complex 1-form.By (1.7.1) this complex 1-form is easily identified with the rhs of (1.7.4), aslog |s|2h′ = log |s|2h−2φ; (1.7.5) then follows from (1.7.4) (or from (1.7.3).

remchernricci Remark 1.7.1. It follows from (1.7.2) and from (1.7.4) that for anyholomorphic line bundle L and any hermitian inner product h on L, thecurvature form ρ∇ of the resulting Chern connection ∇ is closed and thatfor any two hermitian inner products on L the resulting Chern curvatureinduces the same element in the de Rham chomology space H2

dR(M,R).This actually has nothing to do with the holomorphic structure of L, onlywith its structure of complex line bundle: for any C-linear connection ∇on L, the curvature R∇ is a complex 2-form and it is easily checked that−i R∇ determines the same element as the former ones in H2

dR(M,R). Theresulting element in H2(M,R) is equal to 2π c1(L), where c1(L) is the realChern class of L. The factor 2π is there to ensure that c1(L) be the image


in H2(M,R) of an element of H2(M,Z), which is the genuine Chern classof L, cf. Section 8.1. More precisely, this normalization has been chosen sothat the Chern class of the hyperplane line bundle O(1) of the complex linebundle P1 — cf. Section 6.1 — be the positive generator of H2(P1,Z) = Z,cf. [100, Chapter1].

1.8. Real vs complex viewpointssversus

The (real) tangent bundle TM of any almost-complex manifold (M,J)can be regarded as a rank m complex vector bundle by identifying the actionof J with the multiplication by the complex number i. As such, TM will bedenoted by (TM, J).

The complex vector bundle (TM, J) is isomorphic to the bundle T 1,0Mof complex vectors of type (1, 0) defined as follows: the operator J extendsby C-linearity to the complexified tangent bundle TM ⊗C and T 1,0M , resp.T 0,1M , are the complex eigensubbundles of TM ⊗ C for the extended op-erator J , corresponding to the eigenvalue i, resp. −i. Then, (TM, J) isidentified with T 1,0M via the map: X → X1,0 := 1

2(X − iJX).If the Nijenhuis tensor of J is identically zero, then J is integrable — cf.

Section 1.1 — meaning that M is a complex manifold, whose structure is de-termined by local holomorphic complex coordinates which are deduced fromeach other by holomorphic transformations. Let z1, . . . , zm be such a systemof holomorphic coordinates on some open set U of M and, for j = 1, . . . ,m,write zj = xj + i yj , where xj , yj are real. Then, x1, y1, . . . , xm, ym forma system of real coordinates on U . Denote by ∂/∂xj , ∂/∂yj the corre-sponding vector fields: as operators acting on functions, these are the par-tial derivatives with respect to the coordinate system xj , yj . Since the zjare holomorphic, we have that ∂/∂yj = J∂/∂xj for all j = 1, . . . ,m. Wethen define ∂/∂zj := 1

2(∂/∂xj − i∂/∂yj) as the (1, 0)-part of ∂/∂xj and

∂/∂zj := 12(∂/∂xj + i∂/∂yj) as the (0, 1)-part of ∂/∂xj .

Remark 1.8.1. Despite the notation, the local complex vector fields∂/∂zj , ∂/∂zj defined that way cannot be viewed as partial derivatives with-out some precaution. Strictly speaking, ∂/∂zj can be viewed as a partialderivative — with respect to the holomorphic coordinate system z1, . . . , zm— only as an operator acting on holomorphic functions, on which the ∂/∂zjacts trivially. On the other hand, the ∂/∂zj , ∂/∂zj can be viewed as par-tial derivatives, with respect to the a priori fictitious coordinate systemz1, . . . , zm, z1, . . . , zm, if functions on U are (mentally) assumed to be therestrictions to U of holomorphic functions defined on some complexifica-tion of U , on which z1, . . . , zm, z1, . . . , zm becomes a genuine holomorphiccoordinate system.

When J is integrable, the complex vector bundle T 1,0M admits a naturalholomorphic structure determined by decreeing that, for any local systemof holomorphic coordinates z1, . . . , zm as above, the corresponding sections∂/∂z1, . . . , ∂/∂zm of T 1,0M are holomorphic. The corresponding Cauchy-Riemann operator ∂ — cf. (1.6.21) — is defined by ∂Z =

∑mj=1 ∂Zj⊗∂/∂zj ,

if Z =∑m

j=1 Zj∂/∂zj , and can then be rewritten as

CRT10MCRT10M (1.8.1) ∂Y Z = [Y 0,1, Z]1,0,

1.8. REAL VS COMPLEX VIEWPOINT 23

for any section Z of T 1,0M and any (real) vector field Y .Via the complex vector bundle isomorphism X 7→ Z = X1,0 = 1

2(X −iJX) form (TM, J) to T 1,0M and its inverse Z 7→ X = Z+Z from T 1,0M toTM , this holomorphic structure can be regarded as a holomorphic structuredefined on (TM, J) itself, whose Cauchy-Riemann operator then reads

CRTMJCRTMJ (1.8.2) ∂YX = 2Re ([Y 0,1, X1,0]1,0),

for any two real vector fields X,Y . By making explicit the real part — andby using the vanishing of N — we easily obtain the following alternativeexpression:

CRintCRint (1.8.3) ∂YX = −1

2J(LXJ)(Y ),

for all (real) vector fields X,Y (see Section 9.3 for a generalization of thisexpression in the non-integrable case).

A (real) vector field X is thus holomorphic as a section of (TM, J)equipped with this holomorphic structure — and will then be called a (real)holomorphic vector field, cf. Section 1.23 — if and only if LXJ = 0. Fromthe foregoing, we infer that X is a (real) holomorphic vector field if andonly if the (1, 0)-part Z = X1,0 of X is holomorphic in the natural sense, i.e.Z =loc

∑mj=1 Zj ∂/∂zj , where the components Zj are holomorphic functions

of the holomorphic coordinates zj .

For a general almost hermitian structure (g, J, ω) on M , the complexvector bundle (TM, J) comes equipped with a hermitian inner product h,defined as follows: the euclidean inner product (·, ·) extends to a C-bilinearinner product on TM ⊗C relatively to which both T 1,0M and its conjugateT 0,1M are isotropic; together with the natural real structure, this determinesa hermitian inner product, h, on TM ⊗ C defined by h(Z1, Z2) = (Z1, Z2),then, by restriction, a hermitian inner product on T 1,0M and a hermitianinner product on (TM, J) via the above identification (TM, J) = T 1,0M .The resulting hermitian inner product on (TM, J), still denoted by h, is re-lated to g and ω by h(X,Y ) = 1

2(g(X,Y )− iω(X,Y )), for any X,Y in TM .Moreover, for any almost-complex structure J , (TM, J) admits a canoni-cal Cauchy-Riemann operator, namely the operator determined by (1.8.2),which is (1.8.3) if J is integrable and can be expressed by (9.3.1)-(9.3.2)-(9.3.11) in the general case. The corresponding Chern connection — cf.Proposition 1.6.1 — is called the canonical Chern connection of (M, g, J, ω)and the corresponding covariant derivative is denoted by ∇. We then have:

propchernLC Proposition 1.8.1. For any almost hermitian manifold (M, g, J, ω), thecanonical Chern connection and the Levi-Civita connection coincide if andonly if the structure is Kahler.

Proof. If the almost hermitian structure is Kahler, then J is inte-grable and the Levi-Civita connection D preserves J — it is then a C-linear connection — and preserves g, hence the hermitian inner producth = 1

2(g − iω); moreover, a simple computation of the rhs of (1.8.3) shows

that −12J(LXJ)(Y ) = −1

2J [X, JY ]− 12 [X,Y ] = 1

2(DYX + JDJYX); in theKahler case, (1.8.3) then reads:

CRDCRD (1.8.4) ∂ = D0,1;


from the uniqueness of the Chern connection ∇, we infer that D = ∇.Conversely, if ∇ = D, then DJ = ∇J = 0, meaning that the structure isKahler.

1.9. The type of exterior formssstype

The complexified cotangent bundle also splits as a direct sum T ∗M⊗C =Λ1,0M ⊕ Λ0,1M , where Λ1,0M , resp. Λ0,1M , denotes the vector bundle ofcomplex covectors α such that α(JX) = iα(X), resp. α(JX) = −iα(X).With respect to the action of J on T ∗M = Λ1M defined in Section 1.2 andextended into a C-linear action of J on Λ1M ⊗ C, Λ1,0M and Λ0,1M arethen the eigen subbundles relative to the eigenvalues −i and i respectively.We deduce the following isomorphism of complex vector bundles:

typetype (1.9.1) Λ•M ⊗ C = ΛC(T ∗M ⊗ C) = Λ∗,0M ⊗ Λ0,∗M,

where Λ∗,0M = ΛC(Λ1,0M) (for any complex vector space or complex vec-tor bundle E, we agree that ΛC(E) denotes the corresponding full exte-rior algebra with respect to C); similarly, Λ0,∗M = ΛC(Λ0,1M); we thenhave Λ∗,0M = ⊕mr=0Λr,0M and Λ0,∗M = ⊕ms=0Λ0,sM , where Λr,0M is nat-urally identified with the bundle of complex-valued r-forms φ such thatφ(JX1, . . . , Xr) = iφ(X1, . . . , Xr) for any r (complex) vector fieldsX1, . . . , Xr;similarly, elements ψ of Λ0,sM are complex valued s-forms ψ such thatψ(JX1, . . . , Xs) = −iψ(X1, . . . , Xs). Elements of Λr,0M , resp. of Λ0,sM ,are called (complex) exterior forms of type (r, 0), resp. of type (0, s); moregenerally, elements of Λr,sM := Λr,0M⊗Λ0,sM are called (complex) exteriorforms of type (r, s); each (complex) p-form ψ can then be written in a uniqueway as ψ =

∑r+s=p ψ

r,s, where ψr,s denotes the component of type (r, s) of

ψ. To get an easy characterization of elements of Λr,sM in Λr+sM ⊗ C, itis convenient to introduce the operator C defined by

CC (1.9.2) (Cψ)(X1, . . . , Xp) = −p∑j=1

ψ(X1, . . . , JXj , . . . , Xp),

for any complex valued p-form ψ. It is easily checked that ψ is of type (r, s),with r+s = p, if and only if Cψ = (s−r)iψ, and we then have: Jψ = is−rψ.

If J is integrable, and z1, . . . , zm is a local sytem of holomorphic co-ordinates defined on some open set U , the dzj , resp. dzj , are 1-forms oftype (1, 0) and form a frame of Λ1,0M on U ; similarly, the dzj form aframe of Λ0,1M ; it follows that the dzj1 ∧ . . . ∧ dzjr ∧ dzk1 ∧ . . . ∧ dzks ,with j1 < . . . < jr and k1 < . . . < ks form a frame of Λr,sM on U forany 0 ≤ r, s ≤ m. For each r = 0, 1, . . . ,m, Λr,0M has a natural holo-morphic structure determined by requesting that the dzj1 ∧ . . . ∧ dzjr forma holomorphic frame for any holomorphic coordinate system z1, . . . , zm, cf.Section 1.6. In particular, Λ1,0M — which, as a complex vector bundle, isisomorphic to (T ∗M,−J) — is the (holomorphic) C-dual of the (holomor-phic) tangent bundle (TM, J) = T 1,0M , whereas Λm,0M is the so-calledcanonical line bundle of (M,J), which will be also denoted by KM .

rempp Remark 1.9.1. A real exterior p-form ψ cannot be of type (r, s) exceptif r = s and ψ is then J-invariant. In these notes, unless otherwise specified,

1.10. RIEMANNIAN AND SYMPLECTIC HODGE OPERATORS 25

Λr,rM will be implicitely understood as the bundle of real exterior forms oftype (r, r).

1.10. Riemannian and symplectic Hodge operatorssshodge

In general, the riemannian Hodge operator ∗ of an oriented riemannianmanifold (M, g) of dimension n is defined by

(1.10.1) ∗ψ = ψ]yvg,

for each p-form ψ, where the inner product is understood as follows: when-

ever ψ = α1∧ . . .∧αp, then ψ] = α]1∧ . . .∧α]p and (ψ]yvg)(Xp+1, . . . , Xn) =

vg(α]1, . . . , α

]p, Xp+1, . . . , Xn). Alternatively,

hodgestarhodgestar (1.10.2) ψ1 ∧ ∗ψ2 = (ψ1, ψ2) vg,

for all p-forms ψ1, ψ2. The Hodge operator ∗ can be defined on multi-vectorsas well and is then conveniently determined by

(1.10.3) ∗(e1 ∧ . . . ∧ ep) = ep+1 ∧ . . . ∧ en,

for any p and any orthonormal, positively oriented basis at any point of M .It readily follows that ∗ is a (pointwise) isometry and that:

∗2 ψ = (−1)p ψ, if n is even,

∗2 ψ = ψ, if n is odd,squarestarsquarestar (1.10.4)

for any p-form ψ.In the almost hermitian context, recall — see (1.2.6) — that the action

of the almost-complex structure J has been extended to the space of p-forms,for any p; we then have:

lemmaJ* Lemma 1.10.1. The riemannian Hodge operator ∗ and and the operatorJ acting on exterior forms commute.

Proof. Let x be any point of M and e1, . . . , en be any orthonormal,positively oriented basis of TxM . Let ψ be any (n − p)-form on M . Then∗ψ is a p-form, whose value at x is entirely defined by its action on any pelements of this basis, which we can chose equal to e1, . . . , ep, and we thenhave (∗ψ)(e1, . . . , ep) = ψ(ep+1, . . . , en). It follows that (J ∗ψ)(e1, . . . , ep) =ψ(J−1ep+1, . . . , J

−1en). Now, J−1e1, . . . , J−1en is again an orthonormal,

positively oriented basis of TxM (as for any invertible skew symmetric linearoperator, the determinant of J is positive, hence equal to +1); we thenlikewise get (∗Jψ)(e1, . . . , ep) = ψ(J−1ep+1, . . . , J

−1en).

The symplectic Hodge operator is similarly defined by

(1.10.5) ∗ωψ = ψ]ωyvg,

where the operators ]ω : T ∗M → TM and its inverse [ω : TM → T ∗M havebeen defined in Section 1.2. We thus have:

starsstars (1.10.6) ∗ω = J∗ = ∗J,

so that

squarestaromegasquarestaromega (1.10.7) (∗ω)2 = 1,


(by Lemma 1.10.1, J and ∗ commute and, when acting on p-forms, are bothof square (−1)p, as n = 2m is even).

The following identity is easily checked:

staromegastaromega (1.10.8) ∗ωr

r!= ∗ω

ωr

r!=

ωm−r

(m− r)!,

for each r = 0, . . . ,m.For a general riemannian manifold (M, g), the codifferential δ, acting on

exterior forms, is defined as the formal adjoint of d, i.e. is determined by

deltadelta (1.10.9) 〈dφ, ψ〉 = 〈φ, δψ〉,

for any (p − 1)-form φ and any p-form ψ, both compactly supported for(1.10.9) to make sense. Now, by (1.10.2), we have that

(dφ, ψ) vg = dφ ∧ ∗ψ= d(φ ∧ ∗ψ) + (−1)pφ ∧ d(∗ψ)

= d(φ ∧ ∗ψ) + (−1)pφ ∧ ∗ ∗−1 d(∗ψ)

= d(φ ∧ ∗ψ) + (φ, (−1)p ∗−1 d ∗ ψ) vg.

Since d(φ ∧ ∗ψ) integrates to zero on M , we eventually get

deltaevendeltaeven (1.10.10) δψ = − ∗ d ∗ ψ,

if n is even, and

deltaodddeltaodd (1.10.11) δψ = (−1)p ∗ d ∗ ψ,

if n is odd, for any p-form ψ. From the above computation, we then get thepointwise adjunction formula:

adjunctionadjunction (1.10.12) (dφ, ψ)− (φ, δψ) = ∗d(φ ∧ ∗ψ) = (−1)pδ(φ]yψ) = −δα,

for any (p− 1)-form φ and any p-form ψ, where α is the 1-form defined by

α(X) = (X[ ∧ φ, ψ) = (φ,Xyψ).In terms of the Levi-Civita connection D, the codifferential δ admits

the following expression: δψ = −∑n

j=1Dejψ(ej , ·, . . . , ·), for any auxiliary

orthonormal frame e1, . . . , en, i.e.

codifframecodifframe (1.10.13) δψ = −n∑i=1

eiyDeiψ.

For any vector bundle E overM , equipped with a fiberwise inner product— euclidean or hermitian according as E is real or complex — and with alinear connection ∇ which preserves the inner product, we can similarlydefine a codifferential δ∇ relative to ∇ as the formal adjoint of d∇. If D∇

denotes the connection induced by ∇ and the Levi-Civita connection Dacting on the space of E-valued exterior forms, cf. Section 1.6, we then have

deltanabladeltanabla (1.10.14) δ∇Φ = −n∑i=1

eiyD∇eiΦ,

1.11. TWISTED EXTERIOR DIFFERENTIAL AND TWISTED CODIFFERENTIAL 27

for any E-valued exterior form Φ and any auxiliary orthonormal framee1, . . . , en. For decomposed E-valued exterior forms Φ = s ⊗ ψ, we thenhave

deltanabla-dcdeltanabla-dc (1.10.15) δ∇(s⊗ ψ) = −n∑i=1

eiy∇eis⊗ ψ + s⊗ δψ.

remdelta Remark 1.10.1. The rhs of (1.10.13) makes sense when we substituteany p-multilinear form, i.e. any section ψ of ⊗pT ∗M , for any p: then(1.10.13) can be viewed as a definition of δψ. For any (p − 1)-multilinearform φ and any p-multilinear form ψ, both compactly supported, we thenhave

(1.10.16) 〈Dφ,ψ〉 = 〈φ, δψ〉,where the global inner product is relative to the tensorial inner product in⊗pT ∗M , cf. Section 1.3. This means that δ then coincides with the adjoint,D∗, of the Levi-Civita covariant derivative D.

When applied to an exterior form, δ is the adjoint of the exterior deriv-ative d as a section of the bundle ΛM of exterior forms, equipped with thespecific inner product of exterior forms — cf. Section 1.3 — and the adjointof the Levi-Civita covariant derivative D as a section of ⊗T ∗M , equippedwith the tensorial inner product. The coherence of the two interpretation isthen guaranteed by (1.3.5).

Remark 1.10.2. The divergence, divX, of a vector field is defined by

(1.10.17) LXvg = divX vg;

we thus have divX = −δX[.

1.11. Twisted exterior differential and twisted codifferentialsstwisted

The twisted exterior differential dc is defined by

(1.11.1) dcψ = JdJ−1ψ,

for any p-form ψ, where J−1 = (−1)pJ is the inverse of J acting on p-forms.In particular, we have: dcf = Jdf , i.e. (dcf)(X) = −df(JX), for eachfunction f .

The adjoint of dc, denoted by δc, is the twisted codifferential, defined by:

deltac-defdeltac-def (1.11.2) δc = JδJ−1

We then have:

propbryant Proposition 1.11.1. The twisted codifferential δc is entirely determinedby the symplectic form ω only. More precisely, for any p-form ψ, we have:

deltac1deltac1 (1.11.3) δcψ = (−1)p+1 ∗ω d ∗ω ψ.Alternatively, for any linear connection D on TM which is torsion-free andpreserves ω, we have:

deltac2deltac2 (1.11.4) δcψ = −n∑i=1

eiyDeiψ,

for any local frame e1, . . . , en of TM , where the symplectic dual framee1, . . . , en is determined by ω(ei, ej) = δij.


Proof. The first assertion follows from any one of the explicit two ex-pressions (1.11.3) or (1.11.4) of δc, whereas (1.11.3) is a direct consequenceof the definition (1.11.2) of δc, by using (1.10.10) and (1.10.6). From (1.3.5)— which, as already observed, holds when, in the rhs, the Levi-Civita con-nection Dg is replaced by any torsion-free linear connection, in particularby D — we infer that:

δcψ = −(−1)p ∗ω d ∗ω ψ

= −(−1)p ∗ωn∑i=1

e∗i ∧ Dei(∗ωψ)

= −(−1)pn∑i=1

∗ω(e∗i ∧ ∗ωDeiψ)

= −n∑i=1

eiyDeiψ.

remsymplecticdual Remark 1.11.1. By defining 〈ψ1, ψ2〉ω :=∫M ψ1 ∧ ∗ωψ2, for any two

p-forms ψ1, ψ2, we get a non-degenerate pairing on the space of (compactlysupported) real p-forms, determined by the symplectic form ω; when ω =g(J ·, ·), this symplectic pairing can be alternatively defined by 〈ψ1, ψ2〉ω =〈ψ1, Jψ2〉: it is anti-symmetric if p is odd, symmetric if p is even and, in thelatter case, is never definite. We then have

dsymplectaddsymplectad (1.11.5) d∗ω = −δc,

where d∗ω denotes the symplectic adjoint of d, defined by the symplecticadjunction relation 〈dφ, ψ〉ω = 〈ψ, d∗ωψ〉ω. We similarly get:

deltacsymplectaddeltacsymplectad (1.11.6) (δc)∗ω = d.

Notice that the operation of symplectic adjunction is anti-involutive in thiscase.

remJframe Remark 1.11.2. In the Kahler setting, we may choose the Levi-Civitaconnection Dg in place of D in the rhs of (1.11.4); morever, if e1, . . . , en isorthonormal, the symplectic dual frame is clearly the frame Je1, . . . , Jen;(1.11.4) then reads

JcodifframeJcodifframe (1.11.7) δcψ = −n∑i=1

JeiyDgeiψ.

We easily deduce from (1.3.5) the following expression of dc:

JdifframeJdifframe (1.11.8) dcφ =

n∑i=1

Je∗i ∧Dgeiψ,

for any frame e1, . . . , en of TM ; this expression is in fact valid on anycomplex manifold (M,J) when J is integrable and Dg is replaced by anytorsion-free linear connection which preserves J (easy verification).


The twisted exterior differential dc is entirely determined by J and canbe defined for any almost-complex structure J . We thus obtain the followingalternative integrability criterion for J :

propddcdcd Proposition 1.11.2. The almost-complex J is integrable if and only ifd and dc anticommute, i.e. if and only if

ddcddc (1.11.9) ddc + dcd = 0.

This happens if and only if ddcf + dcdf = 0 or, equivalently, if and only ifddcf is J-invariant, for any function f .

Proof. The operator ddc+dcd, acting on exterior forms, is a derivationfor the exterior product, i.e. satisfies (ddc + dcd)(φ ∧ ψ) = (ddc + dcd)φ ∧ψ + φ ∧ (ddc + dcd)ψ for any two exterior forms φ, ψ — easy verification —and obviously commutes with d; it is then entirely determined by its actionon functions. By using (1.3.3), for any (real) function f and any two vectorfields X,Y , we get (ddcf)X,Y = X · dcf(Y ) − Y · dcf(X) − dcf([X,Y ]) =−X ·(JY ·f)+Y ·(JX ·f)+J [X,Y ]·f , whereas (dcdf)X,Y = −(ddcf)JX,JY =−JX · (Y · f) + JY · (X · f)− J [JX, JY ] · f ; by comparing with (1.1.2), weinfer

ddc+dcdddc+dcd (1.11.10) (ddcf + dcdf)X,Y = (ddcf − J(ddcf))X,Y = 4dcf(N(X,Y ))

where, we recall, N denotes the Nijenhuis tensor of J defined by (1.1.2).It readily follows that ddc + dcd acts trivially on functions, hence on thewhole space of exterior forms, if and only if N is identically zero. The lastassertion again readily follows from (1.11.10).

remdeldelbar Remark 1.11.3. When J is integrable, we have that

(1.11.11) d = ∂ + ∂,

where ∂, ∂ are determined as follows: for any form ψ of type (r, s), ∂ψ and∂ψ are defined by:

(1.11.12) ∂ψ = (dψ)r+1,s, ∂ψ = (dψ)r,s+1

(it easily follows from (1.3.3) that, when J is integrable, i.e. when T 1,0Mand T 0,1M are closed for the bracket, dψ has no other components than theabove two). We then have

(1.11.13) dc = i(∂ − ∂),

hence,

delddcdelddc (1.11.14) ∂ =1

2(d+ idc), ∂ =

1

2(d− idc),

(1.11.15) ∂2 = ∂2 = ∂∂ + ∂∂ = 0,

and

(1.11.16) ddc = 2i∂∂.

The operator ∂∗, ∂∗ are defined as the formal adjoints of ∂, ∂ with respectto the global hermitian inner product defined on the space of complex valuedexterior forms, cf. Section 1.8; we then have

deldeltadeldelta (1.11.17) ∂∗ =1

2(δ − iδc), ∂∗ =

1

2(δ + iδc),


(1.11.18) (∂∗)2 = 0, (∂∗)2 = 0, ∂∗∂∗ + ∂∗∂∗ = 0.

The definition of the twisted differential dc readily extends to E-valuedexterior forms over an almost-complex manifold (M,J) for any (real) vectorbundle E over M equipped with a linear connection ∇. We then define dc,∇

by: dc,∇Φ = Jd∇J−1Φ, for any E-valued p-form Φ; here Φ is viewed as anelement of E ⊗ΛpM and the action of J is simply induced by the action ofJ on ΛpM defined by (1.2.6). For any decomposed element s ⊗ φ, where sis a section of E and φ a scalar p-form, we then have

dnablacdefdnablacdef (1.11.19) dc,∇(s⊗ φ) =n∑i=1

∇eis⊗ Je∗i ∧ φ+ s⊗ dcφ

and

(1.11.20) (dc,∇)2Φ = −J(R∇) ∧ Φ.

By choosing any auxiliary connection D on M , which is torsion-free andpreserves J , and by denoting D∇ the induced linear connection on E⊗ΛpM ,(1.11.19) can be rewritten as:

(1.11.21) dc,∇Φ =

n∑i=1

Je∗i ∧D∇eiΦ,

as for dc, cf. (1.11.8). The action of dc,∇, as well as the action of d∇, will beimplicitly extended to complex E-valued exterior forms, i.e. to sections of(E ⊗ Λ•M)⊗C, which is also the tensor product over C of E ⊗C with thebundle Λ•M ⊗ C of complexified exterior forms; if E is already a complexvector bundle and ∇ is a C-linear connection, we agree that the space ofE-valued exterior forms on which d∇ and dc,∇ act is the space of sections ofE ⊗C (Λ•M ⊗ C).

When ∇ preserves a fiberwise inner product on E — euclidean or her-mitian, according as E is a real or a complex vector bundle — and M hasan almost hermitian structure (g, J, ω), we can similarly define a twistedcodifferential δc,∇ acting on E-valued exterior forms, as the formal adjointof dc,∇; with the above notation, we then have

(1.11.22) δc,∇(s⊗ φ) = −n∑i=1

∇eis⊗ Jeiyφ+ s⊗ δcφ.

If the structure is almost-Kahler, i.e. if ω is symplectic, then, we also have

(1.11.23) δc,∇Φ = −n∑i=1

eiyD∇eiΦ,

with the same notation of (1.11.4), for any linear torsion-free connection Don M which preserves ω.

In the context of E-valued exterior forms, Proposition 1.11.2 generalizesas follows:

propddcdcdnabla Proposition 1.11.3. Let (M,J) be a complex manifold and E a (real)vector bundle over M , equipped with a linear connection ∇ — C-linear if Eis a complex vector bundle. We then have

ddcdcdnabladdcdcdnabla (1.11.24) (d∇dc,∇ + dc,∇d∇)Φ = −C(R∇) ∧ Φ,


for any E-valued exterior form Φ, where C denotes the operator defined by(1.9.2), i.e.

(1.11.25) C(R∇)X,Y = −R∇JX,Y −R∇X,JY .

In particular, d∇dc,∇ + dc,∇d∇ = 0 if and only if R∇ is of type (1, 1).

Proof. It is sufficient to check (1.11.24) when Φ = s⊗φ is a decomposedE-valud p-form. from (1.11.19), we easily infer

(d∇dc,∇ + dc,∇d∇)(s⊗ φ) =n∑

i,j=1

R∇ei,ejs⊗ e∗j ∧ Je∗i ∧ φ

+ s⊗ (ddc + dcd)φ,

where (ddc + dcd)φ = 0, as J is integrable, whereas the curvature term inthe rhs is clearly equal to −C(R∇). For 2-forms, the kernel of C is the spaceof forms of type (1, 1); the last assertion then follows from (1.11.24).

rembarpartialE Remark 1.11.4. When E = (E, h) is a hermitian holomorphic vectorbundle over a complex manifold (M,J) and ∇ is the corresponding Chernconnection — cf. Section 1.6 — then the extension of the Cauchy-Riemannoperator ∂E to the E-valued exterior forms of type (0, q), as defined by(1.6.19, coincides with the action of the operator 1

2

(d∇− idc,∇

). It can then

be extended to the whole space of E-valued exterior forms, of any type, bysimply setting:

barpartialddcbarpartialddc (1.11.26) ∂E =1

2(d∇ − i dc,∇).

We similarly extend the action of ∂∇ = ∇1,0 to the space of E-valued exte-riort forms by setting:

partialddcpartialddc (1.11.27) ∂∇ =1

2(d∇ + i dc,∇).

Since the Chern curvature R∇ is J-invariant, d∇dc,∇ + dc,∇d∇ = 0 — cf.Proposition 1.11.3 — whereas (dc,∇)2 Φ = −J(R)∧Φ = −R ∧Φ = (d∇)2 Φ.This implies (∂E)2 = 1

4

((d∇)2 − (dc,∇)2 − i(d∇dc,∇ + dc,∇d∇)

)= 0 and,

similarly, (∂∇)2 = 0.

rem-pforms Remark 1.11.5. For any non-negative integers p, q, the space Γ(E ⊗Λp,qM) of E-valued exterior forms of type (p, q) is naturally identified withthe space Γ

((E ⊗ Λp,0M) ⊗ Λ0,qM) of E ⊗ Λp,0M -valued exterior forms

of type (0, q), where Λp,0M is viewed as a holomorphic vector bundle, viathe natural holomorphic structure of the tangent space TM ∼= T 1,0M , cf.Section 1.6. Also recall that when Λp,0M is equipped with the hermitianinner product induced by any Kahler metric on M , the corresponding Chernconnection is the (induced) Levi-Civita connection. It is then easily checkedthat the extension of the operator ∂E defined by (1.11.26) on Γ(E⊗Λp,qM)

coincides with the natural extension of the holomorphic structure ∂E⊗Λp,0M

of the complex vector bundle E⊗Λp,0 to Γ((E⊗Λp,0M)⊗Λ0,qM), as definedby (1.6.19).


1.12. J-invariant 2-forms, hermitian operators and Chern formssshermitian

Any (real) 2-form ψ can be regarded as a skew-symmetric endomor-phism of TM , still denoted by ψ, via the metric g: ψ(X,Y ) = g(ψ(X), Y ).The action of J defined in Section 1.2 has then the following expression:ψ → J ψ J−1, so that ψ is J-invariant if and only if the correspond-ing operator commutes with J . The vector bundle of J-invariant elementsof Λ2M coincides with the bundle of (real) 2-forms of type (1, 1) and istherefore denoted by Λ1,1M . Any element ψ of Λ1,1M can be viewed asa skew-hermitian endomorphism of (TM, J, h) and can thus be written asψ = JΨ, where Ψ is a hermitian endomorphism of (TM, J, h).

Let P (t) = PΨ(t) =∑m

r=0(−1)rσr(φ)tm−r be the characteristic poly-nomial of Ψ, where, for r = 1, . . . ,m, σr(φ) denotes the r-th elementarysymmetric function of the m (real) eigenvalues of Ψ at each point, andσ0(φ) ≡ 1.

Each σr is a homogeneous real polynomial function of degree r definedon Λ1,1M . The corresponding polarized form, still denoted by σr, is then ar-multilinear form defined on Λ1,1M .

The linear form σ1 is also called the trace of ψ with respect to ω, denotedtrω(ψ) or, simply, tr(ψ), whereas σm coincides with the pfaffian, defined by

(1.12.1) ψm = pf(ψ)ωm.

It follows that P (t) = pf(tω − ψ).By developping this expression in powers of t, we thus get the following

expression for the polarized σr:

sigmarsigmar (1.12.2) σr(ψ1, . . . , ψr)ωm =

m!

(m− r)!r!ψ1 ∧ . . . ∧ ψr ∧ ωm−r,

for any sections ψ1, . . . , ψr of Λ1,1M .Denote by Lω and Λω, or simply L and Λ if ω is understood, the adjoint

pair of operators acting on exterior forms, defined by

elel (1.12.3) L(ψ) = ω ∧ ψ, Λ(ψ) = ω]yψ = ω]ωyψ.

Then, by using (1.10.8) we easily check that (1.12.2) can also be writtenas

sigmarbissigmarbis (1.12.4) σr(ψ1, . . . , ψr) =1

(r!)2Λr(ψ1 ∧ . . . ∧ ψr).

In particular, the quadratic form σ2 can be written as

sigma2sigma2 (1.12.5) σ2(ψ1, ψ2) =1

2(trψ1trψ2 − (ψ1, ψ2)).

Notice that Λ1,1M can be regarded as the adjoint bundle of the Kahlerstructure, i.e. the vector bundle associated to the adjoint action of theunitary group U(m) on its Lie algebra u(m), and that the σr can be viewedas U(m)-invariant polynomial functions defined on u(m). The correspondingcharacteristic class is then the r-th Chern class of (TM, J) (more details inAppendix A).

1.13. THE HODGE-LEPAGE DECOMPOSITION 33

1.13. The Hodge-Lepage decompositionsslepage

The operators L,Λ defined in Section 1.12 are related to each other by

Lambda-star-LLambda-star-L (1.13.1) Λ = ∗L ∗−1 = ∗ω L ∗ω .This can be easily checked by using the following explicit expressions of L,Λ:

LLambda-alterLLambda-alter (1.13.2) Lψ =1

2

n∑i=1

e[i ∧ Je[i ∧ ψ, Λψ =1

2

n∑i=1

JeiyJeiyψ,

for any auxiliary (local) orthonormal frame e1, . . . , e2m, and the generalriemannian identity:

X-psiX-psi (1.13.3) Xyψ = ∗(X[ ∧ ∗ψ),

for any exterior form ψ and any vector field X, on any even-dimensional rie-mannian manifold (on a manifold of dimension n, the right hand side must

be multiplied by (−1)n(p−1) whenever ψ is of degree p), whose easy verifi-cation is left to the reader. Recall that in (1.13.1), ∗−1 = (−1)p, whereas∗ω = ∗−1

ω , cf. (1.10.4)–(1.10.7).

It easily follows from (1.12.3) that

LambdaLLambdaL (1.13.4) [Λ, L] = H,

where H is defined by

HH (1.13.5) H(ψ) = (m− p)ψ,for any p-form ψ. We also easily check that

HLambdaLHLambdaL (1.13.6) [H,Λ] = 2Λ, [H,L] = −2L.

These are the commutation rules of the standard generators of the (com-plex) Lie algebra sl(2,C), via the identifications:

(1.13.7) Λ =

(0 10 0

), L =

(0 01 0

), H =

(1 00 −1

).

Now, L,Λ, H are zero order operators and, upon extending their action tocomplex exterior forms by C-linearity, can be viewed as algebraic operatorsacting on the complex vector space V :=

∑2mp=0 V

p, with V p = ΛpR2m ⊗ C.The kernel of Λ acting on V is denoted by V0. The elements of V0 are calledprimitive. Notice that the degree of a non-zero primitive form cannot begreater than the complex dimension m of M . Indeed, if ψ is any primitiveform of degree p, with p > m, then, by (1.13.4), ΛLψ = (m − p)ψ, hence|Lψ|2 = (m − p) |ψ|2, with (m − p) < 0, so that ψ = 0. By a similarargument, we easily check that the map L : V p → V p+2 is injective forany p < m; indeed, if ψ is any p-form, with p < m and Lψ = 0, then,[Λ, L]ψ = −LΛψ = (m−p)ψ, hence |Λψ|2 = (p−m) |ψ|2, with (p−m) < 0,so that ψ = 0. Equivalently, the map Λ : V p+2 → V p is surjective wheneverp < m. We thus have

primitiveprimitive (1.13.8) V0 =m∑p=0

V p0 ,

where V p0 denotes the space of primitive p-forms of degree p. Moreover,

V 00 = V 0, V 1

0 = V 1, whereas, for p = 2, . . . ,m, dimCVp

0 =(

2mp

)−(

2mp−2

).


Irreducible (complex) linear representations of sl(2,C) are indexed bynon-negative integers. For each integer r ≥ 0, the corresponding represen-tation space S(r) ' Cr+1 is the r-th (complex) symmetric tensor power of

the natural representation space S(1) = C2 and can be described as follows:in S(r) there is, up to a multiplicative factor, a unique element, v0, suchthat Λv0 = 0 and Hv0 = r vr; then, v0, v1 = Lv0, v2 = L2v0, . . . , vr = Lrv0

form a basis of S(r), such that H(vj) = (r − 2j) vj , for each 0 ≤ j ≤ r(vr is a dominant vector, corresponding to the dominant weight r, andr, r − 2, . . . ,−r + 2,−r are the weights of the representation).

When V is viewed as a sl(2,C)-representation space, the isotypic com-

ponent relative to S(r), say V (r), is thus isomorphic to S(r) ⊗ V m−r0 — each

element φ of V (r) is then written in a unique way as φ =∑r

j=0 Ljφj , where

φj is is a primitive form of degree m− r — and we eventually get

(1.13.9) V =m∑r=0

V (r) ∼=m∑r=0

S(r) ⊗ V m−r0 ,

where each V m−r0 appears as a parameter space, on which the action of

sl(2,C) is trivial, and whose dimension is the multiplicity of S(r) in V .Each element of V , in particular each complex-valued p-form ψ, is ex-

pressed in a unique manner as the sum of elements in V (r). We thus get theHodge-Lepage decomposition1 of any (complex) p-form ψ:

lepagelepage (1.13.10) ψ =

j=[ p2

]∑j=max(0,[ p−m

2])

Ljψj ,

where ψj is a (uniquely defined) primitive form of degree p−2j (more detailscan be found, e.g. in Chapter V, Section 3 of [193]).

rem-primitive Remark 1.13.1. It follows from the foregoing that at each point x ofM , any non-zero primitive form φ0 of degree p, together with the (m − p)exterior forms φ1 = Lφ0, . . . , φm−p = Lm−pφ0, form a base of a (m− p+ 1)-

dimensional subspace of ΛpxM , isomorphic to S(m−p), so that

Lmax-primitiveLmax-primitive (1.13.11) Lm−p+1φ0 = 0,

whereas, by (1.13.4),

Lambda-primitiveLambda-primitive (1.13.12) Λφr = r(m− p− r + 1)φr−1,

r = 1, . . . ,m− p.The following identity appears in A. Weil’s book [192] as Theoreme 2

of the Chapitre I, and, with a different argument due to Henryk Hecht,as Theorem 3.16 in Chaper V, Section 3 of R. O. Wells’ book [193]. Theelementary argument given here is due to A. Moroianu.

Proposition 1.13.1. For any primitive p-form ψ, 0 ≤ p ≤ m, and forany integer r, 0 ≤ r ≤ m− p, we have

wellswells (1.13.13) ∗Lrψ = (−1)p(p−1)

2r!

(m− p− r)!Lm−p−rJψ

1 also called Lefschetz decomposition in the literature. This decomposition appears inFN-Lepage[192, Chapitre I] as Theoreme 3.

1.13. THE HODGE-LEPAGE DECOMPOSITION 35

or, equivalently, the purely symplectic identity:

wells-omegawells-omega (1.13.14) ∗ω Lrψ = (−1)p(p+1)

2r!

(m− p− r)!Lm−p−rψ.

Proof. We first notice that (1.13.14) readily follows from (1.13.13) and(1.10.6) — and vice versa — as J obviously commutes with L. We then onlyneed to check (1.13.13). By a first easy induction on r, it is furthermoresufficient to check (1.13.13) for r = 0, i.e. to check

wells-0wells-0 (1.13.15) ∗ψ =(−1)

p(p−1)2

(m− p)!Lm−pJψ,

which is the same as

wells-omega-0wells-omega-0 (1.13.16) ∗ω ψ =(−1)

p(p+1)2

(m− p)!Lm−pψ.

Indeed, assuming that r ≥ 1 and that (1.13.13) holds when r is replaced byr − 1, by using (1.13.1) and (1.13.12), we get:

∗Lrψ = Λ ∗ Lr−1ψ

(−1)p(p−1)

2(r − 1)!

(m− p− r + 1)!ΛLm−p−r+1Jψ

(−1)p(p−1)

2(r − 1)!

(m− p− r + 1)!r(m− p− r + 1)Lm−p−rJψ

(−1)p(p−1)

2r!

(m− p− r)!Lm−p−rJψ.

Finally, we check (1.13.15) by a induction on the degree of the primitivep-form ψ, by observing that if, p ≥ 1, Xyψ is then a primitive (p− 1)-formfor any vector field X (easy consequence of (1.13.2)). Notice that, for p = 0,(1.13.15) reduces to the well-known identity ∗ 1 = ωm

m! . The induction themgoes as follows. If p ≥ 1, for any vector field X we infer from (1.13.3) that

(1.13.17) ∗ (Xyψ) = (−1)p−1X[ ∧ ∗ψ,

whereas, by the induction hypothesis applied to the (p − 1)-primitive formXyψ, we get

∗ (Xyψ) =(−1)

(p−1)(p−2)2

(m− p+ 1)!ωm−p+1 ∧ (JXyJψ)

(−1)(p−1)(p−2)

2

(m− p+ 1)!

[JXy(ωm−p+1 ∧ Jψ) + (m− p+ 1)X[ ∧ ωm−p ∧ Jψ

]=

(−1)(p−1)(p−2)

2

(m− p)!X ∧ ωm−p ∧ Jψ

(1.13.18)

because of (1.13.11). By comparing the above two expressions of ∗ (Xyψ)

we obtain that X ∧ ∗ψ = X ∧[(−1)

p(p−1)2

1(m−p)! L

m−pJψ]

for any vector

field X; this clearly implies (1.13.15).


rem-hecht Remark 1.13.2. H. Hecht’s argument reproduced in Wells’ book [193]uses the fact that the above sl(2,C)-action on the space of exterior forms canbe integrated into a C-linear action of the group SL(2,C and that the actionof the so-called Weyl operator w = exp( i π2 (Λ+L)), associated to the elemnt

exp( i π2 (X + Y )) =

(0 ii 0

)of SL(2,C), is then the same as the action of

the symplectic Hodge operator ∗ω up to some factor; more precisely, it canbe checked that

(1.13.19) ∗ω ψ = ip2−mw(ψ),

for any p-form ψ, cf. Lemma 3.15 and the proof of Theorem 3.16 in [193,Chapter V].

rem-algebraic Remark 1.13.3. The identities in Sections 1.12 and 1.13 are purely al-gebraic, i.e. involve no derivatives, hence are valid on any almost hermitianmanifold.

1.14. Kahler identitiesssidentity

In spite of the name Kahler identities commonly given to identities(1.14.1), these actually hold in the symplectic framework, i.e. for any almosthermitian structure (g, J, ω) such that ω is a symplectic 2-form but J is notassumed to be integrable, cf. Remark 1.1.22.

propKI Proposition 1.14.1. On any symplectic almost hermitian manifold(M, g, J, ω), the following identities hold:

kaehlerkaehler (1.14.1) [Λ, dc] = δ, [Λ, d] = −δc,

and

kaehleradjointkaehleradjoint (1.14.2) [δc, L] = d, [δ, L] = −dc,

where both sides are operators acting on exterior forms and the bracketsdenote commutators.

Proof. Since L and Λ (obviously) commute with J , any identity in(1.14.1) implies the other one; similarly, any identity in (1.14.2) implies theother one. Moreover, two of these four identities, namely the second onein (1.14.1) and the first one in (1.14.2), are purely symplectic, i.e. involveoperators determined by ω only — cf. proposition 1.11.1 — and, by (1.11.5)-(1.11.6), are symplectically adjoint to each other (whereas each identity in(1.14.1) is metrically adjoint to the corresponding one in (1.14.2)). It is thensufficient to check any one of them, say the first identity of (1.14.2), which isan easy consequence of (1.11.4), via the following simple computation (where

2I personally learned of this fact from R. Bryant many years ago.

1.14. KAHLER IDENTITIES 37

D denotes any auxiliary torsion-free, ω-preserving linear connection):

[δc, L]ψ = δc(ω ∧ ψ)− ω ∧ δcψ

=n∑i=1

(−eiyDei(ω ∧ ψ) + ω ∧ eiyDeiψ)

=n∑i=1

(−eiy(ω ∧ Deiψ + ω ∧ eiyDeiψ)

=n∑i=1

e∗i ∧ Deiψ = dψ,

by using eiyω = −e∗i , which readily follows from the definition of the sym-plectic dual frame e1, . . . , en, as in Proposition 1.11.1 (here, as usual,e∗1, . . . , e∗n stands for the algebraic dual coframe of the chosen auxiliaryframe e1, . . . , en).

propdcdelta Proposition 1.14.2. On any almost-Kahler manifold, the following iden-tities hold:

dcdeltadcdelta (1.14.3) δdc + dcδ = 0, δcd+ dδc = 0.

Proof. Direct consequenc of Proposition 1.14.1: by (1.14.1), δdc =[Λ, dc]dc = −dcΛdc, whereas dcδ = dc[Λ, dc] = dcΛdc; similarly, δcd =−[Λ, d]d = dΛd, whereas dδc = −d[Λ, d] = −dΛd. Notice that the secondidentity in (1.14.3) is purely symplectic.

In the context of E-valued exterior forms, Proposition 1.14.1 remainsformally the same:

propKIE Proposition 1.14.3. For any symplectic almost hermitian manifold(M, g, J, ω) and for any vector bundle E, equipped with a fiberwise innerproduct h and an h-compatible linear connection ∇, the following identitieshold:

kaehlerEkaehlerE (1.14.4) [Λ, dc,∇] = δ∇, [Λ, d∇] = −δc,∇,

and

kaehleradjointEkaehleradjointE (1.14.5) [δc,∇, L] = d∇, [δ∇, L] = −dc,∇,

where both sides are operators acting on E-valued exterior forms.

Proof. The four differential operators d∇, dc,∇, δ∇, δc,∇ have been de-fined in Sections 1.6 and 1.11, whereas operators L and Λ readily extendto E-valued forms by: L(s ⊗ ψ) = s ⊗ Lψ and Λ(s ⊗ ψ) = s ⊗ Λψ. Inorder to check that these identities remain valid at some point x of M , it issufficient to compare the actions of these operators on Ψ =

∑rk=1 sr ⊗ ψr,

where s1, . . . , sr is a local frame of E and the sk’s have been chosen insuch a way that ∇sk vanishes at x; then, (1.14.4)-(1.14.5) readily followfrom (1.14.1)-(1.14.2).

rem-deltadc


Remark 1.14.1. Proposition 1.14.2 does not extend to E-valued exteriorforms in such a simple way, as (d∇)2 and (dc,∇)2 are not zero in general. Wehave instead: (

δ∇dc,∇ + dc,∇δ∇)

Φ = −Λ(JR∇ ∧ Φ

)+ JR∇ ∧ ΛΦ,

(δc,∇d∇ + d∇δc,∇

)Φ = Λ

(R∇ ∧ Φ

)−R∇ ∧ ΛΦ,

ddcdcdnablaEddcdcdnablaE (1.14.6)

where J(R∇) is defined by J(R∇)X,Y = R∇JX,JY . Indeed, from (1.14.5) we

infer δ∇dc,∇ + dc,∇δ∇ = [Λ, dc,∇]dc,∇ + dc,∇[Λ, dc,∇] = [Λ, (dc,∇)2], whereas

from (1.6.7) we get (dc,∇)2Φ = −J(R∇ ∧ J−1Φ

)= −(JR∇) ∧ Φ; similarly,

δc,∇d∇+d∇δc,∇ = −[Λ, d∇]d∇−d∇[Λ, d∇] = −[Λ, (d∇)2], whereas (d∇)2Φ =−R∇ ∧ Φ.

1.15. Laplace and twisted Laplace operatorssslaplace

For any riemannian manifold (M, g), the riemannian Laplace operator,or simply laplacian, ∆, acting on exterior forms, is defined by ∆ = δd+ dδ.

For any g-compatible almost-complex structure J , we define the twistedLaplace operator ∆c by ∆c = δcdc + dcδc = J∆J−1.

In the case when J is integrable, we also define the Dolbeault Laplaceoperators or Dolbeault laplacians, and , defined by

(1.15.1) := ∂∗∂ + ∂∂∗, := ∂∗∂ + ∂∂∗,

where the operators ∂, ∂ and their formal adjoints ∂∗, ∂∗ with respect tothe hermitian global inner product have been introduced in Remark 1.11.3.Clearly, both and preserve the type of exterior forms, cf. Section 1.9.We then have

prop-laplace Proposition 1.15.1. On any compact Kahler manifold, we have

laplacelaplace (1.15.2) ∆ = ∆c = 2 = 2.

In particular, ∆ preserves the type of exterior forms, meaning that

DeltatypeDeltatype (1.15.3) (∆ψ)r,s = ∆(ψr,s),

for any p-form ψ and any r, s with r + s = p.

Proof. By using the Kahler identities (1.14.1), combined with (1.11.9),we get

∆c = dcδc + δcdc

= dc[d,Λ] + [d,Λ]dc use δc = [d,Λ]

= dcdΛ− dcΛd+ dΛdc − Λddc

= −ddcΛ− dcΛd+ dΛdc + Λdcd use ddc + dcd = 0

= d(Λdc − dcΛ) + (Λdc − dcΛ)d,

= dδ + δd = ∆, use [Λ, dc] = δ,

1.15. LAPLACE AND TWISTED LAPLACE OPERATORS 39

whereas, by (1.11.14)–(1.11.17), we get

=1

4(d+ idc)(δ − iδc) + (δ − iδc)(d+ idc)

=1

4(dδ + δd+ dcδc + δcdc)

+i

4(−δcd− dδc + dcδ + δdc)

=1

4(dδ + δd+ dcδc + δcdc) use (1.14.3)

=1

4(∆ + ∆c) =

1

2∆ =

1

2∆c,

and similarly for .

From Proposition 1.15.1, we readily infer the following well-known topo-logical obstructions to the existence of Kahler structures in the compactcase:

propbetti Proposition 1.15.2. The odd order Betti numbers, b2k+1, of any com-pact Kahler manifold are even, whereas the even order Betti numbers, b2k,are all positive.

Proof. For any compact riemannien manifold (M, g) the classical Hodgetheory asserts that the (complex) dimension of the kernel of ∆, acting onthe space of complex p-form, i.e. the space, Hpg, of complex g-harmonicp-forms, is finite and a topological invariant, equal to the dimension, bp,of the cohomology space Hp(M,R), called the p-th Betti number, cf. e.g.[193]. Because of (1.15.3), when g is Kahler, Hpg splits as a direct sumHpg = ⊕r+s=pHr,sg , where Hr,sg denotes the space of harmonic forms of type(r, s), for all pairs of non negative integers r, s such that r + s = p; we thenhave bp =

∑r+s=p h

r,s, where the hr,s — the Hodge numbers of (M,J) —

denote the (complex) dimension of Hr,sg . Since ∆ is real, the conjugationinduces a C-antilinear isomorphism from Hr,s to Hs,r for any such pair r, s,

so that hr,s = hs,r. If p = 2k + 1 is odd, we then get bp = 2∑k

r=0 hr,2k+1−r

is even. The last assertion of Proposition 1.15.2 follows from the fact that,for each k = 1, . . . ,m, Ωk := [ωk] is not zero, as Ωm 6= 0 (recall that ωm/m!is the volume form of g, hence does not integrate to zero).

Proposition 1.15.3. For any Kahler manifold, we have the followingidentities:

DeltaLDeltaL (1.15.4) [∆, L] = 0, [∆,Λ] = 0.

In particular, for any harmonic exterior form ψ, Lψ = ω ∧ ψ and Λψ areharmonic as well.

Proof. Since [∆,Λ]∗ = −[∆, L], the second identity in (1.15.4) followsfrom the first one. Since the Kahler form ω is closed, we have [d, L] = 0. Byusing the second identity in (1.14.2), we then get

(1.15.5) [∆, L] = −(ddc + dcd),

on any almost-Kahler manifold. In the Kahler case, we have ddc + dcd = 0by Proposition 1.11.2, hence [∆, L] = 0. The last assertion readily followsfrom (1.15.4).


remcalabieckmann Remark 1.15.1. Proposition 1.15.2 provides an easy non-existence cri-terion of Kahler metrics on a given compact complex manifold (M,J). Forexample, it directly follows from Proposition 1.15.2 that the so-called Calabi-Eckmann complex manifolds, whose underlying spaces are products of odd-dimensional spheres S2p+1 × S2q+1, with p ≥ 0 and q > 0, admit no Kahlermetric, cf. [48].

remlaplacian Remark 1.15.2. For any (real) function f the Kahler identities (1.14.1)combined with the general identity Λ(ddc+dcd)f = 0, which readily followsfrom (1.11.10), imply

deltafdeltaf (1.15.6) ∆f = ∆cf = −Λ(ddcf) = Λ(dcdf),

on any symplectic almost-hermitian manifold.

1.16. The Akizuki-Nakano identitysakizuki

For any vector bundle E over M , equipped with a fiberwise inner prod-uct h and an h-compatible linear connection ∇, we can define a riemann-ian Laplace operator ∆∇ := δ∇d∇ + d∇δ∇ and a twisted Laplace operator∆c,∇ := δc,∇dc,∇ + dc,∇δc,∇ = J∆∇J−1. If, moreover, E = (E, h) is ahermitian holomorphic vector bundle over a complex manifold (M,J), and∇ is the corresponding Chern connection, we can similarly define the Dol-

beault Laplace operator by E

= (∂E)∗∂E + ∂E(∂E)∗, as well as ∇ =(∂∇)∗∂∇ + ∂∇(∂∇)∗, where the operators ∂E and ∂∇, acting on E-valuedexterior forms of any type were defined by (1.11.26)–(1.11.27) in Remark1.11.4 of Section 1.6. If, moreover, the almost hermitian structure (g, J, ω)is Kahler, we have:

prop-akizuki Proposition 1.16.1. Let (E, h) be a hermitian holomorphic vector bun-dle over a Kahler manifold (M, g, J, ω). Then,

laplaceElaplaceE (1.16.1) ∆∇ = ∆c,∇ = ∇ +E,

and

laplaceE-chernlaplaceE-chern (1.16.2) ∇ =1

2

(∆∇ − i [Λ, R∇]

),

E=

1

2

(∆∇ + i [Λ, R∇]

),

where R∇ denotes the curvature of the Chern connection ∇ of (Eh). Inparticular:

akizukiakizuki (1.16.3) E −∇ = i [Λ, R∇],

with

akizuki-expakizuki-exp (1.16.4) [Λ, R∇] Φ = Λ(R∇ ∧ Φ)−R∇ ∧ ΛΦ,

for any E-valued exterior form Φ.

Proof. The proof of (1.16.3)–(1.16.4) is quite similar to the proof ofProposition 1.15.1, by using: (i) Proposition 1.11.3, which guarantees thatd∇dc,∇+dc,∇d∇ = 0, as R∇ is J-invariant; (ii) the extended Kahler identitiesof Proposition 1.14.3, which are formally the same as in the scalar case; (iii)the identity (1.14.6), which replaces the identity (1.14.3) used in the scalar

1.17. THE ddc-LEMMA 41

case. Notice that i[R∇,Λ] is a hermitian endomorphism of E ⊗ Λ•M . Wethus have

∆c,∇ = dc,∇δc,∇ + δc,∇dc,∇

= dc,∇[d∇,Λ] + [d∇,Λ]dc,∇ use δc,∇ = [d∇,Λ]

= dc,∇d∇Λ− dc,∇Λd∇ + d∇Λdc,∇ − Λd∇dc,∇

= −d∇dc,∇Λ− dc,∇Λd∇ + d∇Λdc,∇ + Λdc,∇d∇ use d∇dc,∇ + dc,∇d∇ = 0

= d∇(Λdc,∇ − dc,∇Λ) + (Λdc,∇ − dc,∇Λ)d∇,

= d∇δ∇ + δ∇d∇ = ∆∇ use [Λ, dc,∇] = δ∇,

and

∇ =1

4(d∇ + idc,∇)(δ∇ − iδc,∇) + (δ∇ − iδc,∇)(d∇ + idc,∇)

=1

4(d∇δ∇ + δ∇d∇ + dc,∇δc,∇ + δc,∇dc,∇)

+i

4(−δc,∇d∇ − d∇δc,∇ + dc,∇δ∇ + δ∇dc,∇)

=1

2(∆∇ + ∆c,∇)

− i

2[Λ, R∇] use (1.14.6) together with JR∇ = R∇

=1

2∆∇ − i

2[Λ, R∇].

Similarly

E

=1

4(d∇ − idc,∇)(δ∇ + iδc,∇) + (δ∇ + iδc,∇)(d∇ − idc,∇)

=1

4(d∇δ∇ + δ∇d∇ + dc,∇δc,∇ + δc,∇dc,∇)

− i

4(−δc,∇d∇ − d∇δc,∇ + dc,∇δ∇ + δ∇dc,∇)

=1

2∆∇ +

i

2[Λ, R∇].

rem-akizuki Remark 1.16.1. The identity (1.16.3) is known as the Akizuki–Nakanoidentity, cf. [2], [66].

1.17. The ddc-lemmassddc

For any p-form ψ defined on an oriented, compact riemannian manifold,the Hodge decomposition of ψ is written as follows

hodge1hodge1 (1.17.1) ψ = ψH + ∆Gψ = ψH + dδGψ + δdGψ,where ψH is harmonic and G is the Green operator for the laplacian ∆.Since, ∆ = ∆c, the corresponding Green operators coincide as well and wealso have

hodge2hodge2 (1.17.2) ψ = ψH + ∆cGψ = ψH + dcδcGψ + δcdcGψ.The ddc-lemma then reads (see [64])


lemmaddc Lemma 1.17.1. Let (M, g, J, ω) be a compact Kahler manifold. Thenthe Hodge decomposition of any real ddc-closed p-form ψ can be written as

(1.17.3) ψ = ψH + dδGψ − dcδδcdG2ψ

where ψH is harmonic.

Proof. From (1.17.1) and from ddcψ = 0, we infer 0 = ddcδdGψ =−dδdcdGψ, so that δdcdGψ = −δddcGψ = 0. It follows that ddcGψ = 0.The harmonic part of dGψ with respect to ∆, hence also with respect to∆c = ∆, equals 0. From (1.17.2), we then infer dGψ = dcδcGdGψ, so thatδdGψ = −dcδδcGdGψ = −dcδδcdG2ψ. (In this proof, we have used theidentities δdc = −dcδ and ddc = −dcd, as well as dG = Gd).


propddc Proposition 1.17.1. (Same hypotheses as in Lemma 1.17.1). Let ψ beany J-invariant p-form satisfying ψ = dφ, for some (p− 1)-form φ. Then,

(1.17.4) ψ = ddcψ,

for some (p− 2)form ψ.

Proof. Since Jψ = ψ, ψ is dc-closed and φ is then ddc-closed. Itthen follows from Lemma 1.17.1 that φ = φH + dδGφ− dcδδcdG2φ, so thatψ = dφ = ddcψ, with ψ = −δδcdG2φ = −δδcG2ψ.

The above proposition is most often used when p = 2 and is then statedas follows:

propddc2 Proposition 1.17.2. Let ψ0, ψ1 be any two real J-invariant closed 2-forms defined on a compact Kahler manifold (M, g, J, ω) and suppose thatψ0, ψ1 determine the same de Rham class in H2

dR(M,R). Then there existsa real function f , uniquely defined up to an additive constant, such that

ddc2ddc2 (1.17.5) ψ1 − ψ0 = ddcf.

Proof. (1.17.5) is a direct application of Proposition 1.17.1 for ψ =ψ1 − ψ0. If ddcf = 0, then, by (1.15.6), ∆f = 0, i.e. f is harmonic, henceconstant, as M is compact.

remddckern Remark 1.17.1. In general, for any complex manifold (M,J), the kernelof the operator ddc acting on C∞(M,R) is the space of those real smoothfunctions f on M which are locally the real part of a holomorphic function.Indeed, if ddcf = 0, then, by the Poincare Lemma, dcf =loc dh, for somelocal (real) function h and F := f + ih is then holomorphic, as ∂F =12(d− idc)(f + ih) = 1

2(df + dch) + i2(dh− dcf) = 0. If (M,J) is compact, it

is true, even in the case that (M,J) is not of kahlerian type, that the kernel ofddc acting on C∞(M,R) is reduced to the space of constant functions. Thisfact cannot be directly deduced from the above local statement, as there isno guarantee a priori that an element f of the kernel of ddc be the real partof a globally defined holomorphic function (which would then be constant,

as M is compact). On the other hand, the operator f 7→ ∆ := −Λddcf isdefined for any almost-hermitian metric g on (M,J); except if the structure

is almost-Kahler, ∆ is different from the riemannian Laplace operator ∆g;it nevertheless shares many properties of ∆g: it is as an elliptic, second

1.17. THE ddc-LEMMA 43

order linear differential operator acting on C∞(M,R), such that ∆(1) = 0;

this is sufficient to conclude that the kernel of ∆ is reduced to the space ofconstants, as M is compact, cf. [103].

A related result is the following

proplocpotential Proposition 1.17.3. Any J-invariant real closed 2-form ψ on a Kahlermanifold (M, g, J, ω) can be locally written as

(1.17.6) ψ =loc ddcf,

for some local real function. For any two such functions f, f ′ defined on asame open subset of M , the difference f ′ − f is locally the real part of aholomorphic function.

Proof. In contrast to the previous proposition, Proposition 1.17.3 is oflocal character and does not require that M be compact. Its proof readilyfollows from the Poincare lemma, which asserts that any closed form islocally exact, and from the Dolbeault lemma, which similarly asserts thatany ∂-closed complex form ϕ of type (p, q), q > 0, is locally of the formϕ = ∂φ, for some (p, q − 1)-form φ, cf. e.g. [156], [93]. The argument thengoes as follows. Since ψ is closed, by the Poincare lemma it can locally bewritten as ψ =loc dα, for some real 1-form α. Since ψ is J-invariant, the(0, 1)-part α0,1 of α is ∂-closed; by the Dolbeault lemma there then existsa complex function F , such that α =loc ∂F = 1

2(d − idc)F , possibly, on a

smaller open set. We then get ψ =loc12d((d − idc)F + (d + idc)F ) = ddcf ,

with f = Im(F ). This proves the first part of the proposition. The secondpart has been proved in Remark 1.17.1.

defi-potential Definition 1. For any Kahler manifold (M, g, J, ω), a (local) Kahlerpotential is a real function φ defined on some open subset U of M , such that

(1.17.7) ω =U ddcφ.

By Proposition 1.17.3, such a potential exists around any point of M ,and is then defined up the addition of a function which is locally the realpart of a holomorphic function. It may happen that (M, g, Jω) admits aglobally defined Kahler potential: for example, denote by (g0, J0, ω0) thestandard flat Kahler structure of the standard hermitian vector space Cm(viewed as a 2m-dimensional real manifold), defined by

(1.17.8) g0(X,Y ) = Re(X,Y ), J0X = iX, ω0(X,Y ) = −Im(X,Y ),

for any u in Cm and any two X,Y in TuCm ∼= Cm, where (·, ·) here standsfor the standard hermitian inner product. It is then an easy exercice tocheck that

(1.17.9) ω0 =1

4ddcr2,

where r2 is the square norm function determined by 〈·, ·〉, meaning that r2

4is a global Kahler potential. Note that the standard flat Kahler structure isinvariant under the natural (linear) action of the unitary group U(m), and,

up to an additive constant, r2

4 is the unique U(m)-invariant (global) Kahlerpotential.


On the other hand, on a compact Kahler manifold (M, g, J, ω), no glob-ally defined Kahler potential can possibliy exist, as the de Rham classΩ = [ω] is then non-zero, see Proposition 1.15.2.

1.18. Riemannian curvature and Bianchi identitiesssbianchi

For a general riemannian manifold, the riemannian curvature R is de-fined as the curvature of the Levi-Civita connection, hence by:

curvaturecurvature (1.18.1) RX,Y Z = D[X,Y ]Z − [DX , DY ]Z,

for all vector fields X,Y, Z (see (1.6.6 and Remark 1.6.1).The riemannian curvature satisfies the algebraic Bianchi identity — or

first Bianchi identity — and the differential Bianchi identity — or secondBianchi identity — respectively defined by

bianchi1bianchi1 (1.18.2) RX1,X2X3 +RX2,X3X1 +RX3,X1X2 = 0,

and

bianchi2bianchi2 (1.18.3) (DX1R)X2,X3 + (DX2R)X3,X1 + (DX3R)X1,X2 = 0,

for any vector fields X1, X2, X3.The differential Bianchi identity (1.18.3) is actually a special case of a

very general identity, which holds for any linear connection ∇ on any vectorbundle E over some manifold M , and which can be written as follows

bianchi2genbianchi2gen (1.18.4) d∇R∇ = 0,

where, we recall, the curvature R∇ of∇ is viewed as a End(E)-valued 2-formon M and we still denote ∇ the induced covariant derivative on End(E).Then, (1.18.4) is a direct consequence of (1.6.7) and of the associativityrelation d∇ (d∇ d∇) = (d∇ d∇) d∇.

It is then as simple exercice to show that (1.18.4) reduces to (1.18.3)when ∇ is the Levi-Civita connection of a riemannian manifold.

The algebraic Bianchi identity (1.18.2) also holds in a larger setting,namely for any torsion-free linear connection defined on TM , and is againdeduced from (1.6.7) in the following way. Let∇ be any linear connection onTM and denote by T∇ its torsion, defined by T∇X,Y = ∇XY −∇YX− [X,Y ];the torsion can then be viewed as a TM -valued 2-form and, by comparingwith (1.6.3), we infer that it can be written as T∇ = d∇I, where I standsfor the identity operartor of TM , viewed as TM -valued 1-form; then (1.6.7)reads as

bianchi1genbianchi1gen (1.18.5) R∇ ∧ I = −d∇T∇,

where the lhs is exactly the lhs of (1.18.2) when R is replaced by R∇: ifT∇ ≡ 0, i.e. if ∇ is torsion-free, we thus get (1.18.2); in the general case,(1.18.5) can be regarded as a generalized algebraic Bianchi identity , whichholds for the curvature of any linear connections on TM .

In the riemannian case, the algebraic Bianchi identity (1.18.2) impliesthe following weak (algebraic) Bianchi identity:

weakbianchiweakbianchi (1.18.6) (RX1,X2X3, X4) = (RX3,X4X1, X2).

1.19. THE RICCI FORM OF A KAHLER STRUCTURE 45

The vector bundle Λ2M := Λ2(T ∗M) of 2-forms, the vector bundleΛ2(TM) of 2-vectors and the vector bundle A(M) of skew-symmetric en-dorphisms of TM — or T ∗M — are all naturally identified with each othervia the riemannian duality induced by g: the riemannian curvature R canthen be viewed in a tautological way as an operator acting on any of them,by just setting (R(X1∧X2), X3∧X4) := (RX1,X2X3, X4). In this role, R willbe called the curvature opeerator and (1.18.6) simply says that the curvatureoperator R is symmetric.

The Ricci tensor, r, of a riemannian manifold is defined by:

ricci-defricci-def (1.18.7) r(X,Y ) = trZ → RX,ZY =n∑j=1

(RX,ejY, ej),

for any auxiliary orthonormal basis e1, . . . , en.Again, the weak algebraic Bianchi identity (1.18.6) readily implies that

the Ricci tensor r is symmetric.The scalar curavature s is defined as the trace of r with respect to g,

namely by

(1.18.8) s = (r, g).

Notice that the derivation of s from r does require the use of the metricg, while the derivation of r from R does not.

From the differential Bianchi identity (1.18.3) we easily deduce the con-tracted Bianchi identity :

bianchicontbianchicont (1.18.9) δr +1

2ds = 0

which relates the codifferential of the Ricci tensor — see Section 1.10 — tothe differential of the scalar curvature of any riemannian metric.

A riemannian metric g of dimension n > 2 is called an Einstein metricif its Ricci tensor r is proportional to g; we then have

einsteineinstein (1.18.10) r =s

ng.

From (1.18.9) we then infer that the scalar curvature s is constant: indeed,if r = s

ng, (1.18.9) simply reads (12 −

1n) ds = 0 which implies ds = 0, as

n 6= 2. If n = 2, (1.18.10) holds for any riemannian metric and we say thatg is an Einstein metric if s is constant: (M, g) is then of constant sectionalcurvature — or Gauss curvature — equal to s

2 .

1.19. The Ricci form of a Kahler structuressricci

In the Kahler context of these notes, the riemannian curvature oper-ator R — cf. Section 1.18 — acts trivially on the J-anti-invariant part ofΛ2M , hence appears as a (symmetric) endomorphism of the J-invariant partΛ1,1M .

Alternatively, the riemannian curvature R of a Kahler manifold can beviewed as a J-invariant 2-form with values in Λ1,1M .

This is consistent with the fact that the Levi-Civita connection D co-incides with the canonical Chern connection of (TM, J) (cf. Proposition


1.8.1), so that R coincides with the curvature of the canonical Chern con-nection of (TM, J), when the latter is regarded as a hermitian holomorphicvector bundle, cf. Section 1.6.

The Ricci tensor r is then J-invariant, i.e. satisfies r(JX, JY ) = r(X,Y ),and the Ricci form is the defined as the associated J-invariant 2-form, de-noted by ρ, defined by:

rho-defrho-def (1.19.1) ρ(X,Y ) = r(JX, Y ).

Notice that the Ricci form ρ is associated to the Ricci tensor r in thesame way as the Kahler form ω is associated to the riemannian meric g. Alsonotice that in terms of the Ricci form, the riemmannian scalar curvature sis expressed by

lambdarholambdarho (1.19.2) s = 2 Λρ.

lemmariccialter Lemma 1.19.1. The Ricci form ρ has the following alternative descrip-tion:

rhoalterrhoalter (1.19.3) ρ =1

2

n∑j=1

(Rej ,Jej ·, ·) = R(ω),

for any auxiliary orthonormal frame e1, . . . , en.

Proof. By using the algebraic Bianchi identity (1.18.2) and the verydefinition (1.18.7), the second expression in (1.19.3) can be rewritten as:

1

2

n∑j=1

(Rej ,JejX,Y ) = −1

2

n∑j=1

((RJej ,Xej , Y ) + (RX,ejJej , Y ))

=1

2

n∑j=1

((Rej ,JXej , Y ) + (RX,ejej , JY ))

=1

2r(JX, Y )− 1

2r(X, JY ) = r(JX, Y ) = ρ(X,Y ).

We then get the third expression by observing that the Kahler form ω is alsoω = 1

2

∑nj=1 ej ∧Jej , for any orthonormal basis e1, . . . , en, viewed as basis

of covectors via the riemannian duality (compare with the more complicatedcomputation (9.4.9) in the almost Kahler case).

In (1.19.3), R(ω) denotes the image of the Kahler form ω by the riemann-ian curvature operator R. Note that due to the symmetry of the curvatureoperator, R(ω) is the same as the real 2-form obtained by taking the traceof −J R, when the latter is viewed as a hermitian operator valued 2-form,i.e.

RomegakaehlerRomegakaehler (1.19.4) R(ω)(X,Y ) = tr(−J RX,Y ),

where the trace is here understood as the trace of a C-linear hermitianoperator (see Section 1.12.

This has the following interpretation. Consider the anti-canonical com-plex line bundle K−1

M := Λm((TM, J)), where Λm((TM, J)) denotes the top(complex) exterior power of (TM, J). This is actually a hermitian holomor-phic line bundle, as (TM, J) is itself a hermitian holomorphic vector bundle(see Sections 1.8, 1.6 and 1.7) and the corresponding Chern connection is

1.19. THE RICCI FORM OF A KAHLER STRUCTURE 47

then the connection induced by the Chern connection of (TM, J); it follows

that the Chern curvature of K−1M , say RK

−1M , is equal to the (complex) trace

of the Chern curvature of (TM, J), which, by Proposition 1.8.1, is the rie-

mannian curvature R; we then have RK−1M

X,Y = tr(RX,Y ); here, tr(RX,Y ) is

understood as the (complex) trace of RX,Y as an anti-hermitian, C-linear

operator acting on (TM, J) ∼ T 1,0M ; RK−1M

X,Y is then a purely imaginary 2-

form and can be alternatively written as RK−1M

X,Y = itr(−J RX,Y ), where

−J RX,Y is hermitian, so that tr(−J RX,Y ) is real; by comparing with(1.19.4) and (1.19.3), we eventually end up with

(1.19.5) RK−1M = i ρ.

We have thus established the following basic series of facts concerning theRicci form:

propriccibasic Proposition 1.19.1. For any Kahler manifold (M, g, J, ω), the Ricciform ρ is the curvature form of the Chern connection of the anti-canonicalline bundle K−1

M relatively to the induced hermitian inner product. In par-ticular ρ is closed and its de Rham class, [ρ], in H2(M,R) is equal to2π c1(M,J), where c1(M,J) is defined as the real Chern class of K−1

M , whichis also the first (real) Chern class of (TM, J). The de Rham class [ρ] of ρis then the same for all Kahler metrics on (M,J).

Moreover, ρ has the following local expression:

locrholocrho (1.19.6) ρ =loc −1

2ddc log

vgv0,

where v0 stands for the volume form of the flat Kahler metric determinedby any choice of a (local) holomorphic coordinate system, and, for any tworiemannian metrics, g0 and g — both Kahler with respect to J — the cor-responding Ricci forms ρ0 and ρ are related by

difrhodifrho (1.19.7) ρ− ρ0 = −1

2ddc log

vgvg0

.

Proof. The first statement follows from the above discussion and fromRemark 1.7.1. The local expression (1.19.6) follows from Proposition 1.7.1 inthe following way. For any point x of M , let z1, . . . , zm be any holomorphiccoordinate system defined on some open neighbourhood U of x in M ; theKahler form of the corresponding flat Kahler structure is then i

2

∑mr=1 dzr ∧

dzr; the corresponding volume form is v0 :=∏mr=1

i2dzr ∧ dzr, whereas from

(1.2.2) we easily infer

volumeformgvolumeformg (1.19.8) vg = 2m|∂/∂z1 ∧ . . . ∧ ∂/∂zm|2 v0.

In (1.19.8), ∂/∂z1 ∧ . . . ∧ ∂/∂zm is a (nowhere vanishing) holomorphic sec-tion of K∗(M) on U and |∂/∂z1 ∧ . . . ∧ ∂/∂zm|2 stands for its square normwith respect to the induced hermitian inner product. Then, (1.19.6) readilyfollows from (1.19.8) and from the general formula (1.7.2), whereas (1.19.7)— where the rhs is now globally defined — is an immediate consequence of(1.19.6).


remrhovolume Remark 1.19.1. From (1.19.6) we deduce that the Ricci form ρ of anyKahler structure (g, J, ω) is entirely determined by the complex structure Jand the volume form vg of g.

propricciharmonic Proposition 1.19.2. The Ricci form ρ of a Kahler manifold (M, g, J, ω)is co-closed if and only if the scalar curvature s of g is constant. If M iscompact, ρ is then harmonic if and only s is constant. If M is compact, ofcomplex dimension m > 1, and if the first Chern class c1(M,J) is related tothe Kahler class Ω := [ω] by c1(M,J) = k

2π Ω, for some real number k, theng is an Einstein metric if and only if s is constant; s is then equal to 2km.

Proof. In the Kahler context, the contracted Bianchi identity (1.18.9)can be rewritten as

deltarhodeltarho (1.19.9) δρ = −1

2dcs

(easy verification); this proves the first assertion. For any compact riemann-ian manifold, an exterior form φ is harmonic — meaning that ∆φ = 0 —if and only if φ is closed and co-closed (direct consequence of ∆ = δd+ dδ,as δ is the (formal) adjoint of d); this proves the second assertion. Ifc1(M,J) = k

2π Ω and ρ is harmonic, ρ and k ω are both harmonic repre-sentatives of 2π c1(M,J), so that ρ = k ω; equivalently, r = k g, meaningthat g is an Einstein metric and k = s

2m .

For a general(connected) compact Kahler manifold, the defect for theRicci form to be harmonic is measured by the Ricci potential, pg, defined —via the ddc-lemma 1.17.1 — by

riccipotentialriccipotential (1.19.10) ρ = ρH −1

2ddcpg,

where ρH denotes the harmonic part of the Ricci form ρ with respect to g,and by

riccipotential-normriccipotential-norm (1.19.11)

∫Mpg vg = 0.

Since the trace of any harmonic, J-invariant 2-form is constant — easy con-sequence of the Kahler identity (1.14.1) — we infer that the Ricci potentialand the scalar curvature are related by

s0s0 (1.19.12) sg − sg = ∆pg,

where s = 1V

∫M sg vg denotes the mean value of sg with respect to vg. In

terms of the Green operator G introduced in Section 1.17, we then have

pggreenpggreen (1.19.13) pg = G(sg − s).By its very definition, the Ricci potential is identically zero if and only if theRicci form is harmonic; by (1.19.12)-(1.19.13), we retrieve the fact alreadyobserved in Proposition 1.19.2 that this happens if and only if sg is constant.

Remark 1.19.2. As explained in Appendix A, the curvature of a Kahlermetric splits into three components: a scalar component, identified with thescalar curvature, a second component, identified with the trace-free part ofthe Ricci tensor, and the Bochner tensor. A Kahler metric whose Bochnertensor is zero is called Bochner-flat, or Bochner-Kahler. The holomorphic

1.20. THE CALABI-YAU THEOREM 49

sectional curvature H of a Kahler metric is the restriction of the sectionalcurvature to orthonormal pairs of the form X,JX; for any unit vector X, wethen have H(X) = g(RX,JXX, JX). The holomorphic sectional curvatureentirely determines the whole curvature and a Kahler metric is of constantholomorphic sectional curvature if and only if it is both Kahler-Einstein andBochner-flat, i.e. its curvature is reduced to its scalar component. Thecurvature is then of the form

RchscRchsc (1.19.14) RX,Y =c

4(X ∧ Y + JX ∧ JY + 2ω(X,Y ) J),

for any vector fields X,Y , where c denotes the (constant) value of the holo-morphic sectional curvature (the rhs has to be regarded as a skew-symmetric,J-linear endomorphism via the metric g). The (constant) scalar curvatures is then given by s = m(m+ 1) c.

1.20. The Calabi-Yau theoremsscy

For any de Rham class Ω in H2dR(M,R), we denote by MΩ the set of

Kahler metrics on (M, g, J, ω) whose Kahler form ω belongs to Ω. If MΩ isnon-empty, Ω is called a Kahler class andMΩ is then an infinite-dimensionalFrechet manifold. More precisely, fix any g0 in MΩ, with Kahler class ω0.Then, by Proposition 1.17.2, the Kahler class, ω, of any g in MΩ can bewritten as ω = ω0 + ddcφ for some real C∞ function, uniquely defined up toan additive constant. We then obtain a natural identification of MΩ withthe space of real C∞ functions φ normalized, say, by

∫M φ vg0 = 0 — see

however the end of Section 4.1 for a more natural normalization — and suchthat ω := ω0 + ddcφ is positive, meaning that g := ω(·, J ·) is a riemannianmetric. Recall — cf. Remark 1.7.1 — that for any Kahler structure (g, J, ω)defined on (M,J), the de Rham class [ρ] of the corresponding Ricci form isequal to 2π c1(M,J), where c1(M,J) stands for the (real) first Chern classof (TM, J). The Calabi-Yau theorem can then be stated as follows

thmcy1 Theorem 1.20.1 (Calabi-Yau theorem, 1st statement). Let (M,J) be acompact complex manifold admitting Kahler metrics and let Ω be any Kahlerclass. Then any (closed) J-invariant 2-form in the de Rham class 2π c1(M,J)is the Ricci form of a unique Kahler metric in MΩ.

A (real) 2m-form v on M is called positive if v has no zero and∫M v > 0,

when M is oriented by J . In view of (1.19.7), the Calabi-Yau can then bere-stated in the following equivalent form

thmcy2 Theorem 1.20.2 (Calabi-Yau theorem, 2nd statement). Let (M,J) bea compact complex manifold admitting Kahler metrics and let Ω be anyKahler class. Then any positive 2m-form v such that

∫M v = Ωm

m! [M ] is thevolume form of a unique Kahler metric in MΩ.

The equivalence of the two statements is established in the followingmanner. Let ω0 be the Kahler form of a fixed element g0 in MΩ, ρ0 andv0 the corresponding Ricci form and volume-form. By Proposition 1.17.2, aJ-invariant 2-form ψ belongs to the de Rham class 2π c1(M,J) if and onlyif it is of the form ψ = ρ0 − 1

2ddcf , for some real function uniquely defined


if we normalize it by

normalize-riccinormalize-ricci (1.20.1)

∫M

(ef − 1)vg0 = 0.

Then, because of (1.19.7), ψ is the Ricci form of some metric g in MΩ ifand only if v := ef v0 is the volume-form of g, and vice versa.

On the other hand, for any g in MΩ, there exists a real function φ,uniquely defined up to an additive constant, such that the correspondingKahler form ω is equal to ω = ω0 + ddcφ and v = efvg0 is the volume-formof g if and only if φ and f are related by the following equation:

monge-amperemonge-ampere (1.20.2) (ω0 + ddcφ)m = ef ωm0 .

The Calabi-Yau theorem can therefore be stated in the following third al-ternative way:

thmcy3 Theorem 1.20.3 (Calabi-Yau theorem, 3rd statement). Let (M,J) be acompact complex manifold admitting Kahler metrics and let Ω be any Kahlerclass. Let g0 be any Kahler metric whose Kahler form ω0 belongs to Ω.Let f be any real function on M which satisfies (1.20.1) where vg0 is thevolume form of g0. Then, there exists a solution φ of (1.20.2) such thatω := ω0 + ddcφ is positive definite and this solution is unique up to anadditive constant.

remcalabiprod Remark 1.20.1. Equation (1.20.2) is a (complex) Monge-Ampere equa-tion. For any point of M , denote by λ1, . . . , λm the (real) eigenvalues of thehermitian operator Φ determined by ω0(Φ(X), Y ) = ddcφ(X,Y ). Then, theMonge-Ampere equation reads

calabiprodcalabiprod (1.20.3)

m∏j=1

(1 + λj) = 1 + σ1(λ) + . . .+ σm(λ) = ef ,

where the σj(λ)’s stand for the elementatry symmetric functions of the λj ’s(in particular, σ1(λ) = −∆0φ, where ∆0 denotes the riemannian Laplaceoperator relative to ω0). The condition that ω0 + ddcφ be positive definiteis that the (1 + λj)’s are positive everywhere. From the general inequality

(∏mj=1 aj)

1/m ≤∑mj=1 ajm , valid for any m positive real numbers a1, . . . , am

3;we then infer

(1.20.4) ∆0(φ) ≤ m(1− efm ).

In particular, if f ≡ 0 — the Monge-Ampere equation then reduces to(ω0 + ddcφ)m = ωm0 — we get ∆0(φ) ≤ 0, which implies that φ is constant.This observation readily implies the uniqueness part in Theorem 1.20.3.

Theorem 1.20.1 was first postulated by E. Calabi in [43], [44] and hassince then be referred to as the Calabi conjecture, until a complete proofwas provided by S.-T. Yau in 1976 [197], [198]. The method of resolution

3 This inequality can be read 1m

∑mj=1 log aj ≤ log

∑mj=1 aj

m, which is itself a spe-fnlog

cialization of the inequality∑mj=1 tj log aj ≤ log

∑mj=1 tjaj , for any m-uple tj ≥ 0 with∑m

j=1 tj = 1; this, in turn, is an easy consequence of the concavity of the graph of the log

function.

1.21. KAHLER-EINSTEIN METRICS 51

suggested by E. Calabi in [43] consists in introducing an auxiliary parame-ter t in [0, 1] and in replacing the Monge-Ampere equation (1.20.2) by thefollowing family, E(t), of Monge-Ampere equations:

t-monge-amperet-monge-ampere (1.20.5) (ω0 + ddcφ)m = eft ωm0 ,

where ft is defined by eft = etf1VΩ

∫M etf vg0

(ft is then a smooth curve joining

f0 ≡ 1 to f in the space, say H0, of real functions on M normalized by(1.20.1)). Then, E(1) is the initial equation, whereas E(0) admits constantfunctions as obvious solutions.

Denote by S ⊂ [0, 1] the set of t such that E(t) admits a solution (this isthen unique up to an additive constant, cf. Remark 1.20.1); S is not empty,as it contains 0, and Theorem 1.20.3 amounts to proving that S contains 1,in fact coincides with the whole segment [0, 1], and this is done by showingthat S is both open and closed in [0, 1].

The proof that S is open in [0, 1] can be outlined as follows. Denote byV the map from MΩ to H0 defined by V : g 7→ log

vgvg0

; then , for any (g, ω)

in MΩ, the tangent space TgMΩ is naturally identified to C∞(M,R)/R,hence to the space, C∞0,g(M,R), of functions in C∞(M,R) which integrateto zero along vg — cf. Section 3.2 — whereas the tangent space of H0 atV(g) is also naturally identified to C∞0,g(M,R) (by differentiating (1.20.1) we

get∫M f ef vg0 = 0, which is the same as

∫M f vg = 0 when f = V(g)). The

derivative of V at ω is then simply the map

VderivativeVderivative (1.20.6) f 7→ −∆gf ,

where ∆g is the riemannian Laplace operator relative to g, cf. (3.2.4) inChapter 3. Now, we already observed that ∆g is an isomophism fromC∞0,g(M,R) to itself, whose inverse is the Green operator G relative to g,introduced in Section 1.17. By introducing appropriate Sobolev spaces andby using the elliptic regularity of ∆g — cf. Section 5.3 for details in a similarcontext — we infer that V is a local diffeomorphism from MΩ to its imagein H0. Now, if φt0 is a solution to E(t0) for some t0 in [0, 1] — meaning thatV(ωt0 + ddcφt0) = ft0 — then, for any t close enough to t0, ft is sufficientlyclose to ft0 in the Frechet topology of C∞0,g0

(M,R) so that ft still lies in the

image of V, i.e. E(t) admits solutions. This completes the (sketch of) proofthat S is open in [0, 1].

Proving that S is also closed, hence coincides with the whole segment[0, 1], requires a priori estimates — independent of the parameter t — forsolutions of Et and their derivatives and has revealed a difficult question,solved by S.-T. Yau in [197]-[198]. Many variants and improvements ofYau’s original proof appeared since then in the literature, see e.g. [19],[173], [178], [185], [112], [33].

1.21. Kahler-Einstein metricsssKE

A closely related problem concerns the search of Kahler-Einstein metrics,i.e. of Kahler structures (g, J, ω) such that g is an Einstein metric, on a fixed(connected) compact complex manifold (M,J) (we implicitely assume that


(M,J) is kahlerian, i.e. admits Kahler metrics). In terms of the Ricci formρ and the Kahler class ω, the Kahler-Einstein condition is then

KEKE (1.21.1) ρ = k ω,

where k is some real number, equal to s2m where s is the (constant) scalar

curvature. By considering the corresponding de Rham classes, this implies

KEdRKEdR (1.21.2) c1(M,J) =k

2πΩ,

which we already met in Proposition 1.19.2. This provides an obvious nec-essary condition for the existence of a Kahler-Einstein metric on (M,J),namely that the Chern class c1(M,J) be equal to zero, or positive — mean-ing that it is a Kahler class for some Kahler metric on (M,J), or negative— meaning that −c1(M,J) is positive.

Moreover, these conditions are pairwise incompatible: If c1(M,J) = 0,c1(M,J) cannot contain a Kahler form ω, as [ω]m[M ] > 0 for any Kahlerclass on (M,J), nor the opposite of a Kahler form for the same reason.Suppose now that c1(M,J) contains a Kahler form, say ω, and the oppositeof a Kahler form, say −ω1; then, [ω] = [−ω1] implies [ω]m[M ] = −([ω1] ∪[ω]m−1)[M ]; but this cannot be, as [ω]m[M ] and [ω1]∪ [ω]m−1)[M ] are bothpositive (for the latter, observe that ω1 can be diagonalized with respect toω, with positive eigenvalues).

It follows that a (kahlerian) compact complex manifold (M,J) withc1(M,J) equal to zero, resp. positive, resp. negative, can only admit Kahler-Einstein metrics with k = 0 — hence Ricci-flat — resp. k > 0, resp. k < 0,and admits no Kahler-Einstein if c1(M,J) does not satisfy any of the abovethree conditions.

Remark 1.21.1. A compact complex manifold (M,J) such that c1(M,J)is positive or negative is a projective manifold, cf. Section 8.1. If c1(M,J)is positive, (M,J) is called a Fano manifold.

If c1(M,J) = 0, then Theorem 1.20.1 guarantees the existence of aunique Ricci-flat Kahler metric in any Kahler class Ω of (MJ).

In view of this, we now assume that (1.21.2) is satisfied for some non-zero real number k. The Kahler class Ω is then well defined by (1.21.2) and,for any arbitrary chosen metric g0, of Kahler form ω0, in Ω, the Ricci formρg0 of ω0 is of the form ρg0 = k ω0− 1

2ddcpg0 , where pg0 is the Ricci potential

defined by (1.19.10)-(1.19.11) in Section 1.19. By Proposition 1.17.2 theKahler form of any other element in Ω is of the form ωφ = ω0 + ddcφ, forsome real function φ, well-defined up to an additive real constant. Denoteby gφ the corresponding riemannian metric and by ρgφ the corresponding

Ricci form. By (1.19.7), ρgφ is given by ρgφ = ρg0 − 12dd

c log (ω0+ddcφ)m

ωm0=

kω0 − 12dd

c(pg0 − log (ω0+ddcφ)m

ωm0), whereas, because of (1.21.2), the Ricci

potential, pgφ , of gφ is determined by ρgφ = kωφ− 12dd

cpgφ and∫M pgφvgφ = 0.

We thus get:

pgphipgphi (1.21.3) pgφ = pg0 + 2kφ+ log(ω0 + ddcφ)m

ωm0,


up to an additive real constant. It follows that ωφ is the Kahler form ofa Kahler-Einstein metric if and and only if φ is a solution to the followingMonge-Ampere equation:

calabikcalabik (1.21.4) (ω0 + ddcφ)m = e−pg0−2kφ−c(φ) ωm0 ,

for some appropriate constant c(φ) depending upon a chosen normalizationof φ and also on the chosen normalization (1.19.11) of the Ricci poten-tial pg0 . The most common option consists in substituting pg0 = pg0 +

log 1VΩ

∫M e−pg0vg0 to pg0 — where VΩ = Ωm

m! [M ] denotes the common to-

tal volume∫M vg for all metrics g in MΩ — so as to have the alternative

normalization for the Ricci potential of g0:

(1.21.5)1

VΩ

∫Me−pg0vg0 = 1,

and in substituting the following Monge-Ampere equation

MAcommonMAcommon (1.21.6) (ω0 + ddcφ)m = e−pg0−2kφ ωm0 ,

where φ is normalized a posteriori by the obvious necessary condition:

normcalabinormcalabi (1.21.7)1

VΩ

∫Me−pg0−2kφ vg = 1

(for an alternative option, see Chapter 7).

Remark 1.21.2. By allowing k = 0 in (1.21.6) and by replacing −pg0 byany real function f normalized by (1.20.1), we recover the Monge-Ampereequation (1.20.2) associated to the general Calabi conjecture.

Like for the Monge-Ampere equation (1.20.2), the strategy of resolutionto (1.21.6), whith k 6= 0, relies on the continuity method, which, for k 6= 0,consists in substituting the following family of Monge-Ampere equations,parametrized by t in [0, 1]:

MAktMAkt (1.21.8) (ω0 + ddcφ)m = e−pg0−2ktφ ωm0 ,

where, again, φ is normalized a posteriori by

norm-MAKtnorm-MAKt (1.21.9)1

VΩ

∫Me−pg0−2ktφ vg0 = 1.

As in Section 1.20, denote by S the set of t in [0, 1] such that (1.21.8) has asolution. By Theorem 1.20.3, 0 belongs to S, which is then non-empty. Asin the case when k = 0, it is easy to check that S is open (cf. Proposition7.3.2 in Chapter 7 when k > 0; the case when k < 0 is similar, even easier.see [17]), whereas the hard work starts when trying to prove that S is closed,hence coincides with the whole closed interval [0, 1]. This was done by T.Aubin, cf. also [197] [198], in the case when k is negative. We then have

thmaubin Theorem 1.21.1 (T. Aubin [17], [18]). let (M,J) be a compact com-plex manifold, whose first (real( Chern class c1(M,J) is negative. Then,(M,J) admits a Kahler-Einstein metric — necessarily of negative scalarcurvature — unique up to scaling.


Proof. Again, the proof of the existence part of Theorem 1.21.1 goesbeyond the scope of this book. The uniqueness part amounts to showingthat the following Monge-Ampere equation

MA-uniqueMA-unique (1.21.10) (ω + ddcφ)m = e−2k φ ωm,

has φ ≡ 0 as unique solution for any k < 0. This is done as follows. Firstnotice that (1.21.10) forces φ to satisfy the condition

MA-condMA-cond (1.21.11)

∫Me−2k φ ωm =

∫Mωm.

Since k < 0, it follows from (1.21.11) that φ(xM ) ≥ 0 at each point xMwhere φ attains its maximum and that φ ≡ 0 whenever φ(xM ) = 0. Now,by an argument already used in Remark 1.20.1, we have that

(1.21.12) ∆φ ≤ m (1− e−2k φm ),

where ∆ denotes the riemnmannian laplacian relative to the chosen reference

metric associated to ω. We thus get 0 ≤ ∆φ(xM ) ≤ m (1− e−2k φ(xM )

m ) ≤ 0,hence φ(xM ) = 0, hence φ ≡ 0.

When k > 0, neither the uniqueness, nor the existence of a solutionto the Monge-Ampere equation (1.21.4) can be expected in general. Theuniqueness of a solution may fail because of the possible existence of au-tomorphisms of (M,J) in the identity component which don’t preserve theKahler-Einstein metric. In view of Remark 1.23.2, this cannot happen ifk = 0 or k < 0, but may certainly happen if k > 0, e.g. if (M,J) is a complexprojective space. On the other hand, it is a deep theorem of S. Bando andT. Mabuchi [23] that Kahler-Einstein metrics on a Fano manifold (M,J), ifany, all belong to a single orbit of Hred in eachMΩ, Ω = 2π

k c1(M,J), k > 0.According to an old result of Y. Matsushima, the existence fails wheneverH(M,J) is not the complexification of one of its maximal compact sub-groups, cf. Theorem 3.6.2 in Section 3.6 below. More information on thesequestions in the Fano case can be found in Section 6.6 and in Chapter 7below.

1.22. Bochner identities and vanishing theoremsssbochner

For any riemannian manifold (M, g), the Bochner formula for 1-forms iswritten as

bochnerbochner (1.22.1) ∆α = δDα+ r(α]),

for any 1-form α, where ∆ = δd + dδ denotes the riemannian Laplace op-erator (here acting on 1-forms), D = Dg the Levi-Civita connection of gand r stands for the Ricci tensor of g — cf. Section 1.18 — viewed as anhomomorphism from TM to T ∗M . There is a similar Bochner formula for(scalar) p-forms and, more generally, for E-valued p-forms, for any vectorbundle E equipped with a fiberwise positive definite inner product and acompatible linear connection ∇ (if E is a complex fiber bundle, we implic-itly assume that the inner product is hermitian and that ∇ is a C-linearconnection). In the latter case, D∇ will denote the linear connection in-duced by ∇ and the Levi-Civita connection on the vector bundle E ⊗ Λ•M

1.22. BOCHNER IDENTITIES AND VANISHING THEOREMS 55

of E-valued exterior forms. As in Sections 1.6 and 1.10, we denote by d∇,δ∇ and ∆∇ = δ∇d∇ + d∇δ∇ the exterior derivative, the codifferential andthe Laplace operator relative to ∇ and the metric g. The general Bochnerformula also involves the so-called rough Laplace operator (D∇)∗D∇, whichcan also be written as δ∇D∇ — cf. Remark 1.10.1. We then have

propbochner Proposition 1.22.1 (General Bochner formula). With the above hypothe-ses, the Laplace operators ∆∇ and δ∇D∇ are related by

bochnerpnablabochnerpnabla (1.22.2) ∆∇Φ = δ∇D∇Φ +R∇p (Φ),

for any E-valued p-form Φ, where R∇p Φ — a universal linear expression in

Φ and the curvature, RD∇

, of D∇ — has the following expression

(R∇p (Φ))X1,...,Xp =

p∑k=1

n∑i=1

(RD∇

Xk,eiΦ)X1,...,ei,...,Xp

=

p∑k=1

n∑i=1

R∇Xk,ei(ΦX1,...,ei,...,Xp) +

p∑k=1

ΦX1,...,r(Xk),...,Xp

−∑

1≤r<s≤p(−1)r+s

n∑i=1

ΦRXr,Xsei,ei,X1,...,Xr,...,Xs...,Xp

mathcalRpmathcalRp (1.22.3)

for any vector fields X1, . . . , Xp and any auxiliary orthonormal local framee1, . . . , en of TM ; in the rhs of (1.22.2) it is understood that the second eisits in place of Xk.

Proof. It is convenient here to introduce the second covariant deriv-ative D∇,2Φ := D∇(D∇Φ), acting on E-valued exterior forms Φ, which isthe covariant derivative of D∇Φ, viewed as a section of T ∗M ⊗

(E ⊗Λ•M

)relative to the connection induced by ∇ and D = Dg. For any decomposedE-valued exterior forms Φ = s⊗ ψ and for any vector fields X,Y on M , wehave: (

D∇,2(s⊗ ψ))X,Y

= ∇2X,Y s⊗ ψ + +s⊗D2

X,Y ψ

+∇Xs⊗DY ψ +∇Y s⊗DXψ.(1.22.4)

It follows that:

D2RD2R (1.22.5) RD∇

X,Y Φ = −D∇,2X,Y Φ +D∇,2Y,XΦ,

whereas

roughrough (1.22.6) δ∇D∇Φ = −n∑i=1

D∇,2ei,eiΦ.

From (1.6.4)-(1.10.14), we then get

(δ∇d∇Φ)X1,...,Xp = −n∑i=1

(D∇eid∇Φ)ei,X1,...,Xp

= −n∑i=1

(D∇,2ei,eiΦ)X1,...,Xp −p∑

k=1

(−1)k−1n∑i=1

(D∇,2ei,XkΦ)ei,X1,...,Xk,...,Xp

,

deltadEdeltadE (1.22.7)


and

(d∇δ∇Φ)X1,...,Xp =

p∑k=1

(−1)k−1(D∇Xkδ∇Φ)X1,...,Xk,...,Xp

=

p∑k=1

(−1)k−1n∑i=1

(D∇,2Xk,eiΦ)ei,X1,...,Xk,...,Xp

ddeltaEddeltaE (1.22.8)

By adding (1.22.7) and (1.22.8), and by using (1.22.5) and (1.22.6), wereadily get the first expression in the rhs of (1.22.2). The second expressionin the rhs of (1.22.2) easily follows by using the algebraic Bianchi identity(1.18.2).

In the Kahler setting, we infer the following general Bochner-like for-mula: Akizuki-Nakano identity (1.16.3), we get

prop-bochner-hol Proposition 1.22.2. Let (E, h) be a hermitian holomorphic vector bun-

dle over a Kahler manifold (M, g, J, ω). Let E

be the corresponding Dol-beault Laplace operator, as defined in Section 1.16. Then,

(1.22.9) 2E

Φ = δ∇D∇Φ + i [Λ, R∇] Φ +R∇p (Φ),

for any E-valued p-form Φ.

Proof. Direct consequence of Proposition 1.22.1 and of the Akizuki-Nakano identity (1.16.3).

rem-scal Remark 1.22.1. In the case when Φ is a scalar p-form — then denotedψ — ∆∇ is the usual riemannian Laplace operator, D∇ = D is the Levi-

Civita connection and RD∇

is then simply the riemannian curvature actingon p-forms, R∇p (Φ) will be simply denoted Rp(ψ) and the above expressionreduces to

Rp(ψ)X1,...,Xp = +

p∑k=1

ΦX1,...,r(Xk),...,Xp

−∑

1≤r<s≤p(−1)r+s

n∑i=1

ΦRXr,Xsei,ei,X1,...,Xr,...,Xs...,Xp.

bochnerpbisbochnerpbis (1.22.10)

In particular, (1.22.2) reduces to (1.22.1) for scalar 1-forms. Notice that(1.22.10) can in turn be rewritten in the following closer form:

bochnerpterbochnerpter (1.22.11) Rp(ψ) = −∑

1≤i<j≤n(ei ∧ ej) ·Rei,ejψ.

This expression involves the induced (skew symmetric) action — here de-noted by a dot · — of the bundle, A(M), of skew-symmetric endomorphimsof TM on ΛpM , defined by

AactionAaction (1.22.12) (A · ψ)X1,...,Xp = −p∑r=1

ψX1,...,AXr,...,Xp ,

for any section A of A(M): at each point x of M this action is an action ofthe Lie algebra Ax(M) of anti-symmetric endomorphisms of TxM , which is


the Lie algebra of the group O(TxM) of orthogonal automorphisms of TxM ,and (1.22.12) is then the natural induced action on ΛpxM = Λp(T ∗xM). In(1.22.11), A(M) is identified with Λ2(TM) via the riemannian duality and

this action then reads: (X∧Y )·ψ = −X[∧(Y yψ)+Y [∧(Xyψ) for any vectorfields X,Y (it is then easy to deduce (1.22.11) from (1.22.10)); moreover,for any vector fields X,Y , RX,Y is a section of A(M), also written R(X∧Y )— cf. Section 1.18 — and we then have: RX,Y ψ = RX,Y ·ψ = R(X ∧Y ) ·ψ;the rhs of (1.22.11) can then be read as −

∑1≤i<j≤n(ei∧ej) · (R(ei∧ej) ·ψ);

in this expression, the ei ∧ ej ’s may be replaced by any orthonormal frameof Λ2(TM), in particular by an orthonormal frame of eigenforms ωκ of R,

for κ = 1, . . . , N = n(n−1)2 ; if λR1 , . . . , λ

RN denote the corresponding (real)

eigenvalues, we thus get

bochnerpquatuorbochnerpquatuor (1.22.13) R(ψ) = −N∑κ=1

λRκ ωκ · (ωκ · ψ).

The main interest of the Bochner formula (1.22.2) is to provide vanishingtheorems. In the remainder of this section, we assume that M is compact.

propbochnervanish1 Proposition 1.22.3. Let (M, g) be a connected, compact riemannianmanifold of dimension n. For any integer p, 0 ≤ p ≤ n, denote by bp thep-th Betti number of M .

(i) If the Ricci tensor r is everywhere semi-positive, as a symmetricendomorphism of TM , then all g-harmonic 1-forms are Dg-parallel; in par-ticular, b1 ≤ n. If, moreover, r is positive at some point, then b1 = 0.

(ii) If the riemannian curvature R is everywhere semi-positive, as a sym-metric endomorphism of Λ2M , then all g-harmonic p-forms are Dg-parallelfor any p; in particular, bp ≤

(np

)for any p. If, moreover, R is positive at

some point, then bp = 0 for 0 < p < n.

Proof. (i) For scalar 1-forms, the Bochner formula reduces to (1.22.1);when M is compact, this implies 〈∆α, α〉 = 〈δDα, α〉+ 〈r(α), α〉; since δ isa formal adjoint of d and of D — cf. Remark 1.10.1 — we infer 〈dα, dα〉+〈δα, δα〉 = 〈Dα,Dα〉+ 〈r(α), α〉, for any (real) 1-form α. If α is g-harmonic,we then get 0 = 〈Dgα,Dgα〉+ 〈r(α), α〉. If r is everywhere non-negative asan endomorphism of TM — hence of T ∗M by riemannian duality — bothterms are non-negative, hence equal to zero; in particular, Dgα vanishesidentically. Since M is connected, α is then determined by its value at anypoint of M ; it follows that b1 ≤ n. If r is positive at some point x, thenα(x) = 0 and α is then identically zero; we then have b1 = 0.

(ii) For scalar p-forms the general Bochner formula (1.22.2) reduces to

bochnerpscalbochnerpscal (1.22.14) ∆ψ = δDψ +Rp(ψ),

where Rp is given by (1.22.10), (1.22.11) or (1.22.13). From (1.22.13), we

readily infer (Rp(ψ), ψ) =∑N

κ=1 λRκ |ωκ ·ψ|2 (recall that the action (1.22.12)

is anti-symmetric). If R is non-negative — meaning that all the eigenvaluesλRκ are non-negative — Rp is then non-negative for any p (notice that R0

and Rn are identically zero). Moreover, if R is positive at some point x,then Rp is also positive at x for any p, 0 < p < n. Indeed, R is positive at xif and only if the λRκ (x) are all positive; it follows that Rp(ψ)(x) 6= 0, unless


ωκ ·ψ(x) = 0 for all κ = 1, . . . , N . This in turns means that A ·ψ(x) = 0, forany A in the Lie algebra Ax(M) of anti-symmetric endomorphisms of TxM .Equivalently, ψ(x) is a fixed element of the action of SO(TxM), cf. above.It is now an elementary and well-known fact that the (special) orthogonalgroup SO(n) acting on ΛpRn has no non-zero fixed element for 0 < p < n.Because of (1.22.14), the rest of the argument is as in (i): if the curvatureoperator R is everywhere non-negative, then Rp is eveywhere non-negativefor any p; any harmonic p-form ψ is then Dg-parallel. If, moreover, R ispositive at some point x, then Rp is positive at x, unless p = 0 or p = n.Then, ψ(x) = 0 and ψ is then identically zero; we then get bp = 0, for anyp 6= 0, n.

Remark 1.22.2. The Bochner formula (1.22.1) for scalar 1-forms can berefined as follows. For any (real) 1-form α, decompose the covariant Dgα— which is a bilinear form — into its anti-symmetric and symmetric parts:we then get Dα = 1

2dα + 12Lα]g (easy verification). The symmetric part

12Lα]g can furthermore be decomposed into its trace part, equal to − 1

nδα g

and its trace-free part, which we denote by 12(Lα]g)0. We eventually get the

following decomposition:

decDalphadecDalpha (1.22.15) Dα =1

2dα− 1

nδα g +

1

2(Lα]g)0,

where the three components are pairwise orthogonal. The vector fields Xsuch that (LXg)0 = 0 are called conformal Killing vector fields: the spaceof conformal Killing vector fields is formally the Lie algebra of the group ofdiffeomorphisms of M which preserve the conformal structure [g] determined

by g; since LXg = (LXg)0 − δX[

n g, a vector field X is Killing if and onlyif it is a divergence-free, conformal Killing vector field. By using (1.22.15),the Bochner formula (1.22.1) can be rewritten as:

bochnerrefinedbochnerrefined (1.22.16)1

2δdα+

(n− 1)

ndδα− 1

2δ(Lα]g)0 = r(α]),

for any (real) 1-form α. We obtain the following proposition, which is the“negative counterpart” of Proposition 1.22.3 (i):

propricnegative Proposition 1.22.4. Let (M, g) be a connected, compact riemannianmanifold of dimension n. If the Ricci tensor r is everywhere semi-negative,then all conformal Killing vector fields are Dg-parallel; if, moreover, r isnegative at some point, then the space of conformal Killing vector fields isreduced to zero.

Proof. Let X be a conformal Killing vector field and denote ξ = X[;by (1.22.16) we then have

1

2δdξ +

(n− 1)

ndδξ = r(X);

since M is compact, we infer 12〈dξ, dξ〉 + (n−1)

n 〈δξ, δξ〉 = 〈r(X), X〉. If r iseverywhere semi-negative, we infer that dξ and δξ both vanish identically,so that Dgξ ≡ 0; if moreover r is negative at some point, ξ must vanish atthis point, hence everywhere.


In general, for any holomorphic vector bundle E over a (compact) com-plex manifold (M,J) and for any non-negative integers p, q, we denote byHp,q(M,E) the q-th cohomology group of the sheaf of holomorphic germsof sections of the holomorphic vector bundle E ⊗ Λp,0M . By the classicalHodge-Kodaira-Dolbeault theory, Hp,q(M,E) is isomorphic to the the space

of E

-harmonic E-valued forms of type (p, q), for any Kahler metric g onM and any hermitian inner product h on E, cf. e.g. [100, Chapter 1]. Wethen have the following classical vanishing theorems:

propkodaira Proposition 1.22.5 (Kodaira-Nakano-Akizuki vanishing theorem). Let(M,J) be a connected, compact Kahler manifold of real dimension n = 2m.Let L be any ample holomorphic line bundle over M . Then,

(i)

kodaira-nakanokodaira-nakano (1.22.17) Hp,q(M,L) = 0, ∀p, q, p+ q > m.

In particular, if KM denotes the canonical line bundle of M , cf. Section 1.9,we have

KN-bisKN-bis (1.22.18) Hq(M,KM ⊗ L) = 0, ∀q > 0.

(ii) For any holomorphic vector bundle E, there is a positive integer k0

such that

serreserre (1.22.19) Hq(M,E ⊗ Lk) = 0, ∀q > 0, ∀k ≥ k0.

Proof. (i) Since L is ample, we can choose h so that R∇ = i ω, where ωdenotes the Kahler form of g. From (1.16.3), and by using (1.13.4)–(1.13.5),we then infer

(1.22.20) L

Φ = ∇Φ− [Λ, L]Φ = ∇Φ + (p+ q −m) Φ,

for any L-valued form Φ of type (p, q). By using the hermitian inner producton L-forms induced by g and h, we infer

(1.22.21)

∫M

(L

Φ,Φ) vg =

∫M

(∇Φ,Φ) vg + (p+ q −m)

∫M|Φ|2 vg,

where∫M (

LΦ,Φ) vg and

∫M (∇Φ,Φ) vg are both non-negative. If Φ is

L

-harmonic and (p+ q > m, we thus get Φ = 0, hence (1.22.17); (1.22.18)readily follows, since Hq(M,K(M)⊗ L) = Hm,q(M,L).

(ii) It is convenient to write E ⊗ Lk as KM ⊗ (E ⊗ Lk), by setting

E = E ⊗ K−1(M), so that Hq(M,E ⊗ Lk) = Hm,q(M, E ⊗ Lk). If L

is equipped with the Chern connection as above and E with any Chern

connection, the resulting Chern curvature, RE⊗Lk

say, on E ⊗ Lk is of the

form RE⊗Lk

= i k ω ⊗ IdE⊗Lk + RE ⊗ IdLk , where ω denotes the Kahlerform of the chosen Kahler metric on M , so that the hermitian operator

i [Λ, RE⊗Lk], acting on the vector bundle (E ⊗ Lk) ⊗ Λ0,qM , is of the form

−k [Λ, L]⊗ IdE⊗Lk + i[Λ, RE ]⊗ IdLk = k q IdE⊗Lk + i[Λ, RE ]⊗ IdLk . If q > 0,there clearly exists a positive integer k0 such that the hermitian operator

k0 q Id+ i[Λ, RE ] be positive definite at each point of M — as M is compact

— and, we then infer from (1.16.3) as before that Hm,q(M, E ⊗ Lk) =Hq(M,E ⊗ Lk) = 0 for any integer k ≥ k0.


propbochnerholo Proposition 1.22.6. Let (M, g, J, ω) a connected, compact almost her-mitian manifold and let (E, ∂E , h) be a hermitian (almost) holomorphic vec-tor bundle over M . Denote by ∇ the corresponding Chern connection — cf.Proposition 1.6.1 — and by R∇ the curvature of ∇. Define K∇ := −2iΛR∇:K∇ is then a hermitian endomorphism, called the Chern endomorphism ofE with respect to ω. Call holomorphic any section of E which belongs tothe kernel of ∂E. We then have:

bochnerholobochnerholo (1.22.22) −Λddc|s|2 = (K∇(s), s)− 2|∇s|2,

for any holomorphic section s of E. In particular, if K∇ is everywhere semi-negative, then all holomorphic sections of E are ∇-parallel. If, moreover,K∇ is negative at some point of M , then E admits no non-zero holomorphicsection.

Proof. Let s be any holomorphic section of E; denote by (·, ·) the her-mitian inner product h and by |s|2 = (s, s) the square norm of s with respectto h. We then have dc|s|2(X) = −(∇JXs, s) − (s,∇JXs) = −i(∇Xs, s) +i(s,∇Xs) for any (real) vector field X (notice that ∇JXs = i∇Xs, as s isholomorphic and ∂E = ∇0,1). We readily infer ddc|s|2(X,Y ) = 2i(R∇X,Y s, s)+

2i(∇Xs,∇Y s)− 2i(∇Y s,∇Xs), for any (real) fields X,Y . Now −Λddc|s|2 =−1

2

∑ni=1 dd

c|s|2(ei, Jei), for any auxiliary orthonormal frame e1, . . . , en,and we thus get (1.22.22). If K∇ is semi-negative, the rhs of (1.22.22) isnon-positive and we then have −Λddc|s|2 ≤ 0 everywhere. Since M is com-pact, this implies that |s|2 is constant — cf. [103] — so that both sides of(1.22.22) vanish identically. This implies ∇s ≡ 0 and (K∇s, s) ≡ 0; more-over, if K∇ is negative at some point of M , then s vanishes at this point,hence is identically zero, as it is ∇-parallel.

1.23. The Lichnerowicz fourth order operatorssD-

On any almost complex manifold (M,J), a (real) vector field is said tobe (real) holomorphic if it is a holomorphic section of (TM, J), i.e. if

LJLJ (1.23.1) LXJ = 0,

where, we recall, LX denotes the Lie derivative along X. Since L[X,Y ] =[LX ,LY ], the space of (real) holomorphic vector fields is closed for thebracket, hence is a Lie subalgebra of the space of all (real) vector fields.It will be denoted by h(M,J) or simply h when (M,J) is understood.

This is only a real Lie algebra in general, but when J is integrable theaction of J turns it into a complex Lie algebra. Indeed, the integrabilityof J ensures that JX belongs to h as soon as X does and we then have[JX, Y ]− J [X,Y ] = −(LY J)(X) = 0 for all pairs of elements X,Y of h, cf.Lemma 1.1.1.

Remark 1.23.1. As already observed in Section 1.8, a (real) vector fieldX on a complex manifold (M,J) is (real) holomorphic if and only if its(1, 0)-part X1,0 = 1

2(X− iJX) is holomorphic in the usual sense as a section

of the (holomorphic) vector bundle T 1,0M .

1.23. THE LICHNEROWICZ FOURTH ORDER OPERATOR 61

If, moreover, g is a Kahler riemannian metric on (M,J), of Levi-Civitaconnection D, (real) holomorphic vector fields can be alternatively chara-terized as follows

lemmaDJ Lemma 1.23.1. A (real) vector field X is holomorphic if and only if thefollowing condition is satisfied:

holhol (1.23.2) DJYX = JDYX,

for any vector field Y .

Proof. Lemma 1.23.1 is a direct consequence of Proposition 1.8.1. Analternative derivation is as follows. For any field of endomorphisms A andany torsion-free connection D the following general identity holds:

LieDALieDA (1.23.3) LXA = DXA− [DX,A],

for any vector field X (here, DX is viewed as the endomorphism: Y →DYX. The fact that D is torsion-free can then be written as LXZ = DXZ−DX(Z), for any pair X,Z of vector fields; this readily implies (1.23.3).

For a general almost hermitian structure (g, J, ω) and for any vector fieldX, we then have

LieJLieJ (1.23.4) LXJ = DXJ − [DX,J ].

In the Kahler case, DJ = 0 so that X is a (real) holomorphic vector field ifand only if

(1.23.5) [DX,J ] = 0;

this is the same as (1.23.2).

For any real 1-form α we denote by D+α, resp. D−α, the J-invariantpart, resp. the J-anti-invariant part, of the covariant derivative Dα (as abilinear form); we thus have:

D+D-D+D- (1.23.6) (D±α)X,Y =1

2((DXα)Y ± (DJXα)JY ).

In the remainder of this section, the structure is assumed to be Kahler. Interms of Lie derivative, D− can then be expressed as follows:

lemmaD- Lemma 1.23.2. For any real 1-form α, we have:

D-D- (1.23.7) D−α = −1

2g(J(Lα]J)·, ·) = −1

2ω((Lα]J)·, ·).

In particular, α is the dual 1-form of a holomorphic (real) vector field if andonly if D−α = 0.

Proof. Simple reformulation of (1.8.3)-(1.8.4).

lemmar Lemma 1.23.3. For any real 1-form α, we have

rr (1.23.8) δD+α− δD−α = r(α]).


Proof. For any vector fields X,Y , we have that ((D+ −D−)α)X,Y =(Dα)JX,JY , and then:

(δ(D+ −D−)α)(X) = −n∑j=1

(D2ej ,Jejα)(JX)

= −1

2

n∑j=1

(D2ej ,Jejα−D

2Jej ,ejα)(JX)

=1

2

n∑j=1

(Rej ,Jejα)(JX)

= ρ(α], JX) = r(α], X).

lemmadeltaD- Lemma 1.23.4. For any real 1-form α, we have:

deltaD+deltaD+ (1.23.9) δD+α =1

2∆α,

and

deltaD-deltaD- (1.23.10) δD−α =1

2∆α− r(α]) =

1

2∆α+ Jα]yρ.

If M is compact, α is harmonic if and only if D+α = 0, and α is the dualof a (real) holomorphic vector field, X = α], if and only if it satisfies

(1.23.11)1

2∆α− r(α]) =

1

2∆α+ Jα]yρ = 0.

Proof. In terms of the operators D±, the Bochner formula (1.22.1)reads:

bochner-pmbochner-pm (1.23.12) δD+α+ δD−α = ∆α− r(α]).

Then, (1.23.9)–(1.23.10) readily follow from (1.23.12)–(1.23.8). If M iscompact, we have that 〈δD±α, α〉 = 〈D±α,Dα〉 = 〈D±α,D±α〉, so thatδD±α = 0 if and only if D±α = 0. The second assertion then readilyfollows from (1.23.9)–(1.23.10) and from Lemma 1.23.2.

remlichne Remark 1.23.2. The characterisation of holomorphic vector fields oncompact Kahler manifolds given in Lemma 1.23.4 is due to A. Lichnerowicz,cf. [135]. It readily implies that a compact Kahler manifold whose Riccitensor r is negative definite admits no non-trivial holomorphic vector field,and that if r is semi-negative any holomorphic vector field is parallel withrespect to the Levi-Civita connection.

lemmadeltadelta Lemma 1.23.5. For any real 1-form α, we have:

deltadeltaalphadeltadeltaalpha (1.23.13) δδD−α =1

2∆δα− (dcα, ρ) +

1

2(α, ds).

In particular, for any function f , we have

deltadeltaddeltadeltad (1.23.14) L(f) := δδD−df =1

2∆2f + (ddcf, ρ) +

1

2(df, ds),

1.23. THE LICHNEROWICZ FOURTH ORDER OPERATOR 63

and

deltadeltadcdeltadeltadc (1.23.15) δδD−dcf = −1

2LKf,

where LK denotes the Lie derivative along the vector field K := Jgradgs.

Proof. (1.23.13) follows from (1.23.10) and the adjunction formula —cf. (1.10.12) —

identityidentity (1.23.16) δ(XyΨ) = (dX[,Ψ)− (X[, δΨ),

for any vector field X and any 2-form Ψ; (1.23.14) and (1.23.15) followeasily by specializing ξ = df and ξ = dcf (notice that δdcf = 0 by Kahleridentities, or by observing that dcf = −δ(fω)).

remoperator Remark 1.23.3. The operator f 7→ L = δδD−df is a self-adjoint, semi-positive differential operator of order 4, acting on (real) functions. It canalso be written as

(1.23.17) L(f) =1

2∆2f − δ(r(df)).

In this form, it has been first introduced by A. Lichnerowicz, see [135], andwe shall then refer to it as the Lichnerowicz fourth order operator or, simply,the Lichnerowicz operator. It will play a prominent role in many parts ofthese notes. Notice that L can be also written as

mathbbLmathbbL (1.23.18) L(f) = (D−d)∗D−df,

where (D−d)∗ denotes the formal adjoint of D−d. In particular, when M iscompact, the kernel of L is the space of (real) functions such that D−df =0, which, by Lemma 1.23.2, is the space of (real) functions f such thatthe gradient gradgf is (real) holomorphic, also called Killing potentials, cf.Section 2.6.

CHAPTER 2

Holomorphic vector fields on a compact Kahlermanifold

scompact

In this chapter, we present some well-known properties of the Lie algebrah of (real) holomorphic vector fields on a complex manifold (M,J), as definedin Section 1.23, in the case when (M,J) is a compact, kahlerian manifold.

Most of these properties rely on the simple Lemma 2.1 in Section 2.1.

2.1. The space of holomorphic vector fieldsssspacehol

lemmaspacehol Lemma 2.1.1. For any compact Kahler manifold (M, g, J, ω), any (real)holomorphic vector field can be uniquely written as

decXdecX (2.1.1) X = XH + grad f + Jgradh,

where XH denotes the riemannian dual of a harmonic 1-form and where fand h are real functions, uniquely defined up to additive constants.

Proof. Let ξ = X[ be the dual 1-form of X with respect to the rie-mannian metric g. From Lemma 1.23.2, we know that X is a real holomor-phic vector field if and only if ξ satisfies D−ξ = 0. It follows that dξ isJ-invariant, so that ddcξ = 0. By Lemma 1.17.1, the Hodge decompositionof ξ then reads:

hodgehodge (2.1.2) ξ = ξH + df + dch,

where ξH is harmonic and f, h are real functions, uniquely defined up toadditive constants.

The real functions f and h appearing in (2.1.2), as well as the resultingcomplex function F := f + ih, will be normalized by

norm-potentialsnorm-potentials (2.1.3)

∫Mfvg =

∫Mhvg =

∫MFvg = 0,

and thus uniquely defined.

defipotential Definition 2. For any (real) holomorphic vector field X in h, the realfunction f appearing in (2.1.1) and normalized by (2.1.3) is called the realpotential of X with respect to g. It will be denoted fXg , or simply fX ifthe metric g is understood. The complex function F = f + i h, normalizedby (2.1.3), is called the complex potential of X with respect to g ad will besimilarly denoted FXg or FX . A complex function F defined on (M,J) iscalled a holomorphic potential with respect to some Kahler metric g if, afternormalization, it is the complex potential with respect to g of some (real)holomorphic vector field.

65

66 2. HOLOMORPHIC VECTOR FIELDS

Notice that for any X in h, the real and complex potentials of X andJX are related by

(2.1.4) fJX = −hX , F JX = i FX .

This is because the space of g-harmonic 1-forms is J-invariant on any com-pact Kahler manifold, cf. Section 2.3 below. By (2.1.2), the Hodge decom-position of Jξ, is then Jξ = JξH − dh+ dcf , with JξH = (Jξ)H .

remassignment Remark 2.1.1. For any fixed (real) holomorphic vector field X and forany chosen Kahler class Ω, the assignment: g → fXg can be regarded as avector field on the space MΩ of Kahler structures on (M,J), with [ω] = Ω,cf. Section 3.1 below.

lemmarealpotential Lemma 2.1.2. For any (real) holomorphic vector field X on a compactKahler manifold (M, g, J, ω), the real potential fXg of X with respect to g isdetermined by

realpotentialrealpotential (2.1.5) LXω = ddcfXg

and the normalization (2.1.3). In particular, fXg = 0 if and only if X isKilling with respect to g, i.e. LXg = 0. Moreover:

bracket-potentialbracket-potential (2.1.6) f [X,Y ]g = LXfYg − LY fXg

for any two (real) holomorphic vector fields X, Y .

Proof. By using the Cartan formula, cf. Section 1.4, we get LXω =d(Xyω) = dJξ = ddcfXg . Moreover, since M is compact, for any functionf , ddcf = 0 if and only if f is constant, hence identically 0 when f isnormalized by (2.1.3); it follows that fXg is entirely determined by (2.1.5).

In particuler, fXg = 0 if and and only if LXω = 0; this, in turn, occurs ifand only if LXg = 0 since LXJ = 0. The identity (2.1.6) follows from

L[X,Y ]ω = LX(LY ω)− LY (LXω)

= LXddcfYg − LY ddcfXg= ddc

(LXfYg − LY fXg

)(2.1.7)

— notice that LX and LY both commute with ddc, as X,Y are (real) holo-morphic — and from∫

M(LXfYg − LY fXg ) vg = 〈dfYg , X[

H + dfXg + dchX〉

− 〈dfXg , Y [H + dfYg + dchY 〉

= 〈dfXg , dfYg 〉 − 〈dfYg , dfXg 〉 = 0.

(2.1.8)

remcomplexpotential Remark 2.1.2. We check in a similar way that, for any X in h,

(2.1.9) LX1,0ω =1

2ddcFXg ,

where X1,0 = 12(X − iJX) denotes the part of type (1, 0) of X (see Section

1.8) and that

(2.1.10) F [X,Y ]g = LX1,0F Yg − LY 1,0FXg .

2.1. THE SPACE OF HOLOMORPHIC VECTOR FIELDS 67

On any complex manifold (M,J), the (real) Lie algebra h of (real) holo-morphic vector fields can be viewed as a complex Lie algebra via the action ofJ . Via the identification X 7→ X1,0 = 1

2(X − iJX) of (TM, J) with T 1,0M ,h, as a complex Lie algebra, is then identified with the space of holomor-phic sections of T 1,0M with respect to the natural holomorphic structure ofT 1,0M , cf. Section 1.8. If M is compact, we then have:

propliegroup Proposition 2.1.1. If (M,J) is compact, h, as a complex Lie algebra,is finite-dimensional and is the Lie algebra of the group, Aut(M,J), of auto-morphisms of (M,J), which is then a finite-dimensional complex Lie group.

Proof. As just recalled, h is naturally identified with the space of holo-morphic sections of T 1,0M , equipped with its natural holomorphic struc-ture ∂ defined in Section 1.8. For any auxiliary J-compatible riemann-ian metric g on M1, the operator ∂ + ∂∗, where ∂∗ denotes the adjointof ∂ with respect to g, as well as the corresponding Dolbeault laplacien = (∂ + ∂∗)2 = ∂∂∗ + ∂∗∂, are elliptic operators, of order 1 and 2 respec-tively, and h can equivalently be regarded as the kernel of either of them.It is then a well-known general fact that the kernel of an elliptic differentialoperator acting on the sections of (real or complex) vector bundle over acompact manifold is finite dimensional. We then deduce the second part ofthe proposition from the general Theorem 3.1 of Chapter I in S. Kobayashi’sbook [120], by observing that h clearly coincide with the space of (real) vec-tor fields which generate global 1-parameter groups in Aut(M,J), as M iscompact.

remelliptic Remark 2.1.3. An alternative argument for Proposition 2.1.1, also tobe found in Kobayashi’s book [120], relies on the fact that any almost-complex structure on a manifold M of dimension n = 2m can be regardedas a Gl(m,C)-structure on M , where the complex linear group Gl(m,C) isviewed as as a subgroup of Gl(2m,R), and on the observation that the Liealgebra gl(m,C) of Gl(m,C) is elliptic, as a subspace of gl(2m,R), meaningthat it contains no matrix of rank 1. According to the general Theorem4.1 in [120, Chapter I]), the group of automorphism of any compact almost-complex manifold (M,J) is then a (finite-dimensional, real) Lie group, whoseLie algebra is clearly the (real) Lie algebra of (real) vector fields X on Msuch that LXJ = 0, which is then finite-dimensional. If J is integrable, thisis a complex Lie algebra, and Aut(M,J) is then a complex Lie group, cf.Theorem 1.1 in [120, Chapter III].

remcompactness Remark 2.1.4. In Proposition 2.1.1, the compactness assumption iscrucial. If (M,J) is not compact, it may happen that Aut(M,J) be in-finite dimensional, e.g. Aut(C2) contains all transformations of the form(z1, z2) 7→ (z1, z2 + F (z2)), for any entire holomorphic function F on C. Itmay also happen that Aut(M,J) be a (finite-dimensional, real) Lie group,but not a complex Lie group, e.g. Aut(D) = PSl(2,R) = Sl(2,R)/ ± 1, ifD denotes the open unit disk in C. In this exemple, h is still a complex Liealgebra but does not coincide with the Lie algebra of Aut(D), due to the fact

1 Such a metric always exists, e.g. for any riemannian metric g on M , g(X,Y ) :=fnhermitian12(g(X,Y ) + g(JX, JY )) is J-compatible.


that not all (real) vector fields in h are complete (compare the generator ofthe rotation group centered at the origin, which is evidently complete, andits image by J , which is evidently not complete). More generally. the groupof automorphisms any bounded domain in Cm+1 is a real, but not a complex,Lie group, cf. e. g. [155].

notation-aut Notation 1. In the sequel the full group of automorphisms of a com-pact complex manifold (M,J) will be denoted by Aut(M,J), or simplyAut(M) is J is understood, whereas the identity component of Aut(M,J),simply called the connected group of automorphisms, will be denoted byH(M,J) — or H(M).

2.2. The space of Killing vector fieldssskilling

On a general riemannian manifold (M, g), a (real) vector field X is calleda Killing vector field if it preserves g, i.e. if

(2.2.1) LXg = 0.

Equivalently, X is Killing if and only if DX is g-skew-symmetric.For any riemannian manifold (M, g), the space of Killing vector field is

a (real) Lie algebra which we shall denote by k(M, g) or, simply, by k. If(M, g) is complete, any Killing vector field is complete, cf. e.g. [121], and kis therefore the Lie algebra of the Lie group Isom(M, g) of automorphismsof (M, g). If moreover, M is compact, Isom(M, g) is also compact.

notation-killing Notation 2. The identity component of Isom(M, g), simply called theconnected isometry group of (M, g), will be denoted by K(M, g), or K(M)when the metric g is understood.

We then have the following general lemma:

lemmakilling Lemma 2.2.1. Let M be any compact riemannian manifold. Let γ beany element of K(M, g) and ψ any harmonic p-form. Then:

(2.2.2) γ∗ψ = ψ.

In particular, for any X in k, we have that

(2.2.3) LXψ = 0.

Proof. Since γ belongs to Isom0(M, g) it induces the identity on allde Rham spaces Hp(M,R). It follows that γ∗ψ belongs to the same deRham class as ψ. On the other hand, γ∗ψ is certainly harmonic, as ψ is anisometry. It follows that γ∗ψ = ψ. Let X be any Killing vector field anddenote by ΦX

t the flow of X ; then, ΦXt belongs to K(M, g) for all values

of t so that LXψ := ddt |t=0

ΦXt∗ψ = 0. Alternatively, by Cartan formula,

LXψ = d(Xyψ) + Xydψ = d(Xyψ), as ψ is closed; but, on a compactriemannian manifold, an exact harmonic p-form is identically equal to 0.


propkilling Proposition 2.2.1. On any compact Kahler manifold, the space of Killingvector fields is a Lie subalgebra of the Lie algebra of (real) holomorphic vectorfields:

inclusioninclusion (2.2.4) k ⊂ h.

2.3. HARMONIC 1-FORMS AND PARALLEL VECTOR FIELDS 69

More precisely, k is the subspace of divergence-free elements of h.

Proof. Let X a Killing vector field. By Lemma 2.2.1, X preserves ω(which is D-parallel, hence harmonic). It then preserves J , hence belongsto h. Any Killing vector field is divergence-free. Conversely, according to(2.1.2) we have δgX

[ = ∆gfXg , for any X in h, so that X is divergence-free

if and only if its real potential fXg is identically zero; according to Lemma2.1.2, this occurs if and only if X is Killing with respect to g. Alternatively,X is divergence-free if and only LXvg = LX ωm

m! = 0, whereas, by (2.1.5) and

(1.12.2)–(1.15.6), LXωm = mωm−1 ∧LXω = −∆gfXg ωm, so that LXvg = 0

if and only ∆gfXg = 0, if and only if fXg = 0.

2.3. Harmonic 1-forms and parallel vector fieldsssharmonic

We start this section by recalling some basic facts concerning the space,harm, of (real) harmonic 1-forms on a compact Kahler manifold (recall that,in general, a p-form φ defined on a Riemannian manifold (M, g) is calledharmonic if ∆φ = 0, where ∆ denotes the Laplacian relative to g).

propalpha Proposition 2.3.1. On any compact Kahler manifold (M, g, J, ω), areal 1-form α is harmonic if and only if it satisfies the following two condi-tions:

harmalphaharmalpha (2.3.1) dα = 0, dcα = 0.

Equivalently, α is harmonic if and only if α1,0 := 12(α+iJα) is holomorphic.

Proof. SinceM is compact, a 1-form α is harmonic if and only if dα = 0and δα = 0. Since the metric is Kahler, we have ∆ = ∆c, cf. Proposition1.15.1, so that a harmonic 1-form α is also dc-closed, hence satisfies (2.3.1).Conversely, for any real 1-form α satisfying (2.3.1), we have dα = 0 and, byusing (1.14.1), δα = Λdcα = 0, hence α is harmonic. From

delbaralphadelbaralpha (2.3.2) ∂α1,0 =1

4(d− idc)(α+ i Jα) =

1

4

(dα+ J dα

)+i

4

(dJα+ J dJα

),

we infer that α1,0 is homomorphic whenever (2.3.1) is satisfied. Conversely,if ∂α1,0 = 0, α1,0 is -harmonic, as it is evidently ∂∗-closed for type reason;by Proposition 1.15.1, it is then harmonic, as well as its real part 1

2α and

its imaginary part 12Jα.

As a direct corollary of Proposition 2.3.1, we get

propgammaalpha Proposition 2.3.2. Let (M,J) be any compact complex manifold, andg any riemannian metric on M which is Kahler with respect of J . Then,harm is independent of g and is closed under the action of J . Moreover, forany element γ of H(M,J) and any element α of harm, we have that

gammaalphagammaalpha (2.3.3) γ∗α = α.

Proof. The first part of the proposition readily follows from Propo-sition 2.3.1 since (2.3.1) as well as (2.3.2) are clearly independent of theKahler metric g. The last statement is quite reminiscent of Lemma 2.2.1and is proved similarly. Any element γ of H(M,J) is homotopic to theidentity, hence induces the identity on H1

dR(M,R). It follows that γ∗α still


belongs to the same de Rham class as α. On the other hand, γ∗α still be-longs to harm, as this space only depends on J , and is then harmonic; wethen have γ∗α = α.

rem-Dharm Remark 2.3.1. It readily follows from 1.23.9 in Lemma 1.23.10 that,on any compact Kahler manifold (M, g, J, ω), a (real) 1-form α is harmonicwith respect to g if and only if

DharmDharm (2.3.4) D+α = 0,

i.e. if and only if Dα is J-anti-invariant. In view of Proposition 2.3.2, thiscondition is actually independent of the chosen Kahler metric g on (M,J).

prop-Dharmonic Proposition 2.3.3. An element X of h is D-parallel if and only if itsriemannian dual 1-form ξ = X[ is harmonic.

Proof. By remark 2.3.1, if ξ is harmonic, Dξ is J-anti-invariant, whereas,by Lemma 1.23.2, Dξ is J-invariant, as X belongs to h; it follows that ξ,hence also X, are parallel. Conversely, by (1.23.2), any parallel vector fieldX belongs to h, and its dual 1-form ξ is parallel, hence harmonic.

The space of (real) parallel vector fields is denoted by a. We just sawthat a is a subspace of h. More precisely, we have

propparallel Proposition 2.3.4. The space a of (real) parallel vector fields is a (com-plex) abelian Lie subalgebra of h and is contained in the centre of h.

Proof. a is clearly abelian, as [X,Y ] = DXY −DYX = 0, if X,Y bothbelong to a, and is clearly contained in h, because of lemma 1.23.1. We nowshow that [Z,X] = 0 for any Z in a and any X in h. If Z is parallel, so isJZ and both vector fields Z, JZ generate a foliation, say F , whose each leafis a 2-dimensional, totally geodesic, flat, complete Kahler submanifold ofM (the latter comes from the fact that all geodesics of the leaf are integralcurves of constant linear combinations of Z and JZ). It follows that theuniversal covering of each leaf of F is isometric to C. Actually, the deRham decomposition of the universal cover M of M is a Kahler productC×N , for some simply-connected Kahler manifold N , and, for each y in N ,C × y is the universal cover of the corresponding leaf. Now, X is of theform X = XH + grad f + Jgradh, where XH is the dual of the harmonic1-form ξH . We know from Lemma 2.2.1 that [Z,XH ] = (LZξH)] = 0,as Z is Killing. On the other hand, since Z preserves g and J , we thenhave [Z,X] = gradLZf + JgradLZh. By substituting JZ to Z, we get[JZ,X] = J [Z,X] = gradLJZf + JgradLJZh = −gradLZh + JgradLZf .We infer LZf − LJZh = 0 and LJZf + LZh = 0 (recall that all thesefunctions are of zero integral, as Z and JZ are divergence-free). This meansthat the complex potential F := f + ih is holomorphic on each leaf of F ,hence determines a holomorphic function on its universal cover. On theother hand, F is a globally defined smooth function on M , hence bounded.It follows that F is constant on each leaf, as well as f and h. From [Z,X] =gradLZf + JgradLZh we then infer [X,Z] = 0.

2.4. THE IDEAL hred AND THE REDUCED AUTOMORPHISM GROUP 71

2.4. The ideal hred and the reduced automorphism groupssideal

For any (real) holomorphic vector field X and for any harmonic 1-formα, the scalar α(X) is the real part of α1,0(X1,0), which is a holomorphicfunction, hence constant as M is compact.

Equivalently, for any X in h and any α in harm we have that

fixfix (2.4.1) LXα = 0,

which readily follows from Proposition 2.3.2 (we can also argue as in Section2.2: α is harmonic if and only if it is the real part of a holomorphic (1, 0)-form; for any X in h, LXα = d(α(X)) is then harmonic and exact, henceidentically zero).

We thus get a R-linear map

(2.4.2) τ : h→ harm∗,

from h to the (real) dual of harm: to each X in h we associate the linearform α 7→ α(X).

Moreover, for any two elements X,Y of h we have that τ([X,Y ]) = 0.Indeed, for each harmonic 1-form α and for any two elements X,Y of h, wehave

α([X,Y ]) = −dα(X,Y ) +X · α(Y )− Y · α(X)

= X · α(Y )− Y · α(X) = 0.

This means that the map τ is a Lie algebra morphism from h to harm∗,when the latter is equipped with the trivial (abelian) Lie algebra structure.

The kernel of τ will be denoted by hred: hred is then an ideal of h andactually satisfies the stronger condition

(2.4.3) [h, h] ⊂ hred.

An element X of h belongs to hred if and only if its dual ξ is of the formξ = df + dch, i.e. if and only if ξH = 0: this readily follows from

α(X) =1

V〈α, ξ〉 =

1

V〈α, ξH〉,

for any harmonic 1-form α (here, and henceforth, V =∫M vg denotes the

total volume of the metric).The inclusion [h, h] ⊂ hred can be made more precise via the following

propbracket Proposition 2.4.1. For any two vector fields X = XH + gradfX +JgradhX and Y = YH + gradfY + JgradhY in h, the bracket [X,Y ] belongs

to hred, hence is of the form [X,Y ] = gradf [X,Y ] + Jgradh[X,Y ], with

f [X,Y ] = dfY (X)− dfX(Y ),

h[X,Y ] = ω(X,Y )− 1

V〈JXH , YH〉+ dfX(JY )− dfY (JX).

bracketbracket (2.4.4)

In particular, if X = gradfX + JgradhX and Y = gradfY + JgradhY

both belong to hred, then

f [X,Y ] = fX , hY + hX , fY ,

h[X,Y ] = hX , hY − fX , fY ,bracket-h0bracket-h0 (2.4.5)


where, we recall, ·, · denotes the Poisson bracket with respect to the Kahlerform ω.

Proof. By using the computation already done in the proof of Proposi-tion 1.5.1, as well as the identity (2.1.5), which determines the real potentialof any element of h, we get

ι[X,Y ]ω = −d(ω(X,Y )) + ιXddcfY − ιY ddcfX

= −d(ω(X,Y ))

+ LXdcfY − LY dcfX

− d(ιXdcfY ) + d(ιY d

cfX)

= −d(ω(X,Y ))

+ dc(LXfY )− dc(LY fX)

− d(ιXdcfY ) + d(ιY d

cfX)

= −dh[X,Y ] + dcf [X,Y ]

(note that LX and LY commute to dc, as X,Y belong to h). This gives(2.4.4) (the constant − 1

V 〈JXH , YH〉— where, we recall, XH , YH denote the

riemannian duals of the harmonic parts of X[, Y [ — has been added toguarantee that

∫M h[X,Y ]vg = 0). Then, (2.4.5) follows easily, by using the

expressions of the Poisson bracket given in Section 1.2.

It turns out that the map τ can be integrated to a Lie group homomor-phism τ from H(M) to the Albanese torus Alb(M) := harm∗/Γ, where Γdenotes the natural image of H1(M,Z) in harm∗, via the map α 7→

∫c α, for

any loop c and any harmonic 1-form α. To explain this fact, we first fixa point x0 of M and we consider the Jacobi map, A, from M to Alb(M)defined as follows [32]: to each x we associate A(x) : α →

∫ xx0α, for each

harmonic 1-form α, where the integral runs over some path form x0 to x;any two such paths differ by a loop, so that A(x) actually lives in Alb(M).For any γ in H(M), we have that

A(γ · x)(α) =

∫ γ·x

x0

α =

∫ γ·x

γ·x0

α+

∫ γ·x0

x0

α mod Γ

=

∫ γ·x

γ·x0

γ∗α+

∫ γ·x0

x0

α mod Γ

=

∫ x

x0

α+

∫ γ·x0

x0

α mod Γ

= A(x)(α) +

∫ γ·x0

x0

α mod Γ

(recall that γ∗(α) = α, cf. Proposition 2.3.2). In short, we have

(2.4.6) A(γ · x) = A(x) + τ(γ) mod Γ

with

(2.4.7) τ(γ) = α→∫ γ·x0

x0

α mod Γ

2.4. THE IDEAL hred AND THE REDUCED AUTOMORPHISM GROUP 73

for each α in harm. Then, τ is a homomorphism from H(M,J) to Alb(M).This homomorphism is actually independent of the choice of x0. Indeed, ifx1 is another base-point, we have∫ γ·x1

x1

α =

∫ x0

x1

α+

∫ γ·x0

x0

α+

∫ γ·x1

γ·x0

α mod Γ

=

∫ γ·x0

x0

α+

∫ x0

x1

α+

∫ x1

x0

γ∗α mod Γ

=

∫ γ·x0

x0

α mod Γ,

for any α in harm and any γ in H(M), again because γ∗α = α. Moreover,the derivative of τ at the identity clearly coincides with τ . It follows thatthe ideal hred is the Lie algebra of a (closed) connected Lie subgroup, sayHred(M,J), of H(M,J), namely the identity component of the kernel of τin H(M,J).

defireduced Definition 3. The group Hred(M,J) — or simply Hred(M) if J is un-derstood — will be referred to as the reduced automorphism group of (M,J).

remlinear Remark 2.4.1. It can be shown that the kernel of τ in H(M,J) has afinite number of connected components and that the reduced automorphismgroup Hred(M,J) is a linear algebraic group, [82, Theorem 5.5], [137, The-orem 3.12].

We end this section with the following alternative characterization ofhred:

propzeroset Proposition 2.4.2. For any compact Kahler manifold (M, g, J, ω), theideal hred coincides with the space of (real) holomorphic vector fields whosezero set is non-empty.

Proof. This theorem was first established by Y. Matsushima [149],[120, Theorems 9.7 and 9.8], in the case when M is a projective manifoldpolarized by a holomorphic line bundle, cf. Section 8.1, then, in the generalKahler case, by J. B. Carrell and D. I. Lieberman in [50]. We here followthe nice elementary argument given by C. LeBrun-S. Simanca in [130].

Let X a (real) holomorphic vector field on M , whose zero set is non-empty; for any (real) harmonic 1-form α, α(X) is constant, hence equalto zero; then, X belongs to hred. Conversely, suppose that X = gradgf +

Jgradgh belongs to hred and denote by c ≥ 0 the minimum of |X|2 on M ;we have to prove that c = 0.

The following argument, taken from [130], holds for any real vectorfield X of the form X = gradgf + Jgradgh with the only assumption thatX commutes with JX = −gradgh + Jgradgf (this assumption is certainlysatisfied if X belongs to hred, as we then have LXJ = 0, so that [X, JX] =J [X,X] = 0); the flows, ΦX

s and ΦJXt , of X and JX — which are both

globally defined as M is compact — commute and thus determine a C-action on M : for any point x0 in M — for convenience, we choose x0 suchthat |X(x0)|2 = c — the C-orbit of x0 is the image of the map Φ : C→ Mdefined by Φ(z) = ΦX

s ΦJXt (x0) = ΦJX

t ΦXs (x0), for any z = s + it in

C; we then define H : C → C by H(z) = F (Φ(z)), with F = f + ih;


then (∂H/∂s)(z) = (X · F )(Φ(z)) and (∂H/∂t)(z) = (JX · F )(Φ(z)). Forany r ≥ 0, denote by Dr the (closed) disk of radius r in C around theorigin and by ∂Dr the boundary circle, of radius r. Then, |

∫∂Dr

H dz| isclearly bounded from above by 2πrC, where C denotes the maximal valueof |F | on M . On the other hand,

∫∂Dr

H dz is also equal to∫DrdH ∧ dz;

from X · F = df(X) + idh(X) = |df |2 + (df, dch) + i(dh, df) and JX · F =df(JX)+idh(JX) = −(df, dh)+i(dh, dcf)−i|dh|2, we readily get dH∧dz =(i∂H/∂s−∂H/∂t) ds∧dt = i|X(Φ(z))|2 ds∧dt. We infer that |

∫DrdH ∧dz|

is bounded from below by πr2c. By comparing the above upper and lowerbounds for

∫∂Dr

H dz =∫DrdH ∧ dz, we eventually get: C ≥ c

2r, for any

r > 0; this, of course, is only possible if c = 0, i.e. X(x0) = 0 (Φ is thenreduced to the constant map Φ(z) = x0).

remzero Remark 2.4.2. The existence of (non-trivial) holomorphic vector fieldswith a non empty zero set on a compact Kahler manifold has strong conse-quences concerning the structure of the underlying complex manifold. Theinterested reader will find a detailed account in S. Kobayashi’s book [120],cf. also [50], [51], [137], [84] for further information.

2.5. The Calabi operator L+ and its conjugatessL

Any element of hred is of the form X = gradf + Jgradh, where f, h arereal functions with zero mean value. Conversely, by Lemma 1.23.2, such a(real) vector field belongs to h, hence to hred, if and only if

conditioncondition (2.5.1) D−(df + dch) = 0.

In general, any J-anti-invariant bilinear form, B, splits as B = B2,0 +B0,2, where the (2, 0)-part B2,0 is defined by B2,0(X,Y ) = 1

2(B(X,Y ) −iB(JX, Y )), and the (0, 2)-part B0,2 is defined by B0,2(X,Y ) = 1

2(B(X,Y )+

iB(JX, Y )). More generally, for any bilinear form B, B2,0, resp. B0,2, willdenote the (2, 0)-part, resp. the (0, 2)-part, of the J-anti-invariant part, B−,of B. In the sequel, for any 1-form α — real or complex — we denote byD2,0α the (2, 0)-part of D−α and by D0,2α its (0, 2)-part. We then easilycheck:

D2,0α =1

2

(D−α+ iD−Jα

)= D−α1,0 = D1,0

(α1,0

),

D0,2α =1

2

(D−α− iD−Jα

)= D−α0,1 = D0,1

(α0,1

).

0202 (2.5.2)

Notice in particular that:

02-J02-J (2.5.3) D2,0Jα = −iD2,0α, D0,2Jα = iD0,2α.

By introducing the complex potential F = f + ih of X, cf. Definition 2, thecondition (2.5.1) is then equivalent to

holo-Fholo-F (2.5.4) D0,2dF = 0.

Indeed, since df + dch is real, the condition (2.5.1) is satisfied if and only ifD2,0(df + dch) = D0,2(df + dch) = 0, whereas, by (2.5.3), D0,2(df + dch) =D0,2dF .

2.5. THE CALABI OPERATOR L+ AND ITS CONJUGATE 75

We then define the Calabi operator L+, acting on complex functions, by

LL (2.5.5) L+(F ) = 2δδD0,2dF = 2(D0,2d)∗D0,2dF,

and the conjugate operator L− by:

barLbarL (2.5.6) L−(F ) := L+(F ) = 2δδD2,0dF = 2(D2,0d)∗D2,0dF,

where (D2,0d)∗, resp. (D0,2d)∗, denotes the adjoint operator of D2,0d, resp.D0,2d, with respect to the natural global hermitian inner products defined onthe space of complex-valued functions and on the space of complex valued-bilinear forms of type (2, 0), resp. of type (0, 2). In particular,

ReL+FReL+F (2.5.7) δδ(D−(df + dch)) = Re(L+(F )),

for any real functions f, h, with F = f + ih.

Both operators L+ and L− are elliptic, self-adjoint, semi-positive fourthorder differential operators. Moreover, it follows from (2.5.2) and fromLemma 1.23.5 that L+ and L− are related to the fourth order Lichnerowiczoperator L = δδD−d defined by (1.23.14), cf. also Remark 1.23.3), by:

LexpLexp (2.5.8) L+(F ) = L(F ) +i

2LKF,

and

barLexpbarLexp (2.5.9) L−(F ) = L(F )− i

2LKF,

for any complex function F , where, we recall, K denotes the hamiltonianvector field K = J grads whose momentum is the scalar curvature (noticethat the additional algebraic operators ± i

2LK are self-adjoint, as the vectorfield K, like any hamiltonian vector field, is divergence-free).

We then have (cf. [135, Section 95]):

proph0 Proposition 2.5.1. A (real) vector field X belongs to hred if and only ifit is of the form X = gradf+Jgradh, where f, h are real functions satisfying

LFLF (2.5.10) L+(F ) = 0,

with F = f + ih. Equivalently, the pair (f, h) is a solution to the system:

syst1syst1 (2.5.11) L(f)− 1

2LKh = 0,

syst2syst2 (2.5.12) L(h) +1

2LKf = 0,

where K := Jgrads.

Proof. We already observed that X = gradgf+Jgradgh belongs to hred

if and only if D0,2dF = 0, where F = f+ ih. This, in turn, implies L+(F ) =0. Conversely, L+(F ) = 0 implies 0 = 〈L+(F ), F 〉 =

∫M |D

0,2dF |2 vg, hence

D0,2dF = 0. Since L = δδD−d is a real operator, by (2.5.8), the conditionL+(F ) = 0 is clearly equivalent to the system (2.5.11)-(2.5.12).

remhol Remark 2.5.1. For any (real) vector field X, the (1, 0)-part X1,0 =12(X − iJX) of X is a holomorphic section of T 1,0M — cf. Sections 1.8

and 2.1 — and the (complexified) riemannian dual of X1,0 is the (0, 1)-part,


ξ(0,1), of the riemannian dual, ξ = X[, of X; we then have ξ(0,1) = 12(ξ−iJξ)

and the Hodge decomposition (2.1.2) then implies:

doldol (2.5.13) ξ(0,1) = ξ(0,1)H + ∂F,

where F = f + ih, ∂ = 12(d − idc) denotes the usual Cauchy-Riemann

operator, acting on functions, and ξ(0,1)H is the (0, 1) part of ξH , which is also

the ∆-harmonic and the -harmonic part of ξ(0,1). In particular, ξ(0,1) is∂-closed and (2.5.13) is the Dolbeault decomposition of ξ(0,1), cf. the proofof Proposition 1.17.3. Then, the map

(2.5.14) F 7→ (i ∂F )]

determines a C-linear isomorphism from the space of complex-valued func-tions F which satisfy L+(F ) = 0 and

∫M Fvg = 0 to the space hred, which,

by Proposition 2.4.1, is a (complex) Lie algebra isomorphism when the ker-nel of L+ is equipped with the Poisson bracket (C-linearly extended to thespace of complex functions). Notice that this shows a posteriori that thekernel of L+ is closed with respect to the Poisson bracket.

2.6. The space of hamiltonian Killing vector fieldssshamkilling

The space kham := hred∩k is the space of hamiltonian Killing vector fields,i.e. the space of Killing vector fields of the form X = Jgradg h = gradωh. Asa direct consequence of Proposition 2.5.1, we then have (see [135, Section96]):

prophamkilling Proposition 2.6.1. A (real) holomorphic vector field X is a hamilton-ian Killing vector field, i.e. belongs to kham, if and only if X is of the formX = Jgradh, where h is a solution to the equation

(2.6.1) L(h) = δδD−dh = 0.

Proof. By definition, any vector fielf X in kham is of the form X =gradωh = J gradgh for some real fonction h. By Proposition 2.5.1, such avector field belongs to h — hence to hred, hence to kham — if and only if his a solution to the system

(2.6.2) δδD−dh = 0, LKh = 0.

Now, the first equation δδD−dh = 0 is equivalent to D−dh = 0 — asM is compact — and, by Lemma 1.23.2, this is already equivalent to Xbeing (real) holomorphic, hence Killing. It then remains to check that thesecond equation is redundant, i.e. a consequence of the first one: this isbecause LKh = (dcs, dh) = −(ds, dch) = −LXs = 0, as X is a Killing vectorfield.

defikilling Definition 4. A real function h is called a Killing potential with re-spect to g if the hamiltonian vector field X := Jgradgh = gradω is a Killingvector field — equivalently, a (real) holomorphic vector field.

Notice that in Definition 4 we don’t demand that h be normalized by∫M h vg = 0, i.e. we don’t exclude constant real functions: these are then

Killing potentials for any Kahler metric.

2.6. THE SPACE OF HAMILTONIAN KILLING VECTOR FIELDS 77

In view of Proposition 2.6.1, for any (compact, connected) Kahler man-ifold (M, g, J, ω) the space of Killing potentials with respect to g coincideswith the kernel of the Lichnerowicz operator L in C∞(M,R).

As in Section 2.5, this shows a posteriori that the kernel of L is closedwith respect to the Poisson bracket.

Proposition 2.6.2 (A. Lichnerowicz [135]). Let (M, g, J, ω) be a com-pact Kahler manifold. Assume that the Ricci tensor rg is positive definite,with rg ≥ k g, where k is a positive real number. Then,

bochner+bochner+ (2.6.3) λ1 ≥ 2k,

where λ1 denotes the smallest positive eigenvalue of the riemannian lapla-cian ∆g acting on functions. Moreover, λ1 = 2k if and only if all eigenfunc-tions relative to λ1 are Killing potentials.

Proof. Let λ be any positive eigenvalue of ∆g and let f be any non-zero real function on M such that ∆gf = λ f . Since λ > 0, f is non-constant and ∆gdf = d∆gf = λ df . By using (1.23.10), with α = df ,by considering the inner product of both sides with df and by integratingover M , we get

∫M |D

−df |2 vg = λ2

∫M |df |

2 vg −∫M rg(gradgf, gradgf) vg ≤(

λ2 − k

) ∫M |df |

2 vg. This readily implies (2.6.3). Moreover, λ = 2k if andonly if D−df = 0, if and only if f is a Killing potential.

Notice that for a general compact riemannian manifold (M, g) withrg ≥ k g, a similar argument using the Bochner formula (1.22.1) insteadof (1.23.10) would imply λ1 ≥ k, instead of the refined Kahler inequality(2.6.3).

rem-lichne Remark 2.6.1. For any real vector field X in h, whose dual 1-form ξwith respect to any Kahler metric g is ξg = ξH +dfXg +dchXg , with the usualnotation, we have

LfXLfX (2.6.4) L(fXg ) = −1

2(dsg, ξg − dfXg ) =

1

2δ(sg (ξg − dfXg )

).

Indeed, sinceX belongs to h, we have δδD−ξg = 0, hence L(fXg ) = −δδD−ξH−δδD−dchXg ; from Lemma 1.23.4, the adjunction identity (1.23.16) and the

Bianchi identity (1.19.9), we infer δδD−ξH = δ(Jξ]Hyρ) = −(JξH , δρ) =12 (ξH , dsg), whereas, by (1.23.15), we have δδD−dch = −1

2KhXg = 1

2 (dsg, dchXg ).

We thus get the first equality in (2.6.4); the second equality follows fromξg−dfXg being co-closed. In particular, if the scalar curvature sg is constant,

then fXg is a Killing potential, as ξg− dfXg is co-closed, and so is hXg as well;

it follows that each component, ξH , dfXg and dchXg , of ξg is the dual of a real

holomorphic vector field, XH , gradgfXg and Jgradgh

Xg respectively. We infer

that JgradghXg = gradωh

Xg belongs to kham, that gradgf

Xg = −Jgradωf

Xg

belongs to Jkham and that XH is Dg-parallel, cf. Proposition 2.3.3. We thusget the decomposition (3.6.1) in Theorem 3.6.1 below.

CHAPTER 3

Calabi extremal Kahler metrics

scalabi

3.1. The space of Kahler metrics within a fixed Kahler classssmomega

In this section, we fix a complex structure J of M and an element Ω ofde Rham space H2

dR(M,R) of M . We denote by MΩ the set of all Kahlermetrics (g, J, ω) whose Kahler form ω belongs to Ω.

Elements of MΩ will be indifferently designated by the metric g or bythe (symplectic) Kahler form ω, or by the pair (g, ω).

We assume that Ω is a Kahler class, i.e. that MΩ is non-empty. It isthen a Frechet manifold of infinite dimension.

More precisely, by the ddc-lemma any two elements ω0 and ω are linkedby

(3.1.1) ω = ω0 + ddcφ,

for some real function φ uniquely defined up to additive constant. In otherwords, for any choice of a base-point ω0, MΩ is identified with the subsetof [φ] in C∞(M,R)/R such that ω0 + ddcφ is positive definite. Here, [φ]denotes the class of φ modulo addititive real constant.

For any g in MΩ, the tangent space TgMΩ is then identified with the

space, A1,10 , of exact J-invariant 2-forms. Via the operator ddc, we eventually

get the following identification:

idid (3.1.2) TgMΩ = A1,10 = C∞(M,R)/R = C∞0,g(M,R),

where C∞(M,R) denotes the space of smooth real functions on M , R thespace of (real) constant functions and C∞0,g(M,R) the space of real functionsf of M such that

nulintnulint (3.1.3)

∫Mf vg = 0

(more explanation about this normalization can be found in Section 3.1).A vector field on MΩ can thus be viewed as an assigment MΩ 3 g 7→

fg ∈ C∞0,g.The connected group of automorphisms H(M,J) acts trivially onH2

dRM ,hence preserves Ω and thus acts onMΩ. More generally, denote by HΩ(M,J),or simply HΩ if J is understood, the subgroup of those elements of Aut(M,J)which preserve Ω. This group may not be connected but has at most a finitenumber of connected components, cf. [82, Theorem 4.8], [137, Proposition2.2].

For convenience, we consider the right action of HΩ(M,J) onMΩ definedby:

(3.1.4) γ · ω = γ∗ω = γ−1 · ω.79

80 3. CALABI EXTREMAL KAHLER METRICS

The infinitesimal action on MΩ of an element X of h is then the vectorfield X determined by

(3.1.5) g 7→ LXω = d(Xyω) = ddcfXg .

Via the identification TgMΩ = C∞0,g, X is then the vector field

hatXghatXg (3.1.6) Xg = fXg ,

where fXg stands for the (real) potential of X with respect to g defined inSection 2.1.

3.2. The Calabi functionalsscalabi

The Calabi functional, C, is the real function defined on MΩ by:

(3.2.1) C(g) =

∫Ms2g vg.

This funtion is clearly HΩ(M,J)-invariant:

(3.2.2) C(γ∗g) =

∫Ms2γ∗gvγ∗g =

∫Mγ∗(s2

gvg) = C(g),

for all γ in HΩ(M,J) and all g in MΩ.

defiextremal Definition 5. A Kahler metric is called extremal — with respect to aKahler class Ω — if it is a critical metric of the Calabi functional C in MΩ.

As observed in the previous section, a vector of MΩ at g is naturallyidentified with a (real) function φ of M satisfying (3.1.3). The correspondingvariation of ω is then:

varomegavaromega (3.2.3) ω = ddcφ.

The corresponding variation of the various elements appearing in thedefinition of the Calabi functional C are then given by the following lemma:

lemmavar Lemma 3.2.1. For any variation ω = ddcφ of g in MΩ, the first varia-tion of the volume form vg, of the Ricci form, of the scalar curvature sg = sand of s vg are given by:

varmuvarmu (3.2.4) vg = −∆φ vg,

varriccivarricci (3.2.5) ρ =1

2ddc∆φ,

varsvars (3.2.6) s = −∆2φ− 2(ddcφ, ρ) = −2δδD−dφ+ (ds, dφ),

and

varsmuvarsmu (3.2.7) ˙(s vg) = δ(−2δD−dφ− sdφ) vg.

Proof. From (1.2.3) and (1.15.6) we infer: vg = ω∧ ωm−1

(m−1)! = Λ(ddcφ) vg =

−∆φ vg. The identity (3.2.5) follows from (1.19.6) and (3.2.4). The scalarcurvature s is deduced from the Ricci form ρ by

somegarhosomegarho (3.2.8)1

2msωm = ρ ∧ ωm−1,

3.2. THE CALABI FUNCTIONAL 81

or, equivalently, by

sLambdarhosLambdarho (3.2.9) s = 2Λρ,

where Λ is defined by (1.12.3) or, alternatively, cf. (1.12.2), by:

LambdaLambda (3.2.10) Λ(ψ)ωm = mψ ∧ ωm−1,

for any 2-form ψ. From (3.2.10), we infer: Λ(ψ)ωm = −mΛ(ψ)ω ∧ ωm−1 +m(m− 1)ψ ∧ ω ∧ ωm−2. By using (1.12.2) and (1.12.5), we thus get:

varLambda-genvarLambda-gen (3.2.11) Λ(ψ) = −(ω, ψ),

for any variation ω of ω, hence

varLambdavarLambda (3.2.12) Λ(ψ) = −(ddcφ, ψ),

in the current situation when ω = ddcφ. From (3.2.5) and (3.2.10), we then

infer: s = 2Λ(ρ)+2Λ(ρ) = −2(ddcφ, ρ)−∆2φ, which is the first expression ofs in (3.2.6), whereas the second expression is readily deduced from (1.23.14).Finally, (3.2.7) is a direct consequence of this expression and (3.2.4).

Remark 3.2.1. In view of the Apte formulae — see Appendix A —any other functional on MΩ involving an invariant quadratic form in thecurvature, like e.g. g 7→

∫M |R|

2 vg, g 7→∫M |r|

2 vg or g 7→∫M |W

K |2 vg, isan affine function of C, hence gives rise to the same critical metrics in MΩ.Also notice that the total volume VΩ :=

∫M vg and the total scalar curvature

SΩ :=∫M sg vg are constant as functions defined onMΩ, hence only depend

on the Kahler class Ω and on the complex structure J ; more precisely, from(1.2.3) and (3.2.8) we readily obtain

VΩ = 〈Ωm

m!, ([M ])〉,

SΩ = 4π 〈c1(M,J) ∪ Ωm−1

(m− 1)!, [M ]〉.

SOmegaVOmegaSOmegaVOmega (3.2.13)

The fact that VΩ and SΩ are constant onMΩ can also be derived from (3.2.4)

and (3.2.7), from which we readily infer∫M vg = 0 and

∫M

˙(sg vg) = 0.

thmdercalabi Theorem 3.2.1 (E. Calabi [45]). The first derivative, C, of the Calabi

functional along any variation ω = ddcφ of g in MΩ is given by

C = −4〈φ, δδD−ds〉

= −4〈D−ds,D−dφ〉.(3.2.14)

A metric g in MΩ is then extremal if and only if the scalar curvature sg isa Killing potential.

Proof. By using (3.2.6) and (3.2.4), as well as (1.23.14), we get

C =

∫M

(2ss− s2∆φ) vg

=

∫M

(− 2s∆2φ− 4s(ddcφ, ρ)− 2s (dφ, ds)

)vg

+

∫M

(2s (dφ, ds)− s2 ∆φ

)vg

= −4 〈s,L(φ)〉 = −4 〈L(s), φ〉,


where, we recall, L = δδD−d = (D−d)∗D−d denotes the fourth order Lich-nerowicz operator defined in Section 1.23, which here appears as the righthand side of (1.23.14). It follows that g is critical in MΩ iff Lsg = 0, iff sgis a Killing potential of g, meaning, we recall, that the hamiltonian vectorfield K := J gradgsg = gradωsg is a Killing vector field (equivalently, a (real)holomorphic vector field), cf. cf. Proposition 2.6.1 in Section 2.6.

3.3. Semi-positivity of the second derivative at critical pointssssecond

Assume that g is an extremal Kahler metric on (M,J), i.e. a criticalpoint of the Calabi functional C in MΩ. The second derivative of C at g isthen given by the following proposition:

propdersecond Proposition 3.3.1. The second derivative of C at a critical point galong the variations ddcφ1 and ddcφ2, where φ1 and φ2 are any two realfunctions on M , is given by

ddotPhifinddotPhifin (3.3.1) C = 8 〈δδD−dφ1, δδD−dφ2〉 − 2 〈LKφ1,LKφ2〉,

where K = Jgradgs.

Proof. Without loss of generality, we can assume that φ1 = φ2 = φ.Since g is extremal, we have D−ds = 0 at g; it follows that the first derivativeof C along the variation ddcφ is given by

ddotPhiddotPhi (3.3.2) C = −4 〈D−dφ, ˙(D−ds)〉,

where ˙(D−ds) denotes the first variation of D−ds along ω = ddcφ at the(extremal) metric g. Since g is extremal, by Lemma 1.23.2 we have

˙(D−ds) = −1

2ω((L ˙gradsJ)·, ·)

= D−(( ˙grads)[).dotD-dsdotD-ds (3.3.3)

It thus remains to compute ˙(grad s) = ˙grad s + grad s, where s is givenby (3.2.6), whereas, for any (fixed) function f , the first variation of grad fis easily deduced from the identity: gradfyω = Jdf , where the rhs is inde-pendent of g. We thus get

vargradvargrad (3.3.4) ( ˙grad f)[ = (dcf)]yddcφ,

so that

( ˙grad s)[ = −d∆2φ− 2d((ddcφ, ρ))

+ (dcs)]yddcφ;

since K is Killing and preserves J , the last term in the rhs can be writtenas follows :

Kyddcφ = LKdcφ− d((dcs, dcφ))

= dcLKφ− d((ds, dφ))

= dc((dcs, dφ))− d((ds, dφ)) ;

3.3. SEMI-POSITIVITY OF THE SECOND DERIVATIVE AT CRITICAL POINTS 83

we thus finally get (by using (1.23.7) again) :

( ˙grad s)[ = −d∆2φ− 2d((ddcφ, ρ))− d((ds, dφ)) + dc((dcs, dφ))

= −2d(δδD−dφ) + dc((dcs, dφ)).(3.3.5)

Then, by (3.3.2) et (3.3.3):

C = 8 〈D−dφ,D−dLφ〉 − 4 〈D−dφ,D−dc((dcs, dφ))〉

= 8 〈Lφ,Lφ〉 − 4 〈D−dφ,D−dc((dcs, dφ))〉

= 8 〈Lφ,Lφ〉 − 4 〈φ, δδD−dc((dcs, dφ))〉.By (1.23.15), we have:

−4 〈φ, δδD−dc((dcs, dφ))〉 = 2〈φ, (d((dcs, dφ)), dcs)〉

= 2 〈d((dcs, dφ)), φ dcs〉

= 2 〈(dcs, dφ), δ(φdcs)〉

= −2 〈(dcs, dφ), (dcs, dφ)〉,

where we have used the fact that dcs is co-closed; since (dcs, dφ) = LKφ, weeventually get (3.3.1).

The fact that C is semi-positive at critical points is not directly visiblefrom (3.3.1). Following Calabi in [45], it is convenient to consider the Calabioperators L+ and L− introduced in Section 2.5, defined by (2.5.5)-(2.5.6).

Recall that the kernel of L+ is the space of complex functions F = f+ih,f, h real, such that X = gradf +Jgradh is a (real) holomorphic vector field,cf. Section 2.5.

In general, L+ and L− are distinct. More precisely, it follows from(2.5.8)-(2.5.9) that L+ and L− coincide if and only if K ≡ 0, if and only ifthe scalar curvature s is constant. More generally,

prop-commute Proposition 3.3.2. If the Kahler structure is extremal, then L+ andL− commute.

Proof. If the Kahler structure is extremal, then K is Killing and pre-serves J ; we thus have the identity

idKidK (3.3.6) LKδδD−dF = δδD−dLKF,for any F . By (2.5.8) and (2.5.9), this directly implies that L+ and L−commute.

We infer:

propsp Proposition 3.3.3 (E. Calabi [45]). The second derivative C at a crit-

ical metric g is semi-positive. The kernel of C is the tangent space at g ofthe orbit of g under the action of Hred(M,J).

Proof. It follows from (2.5.8)-(2.5.9) and from (3.3.6) that (3.3.1) canbe rewritten as

ddotCLddotCL (3.3.7) C = 8 〈L+φ1,L−φ2〉 = 8 〈L−L+φ1, φ2〉.The operators L+ and L− are commuting, elliptic, self-adjoint, semi-positiveoperators on the compact manifold M . This implies that the product


L−L+ = L+L− is itself self-adjoint and semi-positive and that ker(L−L+) =

kerL−+ kerL+ = kerL+ + kerL+. This means that a real function φ belongsto ker(L−L+) if and only if there exists a complex functions F in kerL+ such

that φ = F + F = 2Re(F ). By Proposition 2.5.1 this happens if and only

if φ is the real potential of a (real) holomorphic vector field X belonging tothe ideal hred. Taking into account (3.4.3) which will be proved later, this is

equivalent to φ being the real potential of a (real) holomorphic vector fieldX in h.

As observed by E. Calabi, Proposition 3.3.3 has the following directcorollary:

propcalabiorbit Proposition 3.3.4. The space of extremal Kahler metrics in MΩ isa submanifold whose each connected component is an orbit of the reducedautomorphism group Hred(M,J).

Proof. By Proposition 3.3.3, at any critical point of C the hessian ofC is non-degenerate, even positive definite, in the directions transverse tothe orbit of Hred(M,J). The space of extremal Kahler metrics is then asmooth submanifold of MΩ and, at each point, its tangent space coincideswith the tangent space of the orbit. Each orbit of Hred(M,J) is then open,hence coincides with the corresponding connected component. Accordingto [60], the space of extremal Kahler metrics is actually either empty orconnected.

3.4. Holomorphic vector fields on a compact extremal Kahlermanifold

ssstructure

In this section, we assume that (M, g, J) is a connected, compact Kahlermanifold and that the Kahler structure is extremal, i.e. that the hamiltonianvector field K = Jgradsg is Killing.

The notation are those of Chapter 2. In particular we recall that:

(1) the complex Lie algebra of infinitesimal automorphisms of (M,J),thought of as the Lie algebra of real vector fields preserving J , hasbeen denoted by h and is the Lie algebra of the (complex) Lie groupH(M,J);

(2) the Lie algebra of Killing vector fields, denoted by k, is a (real)Lie subalgebra of h and is the Lie algebra of the (compact) groupK(M, g);

(3) the Lie algebra a of parallel (real) vector fields is a (complex)abelian complex Lie subalgebra of h, contained in k and in thecenter of h;

(4) any element X of h can be uniquely written as

X = XH + grad fX + JgradhX ,

where XH is the dual of a harmonic 1-form and where fX , hX arereal functions, normalized by

∫M fXvg =

∫M hXvg = 0, see Lemma

2.1.1 (fX , resp. FX = fX + ihX , is called the real potential, resp.the complex potential, of X);

3.4. HOLOMORPHIC VECTOR FIELDS 85

(5) the ideal hred of element X of h such that α(X) = 0 for all har-monic 1-forms α coincides with the space of elements of h of theform X = gradf + Jgradh, where F = f + ih satisfies L+F = 0,see Proposition 2.5.1; hred is the Lie algebra of the reduced auto-morphism group Hred(M,J) ;

(6) the space kham = k ∩ hred of hamiltonian Killing vector fields co-incides with the space of (real) vector fields of the form Jgradh,where h is a real function satisfying Lh = 0, see proposition 2.6.1.

The following theorem is due to E. Calabi, cf. [45].

thmdeccalabi Theorem 3.4.1. For any compact extremal Kahler manifold, the com-plex Lie algebra h admits the following orthogonal decomposition:

dechcalabidechcalabi (3.4.1) h = h(0) ⊕ (⊕λ>0 h(λ)),

where h(0) denotes the kernel of LK, i.e. the centralizer of K in h, and,for any λ > 0, h(λ) denotes the subspace of elements X of h such thatLKX = λJX.

The subspace h(0) is a reductive (complex) Lie subalgebra of h; it containsa, kham and Jkham and is actually the orthogonal sum of these three spaces:

dech00dech00 (3.4.2) h(0) = a⊕ kham ⊕ Jkham.

Moreover, each h(λ), λ > 0, is contained in the ideal hred, so that we alsohave the following orthogonal decompositions:

dechdech (3.4.3) h = a⊕ hred,

deckdeck (3.4.4) k = a⊕ kham,

and

dech0dech0 (3.4.5) hred = kham ⊕ Jkham ⊕ (⊕λ>0h(λ)).

Proof. Let X = XH + gradf +Jgradh be any element of h, where XH

is the dual of the harmonic 1-form ξH . By (1.23.10), we have δD−ξH =JXHyρ, so that, by using (1.23.16) and the fact that JξH is also harmonic,hence closed:

δδD−ξH = δ(JXHyρ) = −(δρ, JξH) =1

2(dcs, JξH).

Now, (dcs, JξH) is the inner product of a harmonic 1-form, JξH , and the

dual of a Killing vector field, dcs = K[; it is then constant, hence equal tozero since K has zeros. It follows that δδD−ξH = 0, whence also δδD−(df +dch) = 0, as ξ = ξH + df + Jdh is the dual of a (real) holomorphic vectorfield. By (2.5.7), the latter equality can be read: Re(L+(F )) = 0. Startingfrom JX instead of X, we similarly get Im(L+(F )) = 0, hence, eventually,L+(F ) = 0. By Proposition 2.5.1, this implies that grad f+Jgradh and XH

are separately (real) holomorphic. As already observed in Section 2.3, thisimplies that XH is parallel, hence belongs to a, whereas grad f + Jgradhcertainly belongs to h0. This completes the proof of (3.4.3), hence also of(3.4.4).


Since L− and L+ commute, L− acts on kerL+; it readily follows from(2.5.8) and (2.5.9) that this action coincides with the action of −iLK; sinceL− is self-adjoint and semi-positive, hred splits as

(3.4.6) hred = h(0)red ⊕ (⊕λ>0h

(λ)),

where h(0)red denotes the kernel of LK in hred, whereas, for each λ > 0, h(λ)

denotes the subspace of elements X of h such that LKX = λJX (h(λ) = 0but for a finite, possibly empty, set of λ′s).

Since LXs = 0 for any Killing vector field X, h(0) certainly contains thedirect sum a⊕kham⊕Jkham. In fact, both spaces coincide. Indeed, by (2.5.8)and (2.5.9) the restriction of LK to kerL+ coincides with the restriction ofδδD−d and we conclude by using Proposition 2.6.1. This completes theproof of (3.4.2).

Remark 3.4.1. Decomposition (3.4.1) provides a criterion of non-existenceof extremal metrics. In particular, a compact complex manifold (M,J) forwhich H(M,J) is not reduced to 1 but has no connected compact sub-group apart from 1 cannot have any extremal Kahler metric. Indeed,the latter hypothesis implies that K(M, g) is reduced to 1 for any rie-mannian metric which is Kahler with respect to J ; in particular, if (g, J) isextremal the scalar curvature s of g is necessarily constant; but for (com-pact) Kahler manifolds of constant scalar curvature, K(M, g) = 1 impliesH(M,J) = 1 as directly follows from (3.4.1). Examples of kahlerian com-pact complex surfaces satisfying the above hypotheses, hence admitting noextremal Kahler metric, have been first given by M. Levine, cf. [134].

3.5. Maximality of K(M, g) in H(M,J)ssmaximal

By Theorem 3.4.1, when (g, J) is extremal the kernel h(0) of LK in hcoincides with the sum k+ Jk = a⊕ (kham⊕ Jkham) in h. On the other hand,k = a⊕ k0 is the Lie algebra of the (connected) Lie group Iso0(M, g), whichis a compact (real) subgroup of H(M). The following theorem is the mainresult of [46] and the proof given here closely follows Calabi’s one in [46]:

thmcalabi Theorem 3.5.1 (E. Calabi [46]). For any connected, compact extremalKahler manifold, K(M, g) is maximal among all compact, connected Liesubgroups of H(M,J).

Proof. Let g be the Lie algebra of a connected, compact Lie subgroup,G, of H(M,J) containing K(M, g). Suppose, for a contradiction, that thereexists X in g that does not belong to k. Since k = a⊕ kham, we can assumethat X belongs to Jkham ⊕ (⊕λ>0h

(λ). Let X = X0 +∑

λ>0Xλ be the

corresponding decomposition of X; then, (LK)2rX = −∑λ2rXλ for any

positive integer r. We easily infer that each component of X belongs to g,so that g =

∑λ≥0 g

(λ), with g(λ) = g ∩ h(λ). We thus can assume that X

belongs to some g(λ), λ > 0, or to Jkham ⊂ g(0).Assume that X belongs to some g(λ), λ > 0, and is not zero. Let B

denotes the Killing form of g. Since g is the Lie algebra of a compact Lie


group, B is semi-negative and its kernel coincides with the center of g 1.On the other hand, X belongs to the kernel of B: indeed, for any X1 ing(µ1) and any X2 in g(µ2), with µ1, µ2 positive or 0, [X, [X1, X2]] belongs to

g(λ+µ1+µ2) 6= g(µ2) (notice that g(λ+µ1+µ2) may be reduced to 0). It followsthat tr(adX adX1) = 0 for any X1 in g. This implies that X belongs to thekernel of B, hence to the center of g. On the other hand, [X,K] = −λJX,which is non-zero: a contradiction.

It follows that X necessarily belongs to Jkham. Then, X = grad f forsome (real) function f . By hypothesis, the flow, ΦX

t , of X is containedin a connected, compact subgroup of H(M,J): its closure in H(M,J) isthen a torus T k of dimension k, for some k ≥ 1. If k = 1, the orbits ofX = grad f are all closed; but a gradient has no non-trivial closed integralcurve, as d

dtf(ΦXt (x)) = |X|2

ΦXt (x))≥ 0. It follows that k > 1. But this

is impossible as well. Indeed, fix a point x of M such that Xx 6= 0. Bythe above computation, f(ΦX

t (x)) is an increasing function of t, so thatf(ΦX

t (x)) − f(x) > c, where c is some positive constant, for t > 1, say.On the other hand, since ΦX

t is dense in T k, ΦXt (x) meets any (small)

neighbourhood U of x in M for some t > 1: again a contradiction.We thus conclude that g coincides with k.

rem-gradient Remark 3.5.1. The argument at the end of the proof of Theorem 3.5.1extends to more general situations and actually proves the following state-ment:

Proposition 3.5.1. Let X be a complete vector field defined on somemanifold M and suppose that the flow ΦX is contained in a compact subgroupof DiffM . Then, X cannot be the gradient of a function for any riemannianmetric on M .

Proof. By a well-known theorem of Bochner-Montgomery [35], anylocally compact subgroup of DiffM is a Lie group. We can then argue asabove.

3.6. Kahler metrics with constant scalar curvature andKahler-Einstein metrics

ssconstant

A (compact) Kahler manifold whose scalar curvature is constant is cer-tainly extremal since K = Jgrad s vanishes identically. Theorem 3.4.1 thenimplies:

thmlichnerowicz Theorem 3.6.1 (A. Lichnerowicz [135]). For any compact Kahler man-ifold of constant scalar curvature, the complex Lie algebra of (real) holomor-phic vector fields h splits as

lichnelichne (3.6.1) h = h(0) = a⊕ kham ⊕ Jkham.

1 In general, the Killing form of a Lie algebra g is the (symmetric) bilinear form on gkillingdefined by Bg(a, b) = tr(ada adb), where ad : g→ End(g) stands for the usual ad-actionof g on itself given by the bracket, so that adab = [a, b], and the trace is over R or over C,according as g is a R- or a C-Lie algebra. If g is the Lie algebra of a compact Lie group G,then g can be given a G-invariant positive definite inner product, with respect to whichall ada are then anti-symmetric. It follows that Bg is semi-negative and that a belongs tothe kernel of Bg if and only of ada = 0, i.e. if and only if a belongs to the centre of g.


The decomposition 3.6.1 is a Lie algebra direct sum of the abelian Lie algebraa of parallel vector fields and of the complexification in h of the Lie algebrakham of hamiltonian Killing vector fields. In particular, h is a reductivecomplex Lie algebra.

Proof. (Cf. also Remark 2.6.1). Since K = Jgradgsg = 0, for each

λ > 0, h(λ) = 0 in (3.4.1) and (3.4.1) is then reduced to (3.6.1). We recallthat a Lie algebra, say l, is said to be reductive if its radical, i.e. its maximalsolvable ideal, coincides with its center Z(l): then, either l is abelian, andwe then have l = Z(l), or the derived ideal [l, l] of l is semi-simple; in bothcases, l = Z(l) ⊕ [l, l], cf. e.g. [105]. The Lie algebra of any compactLie group is reductive: in this case, the Killing form is semi-negative, thecenter Z(l) coincides with the kernel of the Killing form and the Killingform of [l, l] is negative-definite. The complexification of the Lie algebra of acompact group is then reductive as well. By (3.6.1), in the case of a compactKahler manifold of constant scalar curvature, h is a Lie algebra direct sumof a, which is abelian, and of the complexification of kham, which is the Liealgebra of a compact group; it is then reductive.

The set of compact Kahler manifold of constant scalar curvature includesthe set of (compact) Kahler-Einstein manifolds, introduced in Section 1.19.We then have

thmmatsushima Theorem 3.6.2 (Y. Matsushima [148]). For any compact Kahler-Einsteinmanifold (M, g, J, ω) of real dimension n = 2m and of (constant) scalar cur-vature s, the complex Lie algebra of (real) holomorphic vector fields h is 0if s < 0, is abelian and reduced to the space a of parallel vector fields if s = 0;if s > 0, h is of the form

matsushimamatsushima (3.6.2) h = k⊕ Jk.

where k = kham denotes the space of Killing vector fields, with then coin-cides with the space of hamiltonian Killing vector fields. In particular, h isreductive.

Moreover, any positive eigenvalue λ of the riemannian laplacian ∆ act-ing on functions satisfies

lambdasmlambdasm (3.6.3) λ ≥ s

m,

and sm is an eigenvalue of ∆ if and only if h 6= 0. In this case, the

eigenspace, E, of sm is closed for the Poisson bracket and the map f →

Zf := gradω f = Jgradg f induces a Lie algebra isomorphism from E tokham.

Proof. If (g, J, ω) is Kahler-Einstein with s < 0, then the Ricci tensorr is negative definite and then h = 0, cf. Remark 1.23.2; if s = 0, thenr ≡ 0 and h = a, cf Remark 1.23.2 again.

If s > 0, r is positive definite and it is a direct consequence of theBochner formula (1.22.1) that a compact riemannian manifold with positivedefinite Ricci tensor has no non-zero harmonic 1-form (this fact holds forany compact riemannian manifold; in the Kahler case it has been refinedby S. Kobayashi who proved that a compact Kahler manifold with positive


definite Ricci tensor is simply-connected [119]; this refinement however isnot needed here). In particular, M has no non-zero parallel 1-form, hence nonon-zero parallel vector field: (3.6.1) then reduces to (3.6.2) and k = a⊕kham

reduces to kham. In particular, h is reductive as the complexification of theLie algebra of the compact Lie group K(M, g). Actually, (3.6.2) can beobtained in a simpler way as a direct applicaton of (1.23.10) which, in theKahler-Einstein case, simply reads

newdeltaD-newdeltaD- (3.6.4) δD−ξ =1

2(∆ξ − s

mξ).

Since M has no non-zero harmonic 1-form, the riemannian dual ξ of anyvector field X in h is of the form ξ = df+dch, with

∫M f vg = 0,

∫M h vg = 0.

From (3.6.4), we then get

(3.6.5) 0 = δD−ξ =1

2d(∆f − s

mf) +

1

2dc(∆h− s

mh).

We infer that gradgf and Jgradg h are separately in h, in fact in Jkham andin kham respectively. We also infer that Jgradgf is a non-zero element of kham

if and only if f is a non-zero eigenfunction of ∆ for the eigenvalue sm . The

map f 7→ gradω0f = Jgradg f then induces a linear isomorphism from the

space, E, of (real) functions f satisfying ∆f = sm f onto the space kham of

hamiltonian Killing vector fields. Moreover, because of (1.2.11) we infer thatE is closed with respect to the Poisson bracket and that the isomorphismE ∼= k is therefore a Lie algebra isomorphism from E, equipped with thePoisson bracket, to the Lie algebra k of Killing vector fields. Finally, byintegrating (3.6.4) over M when ξ = df and f is an eigenfunction f of ∆with respect to any positive eigenvalue λ, we get

(3.6.6)

∫M|D−df |2 vg =

1

2λ (λ− s

m)

∫M|f |2 vg.

We infer that any positive eigenvalue λ of ∆ satisfies (3.6.3) and that sm is

an eigenvalue of ∆ if and only if kham 6= 0.

Theorems 3.6.2 and 3.6.1 provide an obstruction to the existence ofKahler metrics with constant scalar curvature — a fortiori for Kahler-Einsteinmetrics — which can be expressed as follows:

A compact complex manifold whose automorphism group Aut(M,J) isnot reductive admits no Kahler metric with constant scalar curvature.

Examples of such complex manifolds are provided in Section 6.6 and inChapter 10.

remlichnevf Remark 3.6.1. Part of the argument in the proof of Theorem 3.6.2 is aspecial case of the following general result:

proplichnecond Proposition 3.6.1 (A. Lichnerowicz [135]). Let (M, g, J, ω) be a com-pact Kahler manifold, whose Ricci tensor r satisfies

lichnecondlichnecond (3.6.7) r ≥ k g,for some positive real number k. Denote by λ1 the smallest positive eigen-value of the riemannian Laplace operator ∆ and by E1 the correspondingeigenspace. Then,

lambdalichnelambdalichne (3.6.8) λ1 ≥ 2k.


Moreover, if λ1 = 2k then all elements of E1 are Killing potentials.

Proof. Let f be any non-zero element of E1. From (1.23.10) applied

to ξ = df we infer δD−df = λ12 df − r(gradgf). By considering the global

inner product of both sides with df and by using (3.6.7), we get∫M|D−df |2 vg ≤

1

2(λ1 − 2k)

∫M|df |2 vg.

Since f is not constant, we readily infer (3.6.8). Moreover, if equality holds,then D−df = 0, meaning that Jgradgf is (real) holomorphic, hence Killing.

CHAPTER 4

More on the geometry of the space of Kahlermetrics

smore

4.1. The space of relative Kahler potentialssstildeM

The spaceMΩ of Kahler metrics within a fixed Kahler class Ω, as definedin Section 3, is closely related to the space of relative Kahler potentials, Mω0 ,defined as the space of real functions φ on M such that ωφ := ω0 + ddcφis positive definite, for some chosen element ω0 in MΩ (the correspondingriemannian metric will be denoted by g0).

Notice that if φ belongs to Mω0 , so does tφ for each real number t inthe interval [0, 1], as ωtφ = ω0 + tddcφ = (1− t)ω0 + t ωφ.

As a (contractible) open subset of C∞(M,R), Mω0 is a Frechet manifold

modeled on C∞(M,R) and, for each φ in Mω0 , the tangent space TφMω0 isnaturally identified with C∞(M,R). In particular, any f in C∞(M,R) can

be regarded as a constant vector field on Mω0 . The space of constant vector

fields on Mω0 generates the tangent bundle TMω0 in the sense that for any

φ in Mω0 any element of TφMω0 is the value at φ of a (uniquely) definedconstant vector field. Moreover for any two elements f1, f2 of C∞(M,R),the corresponding constant vector fields, still denoted by f1, f2, commute.

We observe that the definition of Mω0 depends upon the choice of a base-point ω0 of MΩ. In particular, the group H(M,J) does not act directly on

Mω0 , since ω0 is not preserved in general by the action of H(M,J) onMΩ.

The additive group R acts freely on Mω0 , by a · φ = φ + a, for any a

in R and MΩ is then naturally identified with the quotient Mω0/R. The

natural projection of Mω0 onto MΩ will be denoted by π.

It may be convenient to realize MΩ as a submanifold of Mω0 , so that

Mω0 is identified with the product MΩ × R. This can be done as follows[74].

We first observe that the (smooth) assignment φ 7→ νφ of a n-form νφon M to each element φ of Mω0 can be viewed as a 1-form, ν say, on Mω0 ,via the pairing

1form1form (4.1.1) νφ(f) =

∫Mf νφ,

for each f in C∞(M,R) = TφMω0 . In particular, let τ be the 1-form on

Mω0 defined by

tautau (4.1.2) τφ(f) =1

VΩ

∫Mf vφ,

91

92 4. MORE ON THE GEOMETRY OF THE SPACE OF KAHLER METRICS

for each φ in Mω0 and each f in TφMω0 = C∞(M,R), where vφ stands forthe volume form of ωφ.

It is easily checked that τ is closed. Indeed, choose any two elementsf1, f2 of C∞(M,R), viewed as (commuting, constant) vector fields on MΩ

(see above). By using (3.2.4), we have that

dτ(f1, f2) = f1 · τ(f2)− f2 · τ(f1) =1

VΩ

∫M

(f1∆f2 − f2∆f1)vφ = 0.

Since Mω0 is contractible, τ admits a unique primitive, denoted by Iω0 ,equal to 0 at φ = 0. We thus have

varbbIvarbbI (4.1.3) (d Iω0)φ(f) =1

VΩ

∫Mfvgφ ,

for any φ in Mω0 and any f in TφMω0 = C∞(M,R), and

(4.1.4) Iω0(φ) =

∫ 1

0τφt(φt) dt,

where t ∈ [0, 1]→ φt is any curve connecting 0 = φ0 to φ = φ1 in Mω0 . By

choosing the curve t 7→ tφ, we have φt = φ for all t, so that:

II (4.1.5) Iω0(φ) =1

VΩ

∫Mφ (

∫ 1

0vtdt),

where vt stands for the volume form of ωt := ω0 + tddcφ, i.e. vt = 1m!(ω0 +

tddcφ)m.

lemmaIalter Lemma 4.1.1. The function Iω0 defined by (4.1.5) has the following al-ternative expression

I-bisI-bis (4.1.6) Iω0(φ) =1

VΩ

∫Mφ

∑mj=0 ω

jφ ∧ ω

m−j0

(m+ 1)!

for any φ in MΩ, with ωφ = ω0 + ddcφ.

Proof. The rhs of (4.1.6) is evidently equal to 0 when φ ≡ 0. It is thensufficient to check that it is a primitive of τ . By differentiating the rhs of(4.1.6), we get

1

VΩ

( ∫Mf

∑mj=0 ω

jφ ∧ ω

m−j0

(m+ 1)!+ φ

∑mj=1 j ω

j−1φ ∧ ωm−j0

(m+ 1)!∧ ddcf

)=

1

VΩ

∫Mf(∑m

j=0 ωjφ ∧ ω

m−j0

(m+ 1)!+

∑mj=1 j ω

j−1φ ∧ ωm−j0

(m+ 1)!∧ ddcφ

)=

1

VΩ

∫Mf

∑mj=0

(ωjφ ∧ ω

m−j0 + j (ωj−1

φ ∧ ωm−j0 ) ∧ (ωφ − ω0))

(m+ 1)!

=1

VΩ

∫Mfωmφm!

= τφ(f).

The functional Iω0 defined by (4.1.5)-(4.1.6) is R-equivariant:

(4.1.7) Iω0(φ+ a) = Iω0(φ) + a,

4.2. AUBIN FUNCTIONALS 93

for any φ in Mω0 and any a in R, as 1VΩ

∫M vt = 1 for all t. In particular,

Iω0 is everywhere regular and thus makes Mω0 into a fibration over R whosefibers — the level sets of Iω0 — are all isomorphic toMΩ and can be viewed

as the graphs of parallel sections of π : Mω0 → MΩ with respect to theconnection determined by τ .

In view of the above, the space MΩ can be naturally regarded as asubspace of MΩ via the identification

don-iddon-id (4.1.8) MΩ = I−1ω0

(0).

Within this identification, any ω in MΩ is unambiguously written as ω0 +ddcφ, where the relative Kahler potential φ is normalized by Iω0(φ) = 0, cf.[74]. For any curve ωt = ω0 + ddcφt inMΩ, with Iω0(φt) = 0, the derivative

φt will then automatically satisfy Condition (3.1.3).The functional Iω0 has been first introduced in Bando-Mabuchi’s paper

[23]. In view of (4.1.8) and of its role in Donaldson’s paper [74], it will bereferred to as the Donaldson gauge functional relative to ω0.

Notice that the map φ 7→ (ωφ = ω0 + ddcφ, Iω0(φ)) is a diffeomorphism

of Frechet manifolds from Mω0 to the product MΩ × R, whose inverse isthe map (ω = ω0 + ddcφ, a) 7→ φ− Iω0(φ) + a.

Remark 4.1.1. If another base-point ω′0 = ω0 + ddcψ is chosen in MΩ,

the space Mω′0of relative Kahler potentials with respect to ω′0 is deduced

from Mω0 by the translation φ 7→ φ′ = −ψ+φ in C∞(M,R). We thus have

(4.1.9) Iω′0(φ′) = Iω0(φ′ + ψ)− Iω0(ψ),

for any φ′ in Mω0′ . Notice that Iω0(ψ) = −Iω′0(−ψ).

4.2. Aubin functionalsssaubin

The functionals Iω0 and Jω0 on MΩ considered in this section wereintroduced by T. Aubin in [20] and have been extensively used since then inthe literature. These depend on the choice of a base-point ω0 and are thendefined by

IaubinIaubin (4.2.1) Iω0(ωφ) =1

VΩ(

∫Mφ vg0 −

∫Mφ vgφ),

Jω0(ωφ) =

∫ 1

0

Iω0(ωtφ)

tdt

=1

VΩ

∫Mφ vg0 −

1

VΩ

∫Mφ(

∫ 1

0vgtφ dt),

JaubinJaubin (4.2.2)

for any ωφ = ω0 + ddcφ in MΩ, where ωtφ = ω0 + tddcφ, 0 ≤ t ≤ 1, isthe linear interpolation between ω0 and ωφ. Here, as usual, gφ, gtφ denotethe corresonding riemannian metrics and vgφ , vgtφ the corresponding volumeforms.

propaubin Proposition 4.2.1 ([17], [23], [186], [187]). The Aubin energy function-als Iω0 and Jω0 admit the following alternative expressions:

IalterIalter (4.2.3) Iω0(ωφ) =1

VΩ

∫Mdφ ∧ dcφ ∧

∑m−1i=0 ωm−i−1

0 ∧ ωiφm!

,


and

JalterJalter (4.2.4) Jω0(ωφ) =1

VΩ


∑m−1i=0 (m− i)ωm−i−1

0 ∧ ωiφ(m+ 1)!

,

for any ωφ = ω0 + ddcφ in MΩ. In particular, Iω0 and Jω0 satisfy

IJinequalityIJinequality (4.2.5) 0 ≤ 1

(m+ 1)Iω0(ω) ≤ Jω0(ω) ≤ m

(m+ 1)Iω0(ω),

for any ω in MΩ, with equality — in any of the above three inequalities —if and only if ω = ω0.

Proof. From (4.2.1) we get

Iω0(ωφ) =1

VΩ

∫Mφ(ω0 − ωφ) ∧

m−1∑i=0

ωm−i−10 ∧ ωiφ

m!

= − 1

VΩ

∫Mφddcφ ∧

m−1∑i=0


m!

=1

VΩ


m−1∑i=0


m!,

which is (4.2.3). Similarly, from (4.2.7) and (4.1.6), we get

Jω0(ωφ) =1

VΩ

∫Mφ(ωm0m!−

m∑i=0

ωm−iφ ∧ ωi0(m+ 1)!

)=

1

VΩ

∫Mφm−1∑i=0

ωi0(ωm−i0 − ωm−iφ )

(m+ 1)!

= − 1

VΩ

∫Mφddcφ ∧

m−1∑i=0

ωi0 ∧ (ωm−1−i0 + ωm−2−i

0 ∧ ωφ + . . .+ ωm−1−iφ )

(m+ 1)!

=1

VΩ


m−1∑i=0

(m− i)ωm−1−i0 ∧ ωiφ

(m+ 1)!,

which is (4.2.4). For the rest of the proof of Proposition 4.2.1 we need thefollowing lemma:

lemmapositive Lemma 4.2.1. For any φ in Mω0 and for each i = 0, . . . ,m−1, we have

norminormi (4.2.6)


ωm−1−i0 ∧ ωiφ(m− 1)!

=

∫M|dφ|2

g(i) vg(i) ,

where g(i) denotes the unique hermitian metric on (M,J) — in general non-

Kahler for 0 < i < m− 1 — whose Kahler form, ω(i), satisfies (ω(i))m−1 =ωm−1−i

0 ∧ ωiφ.

Proof. The existence of ω(i) is easily established by considering, forany x in M , a J-adapted basis e1, Je1, . . . , em, Jem of TxM , with respectto which ω0(x) =

∑mr=1 e

∗r ∧ Je∗r and ωφ(x) =

∑mr=1 λre

∗r ∧ Je∗r (where

e∗1, Je∗1, . . . , e∗m, Je∗m denotes the dual basis of T ∗xM and the λr’s are all

positive real numbers). We then get ω(i)(x) =∑m

r=1 µ(i)r e∗r ∧ Je∗r , with

4.2. AUBIN FUNCTIONALS 95

µ(i)r =

c1/m−1i

σi(λ1,...,λr,...,λm), where σi(λ1, . . . , λr, . . . , λm) stands for the i-th ele-

mentary symmetric function of the (m − 1) numbers obtained by deleting

λr from λ1, . . . , λm, and ci =(m−1i

) ∏ms=1 σi(λ1, . . . , λs, . . . , λm). Observe

that ω(0) = ω0 and ω(m−1) = ωφ, while ω(i) is not closed in general for

i = 1, . . . ,m−2. Then, (4.2.6) readily follows from (ω(i))m−1 = ωm−1−i0 ∧ωiφ.

The uniqueness of ω(i) follows from the general fact that ωm−11 = ωm−1

2 im-plies ω1 = ω2 for any two positive J-invariant 2-forms, ω1, ω2 (easy verifi-cation by diagonalizing ω2 with respect to ω1 as above).

The second part of Proposition 4.2.1 readily follows from (4.2.3)-(4.2.4)upon using (4.2.6), which ensures the positivity of all terms in the rhs of

(4.2.3)-(4.2.4) for any non-constant φ in Mω0 .

The Aubin functionals Iω0 and Jω0 are related to the Donaldson gaugefunctional Iω0 defined in Section 4.1 via the next statement:

propnorm Proposition 4.2.2. For any φ in Mω0, we have

1

VΩ

∫Mφ vg0 = Jω0(ωφ) + Iω0(φ),

− 1

VΩ

∫Mφ vgφ = Iω0(ωφ)− Jω0(ωφ)− Iω0(φ).

IIJIIJ (4.2.7)

In particular, if φ is normalized by

normphinormphi (4.2.8) Iω0(φ) = 0,∫M φvg0 and −

∫M φvgφ are both positive, unless φ ≡ 0, and we have

int-compareint-compare (4.2.9)1

m

∫Mφvg0 ≤ −

∫Mφvgφ ≤ m

∫Mφvg0 ,

with equality in both sides if and only if φ ≡ 0.

Proof. The first equality in (4.2.7) is a direct consequence of (4.1.5)and (4.2.2); the seconde equality then follows from (4.2.1). If φ is normalizedby (4.2.8), we infer

int-normint-norm (4.2.10)1

VΩ

∫Mφ vg0 = Jω0(ωφ), − 1

VΩ

∫Mφ vgφ = (I − J)ω0(ωφ).

Then, (4.2.9) easily follows from (4.2.5). By Proposition 4.2.1, in (4.2.9),equality holds in either inequalities if and only if φ is constant, hence iden-tically zero because of (4.2.8).

remnorm Remark 4.2.1. When MΩ is identified to I−1ω0

(0), cf. Section 4.1), by

Proposition 4.2.2, both functionals φ 7→ 1VΩ

∫M φvg0 and φ 7→ − 1

VΩ

∫M φvgφ

on Mω0 can be viewed as distance functions from ω0 onMΩ, as well as Iω0 ,Jω0 and (I − J)ω0 := Iω0 − Jω0 . By Proposition 4.2.1 and Proposition 4.2.2,all these distance fonctions are pairwise uniformly bounded by constantswhich only depend on m.


Proposition 4.2.3. For any ωφ = ω0 + ddcφ in MΩ, the derivatives ofIω0, Jω0 and (I − J)ω0 at ωφ are given by

dIω0(f) =1

VΩ

( ∫Mf vg0 −

∫Mf vgφ +

∫Mf∆φφ vgφ

),

dJω0(f) =1

VΩ

( ∫Mf vg0 −

∫Mf vgφ

),

d(I − J)ω0(f) =1

VΩ

∫Mf∆φφ vgφ ,

dotIJdotIJ (4.2.11)

for any ωφ = ω0 +ddcφ inMΩ and any f in TωφMΩ = C∞(M,R)/R, where∆φ denotes the riemannian laplacian relative to gφ.

Proof. The first identity follows from (4.2.1) and (3.2.4); the secondidentity follows from (4.2.7) and (4.1.3); the third identity follows from theprevious two ones.

4.3. The Mabuchi metricssmabuchimetric

The space Mω0 comes equipped with a natural (weak) riemannian met-ric, whose inner product at φ is defined by

metricmetric (4.3.1) 〈f1, f2〉φ =

∫Mf1f2 vgφ ,

for any pair f1, f2 of real functions on M , viewed as elements of TφMω0 ,where gφ = π(φ) denotes the metric determined by φ, i.e. the metric whoseKahler form is ω = ω0 + ddcφ. The space MΩ, identified to the level setI−1ω0

(0) in MΩ, cf. Section 4.1, is equipped with the induced riemmannianmetric: at any g in MΩ, the corresponding inner product is then given by(4.3.1) again, for any f1, f2 in TgMΩ, hence for any (smooth) real functionsf1, f2 on M satisfying (3.1.3).

The above riemannian metrics on Mω0 andMΩ has been first introducedand worked out by T. Mabuchi in [142]. They will therefore be referred toas the Mabuchi metrics.

Notice that Mω0 , equipped with its Mabuchi metric, is a riemannianproduct of MΩ, equipped with its Mabuchi metric, with R, equipped withits standard metric, whereas π :MΩ →MΩ is a riemannian submersion.

The Mabuchi metric and the Poisson bracket defined in Section 1.2 arelinked together by:

killinglikekillinglike (4.3.2) 〈f3, f1, f2〉φ + 〈f1, f3, f2〉φ = 0,

for each φ in Mω0 and for any f1, f2, f3 in C∞(M,R), viewed as elements

of TφMω0 , where the Poisson bracket is relative to the symplectic formω = ω0 + ddcφ. This identity readily follows from the following expressionsof the Poisson bracket in the Kahler context:

poissonbispoissonbis (4.3.3) f1, f2 = Λ(df1 ∧ df2) = δ(f1dcf2) = −δc(f1df2).

These in turn are easily checked by using the Kahler identities (1.14.1).

4.4. THE LEVI-CIVITA CONNECTION OF THE MABUCHI METRIC 97

4.4. The Levi-Civita connection of the Mabuchi metricssLCmabuchi

In an infinite dimensional context, there is a priori no guarantee a thata (weak) riemannian metric should admit a Levi-Civita connection, as theusual Koszul formula only provides an element of the dual of the tangentspace, which in general cannot be identified with an element of the tangentspace itself. In the present situation however, a Levi-Civita connection forthe Mabuchi metric exists on Mω0 , then onMΩ = I−1

ω0(0), denoted by D in

both cases, according to the following proposition.

Proposition 4.4.1. The Mabuchi metric on Mω0 admits a unique Levi-Civita connection, D, defined at φ by

mabuchi-connectionmabuchi-connection (4.4.1) Dfξ = ξφ(f)− (dξ, df)g,

for any vector field ξ on Mω0 and any f in TφMω0, where g = π(φ) denotes

the associated metric in MΩ, and ξφ(f) = ddt |t=0

ξφ+tf denotes the variation

of ξ along f at φ.

Proof. We first show that a linear connection on Mω0 which is torsion-free and which preserves the Mabuchi metric is necessarily given by (4.4.1).For that, we fist consider the case when ξ is a constant vector field, thatis to say ξ(φ) = f2 for any φ in Mω0 , where f2 is a fixed fonction on M .

Constant vector fields on Mω0 pairwise commute and the Koszul formula,applied to the constant vector fields f1, f2, f3, then reduces to

2〈Df1f2, f3〉 = f1 · 〈f2, f3〉+ f2 · 〈f1, f3〉 − f3 · 〈f1, f2〉.

By using (3.2.4), we get

2〈Df1f2, f3〉 = −〈f2f3,∆f1〉 − 〈f1f3,∆f2〉+ 〈f1f2,∆f3〉= 〈∆(f1f2)− f1∆f2 − f2∆f1, f3〉= −2〈(df1, df2), f3〉.

In the case when ξ is a constant vector field, Dfξ is then necessarily of theform

(4.4.2) Dfξ = −(dξ, df)g.

For any vector field ξ, and any constant vector field f , f3 on Mω0 , we thenhave (since D should preserve the Mabuchi metric):

〈Dfξ, f3〉 = f · 〈ξ, f3〉 − 〈ξ,Dff3〉

= f ·∫Mξ(φ)f3 vg +

∫Mξ(φ)f(df, df3)g vg

=

∫Mξφ(f)f3 vg −

∫Mξ(φ)f3∆g(f) vg +

∫Mδg(ξ(φ)df)f3 vg

= 〈ξφ(f)f3 − (dξ(φ), df)gf3〉.

This shows that D is necessarily given by (4.4.1). Conversely, it is easy to

check that the rhs of (4.4.1) defines a linear connection on Mω0 , which istorsion-free and preserves the Mabuchi metric.


The induced connection on MΩ, still denoted by D, is also given by(4.4.1) for any vector field ξ on MΩ. Notice that, ξ is then an assignmentg 7→ ξ(g), which associates ξ(g) in C∞0,g(M,R) to any g in MΩ and that, by

(3.2.4), the constraint∫M ξ(g)vg = 0, for any g in MΩ, implies that the rhs

of (4.4.1) obeys the same constraint.The curvature of D is easily deduced from (4.4.1), cf. Proposition 4.9.1

below.

4.5. Group action of H(M,J) on MΩssaction

The (right) action of the group H(M,J) on MΩ defined in Section 3.1clearly preserves the Mabuchi metric. For any X in h the induced vectorfield X on MΩ is then a Killing vector field for the Mabuchi metric andwe therefore expect that the covariant derivative DX be skew-symmetric.Recall that X is given by g ∈ MΩ 7→ fXg , where fXg denotes the real

potential of X with respect to g. In order to compute DX we thus need toknow the variation of fXg and this is given by the following lemma:

lemmatilde Lemma 4.5.1. Let X be any vector field in h. Let g, g be any two metricsin MΩ, of Kahler forms ω and ω = ω + ddcφ. Let ξg, ξg be the dual 1-forms of X with respect to g, g respectively, and ξg = ξH + dfXg + dchXg ,

ξg = ξH +dfXg +dchXg their Hodge decomposition, according toLemma 2.1.1.Then

varxivarxi (4.5.1) ξH = ξH , fXg = fXg + LXφ, hXg = hXg − LJXφ.

In particular, the derivative of fXg and hXg along φ in TgMΩ is given by:

varfhvarfh (4.5.2) ˙fXg = LX φ, ˙hXg = −LJX φ.

Equivalently, if FXg = fXg + ihXg ,

varFvarF (4.5.3) ˙FXg = 2LX1,0F,

where X1,0 = 12(X − iJX) denotes the (1, 0)-part of X.

Proof. We haveξg = −JXyω = −JXyω − JXyddcφ

= ξg − LJXdcφ+ dLXφ= ξg + dLXφ− dc(LJXφ).

The Hodge decomposition of ξg then reads

hodgetildehodgetilde (4.5.4) ξ = ξH + d(fXg + LXφ) + dc(hXg − LJXφ).

It follows that ξH = ξH — this fits with the fact that the space harm ofharmonic 1-forms is independent of g, cf. Section 2.3 — fXg = fXg +LXφ+c1,

hXg = hXg −LJXφ+ c2, where c1 = c1(X, g, g), c2 = c2(X, g, g) are constants

on M , depending on X, g, g, determined by∫M fXg vg = 0,

∫M hXg vg = 0.

By (3.2.4), the first variation of c1 along φ in TgMΩ. when g and X arefixed, is given by

(4.5.5) c1 =1

VΩ

∫M

(fXg ∆φ− LX φ

)vg)

=1

VΩ〈φ,∆fXg − δξg〉 = 0,

4.6. THE EQUATION OF GEODESICS 99

meaning that c1(X, g, g), as a function of g in MΩ, is constant; sincec1(X, g, g) = 0, c1 is then 0 everywhere. We similarly show that c2 = 0.We thus get (4.5.1); (4.5.2) follows immediately, as well as (4.5.3).

propcovX Proposition 4.5.1. For any X in h and any g in MΩ, the covariantderivative DX at g is given by

covhatXcovhatX (4.5.6) Df X = (df, ξg − dfXg )g = −δg(f(ξg − dfXg )),

for any f in TgMΩ, where ξg stands for the dual of X with respect to g.

In particular, DX vanishes at g if and only if JX is a hamiltonian Killingvector field with respect to g. Moreover, for any two f1, f2 in TgMΩ, wehave that

symsym (4.5.7) 〈Df1X, f2〉+ 〈Df2X, f1〉 = 0.

Proof. For convenience, we compute DX on Mω0 . From (4.4.1) and(4.5.2), we infer

Df X = f · fXg − (df, dfXg )

= LXf − (df, dfXg )g

= (df, ξg − dfXg )g

= −δg(f(ξg − dfXg ))

which is (4.5.6) (for the last equality, we used the fact that ξg − dfXg =

ξH +dchXg is co-closed). From the above, we deduce that DX = 0 at g if and

only if ξg − dfXg = 0, hence X = gradgfXg , meaning that JX = gradωf

Xg is

both hamiltonian and (real) holomorphic, hence a hamiltonian Killing vectorfield with respect to g. Identity (4.5.7) follows readily from (4.5.6).

4.6. The equation of geodesicsssgeodesic

Proposition 4.6.1. Let φ(t), t ∈ [0, T ], be a (smooth) curve in Mω0,for some positive real number T (or T = +∞). Then φ(t) is a geodesic withrespect to the Levi-Civita connection D if and only if φ(t) satisfies

geodesicgeodesic (4.6.1) φ(t)− |dφ(t)|2t = 0,

for all t in [0, T ], where |dφ(t)|2t denotes the square norm of dφ(t) withrespect to the metric g(t) = π(φ(t)) for each (fixed) t. Similarly, a curveω(t) = ω0 + ddcφ(t) in MΩ, with Iω0(φ(t)) = 0 for any t in [0, T ], is ageodesic for D if and only if 4.6.1) is satisfied.

Proof. The tangent vector field to the curve φ(t) is ξ(t) := φ(t) at

φ(t) and φ(t) is a geodesic if and only if Dφ(t)φ(t) = 0 for each t in [0, T ].

In the rhs of (4.4.1), the term ξ(φ(t)) is φ(t), so that Dφ(t)φ(t) = φ(t) −(dφ(t), dφ(t))g(t). The curve φ(t) is then a geodesic if and only if (4.6.1) issatisfied. This proves the first assertion. Since the Levi-Civita connectionon MΩ coincides with the restriction of D on I−1

ω0(0), the second assetion

follows.


Remark 4.6.1. For any curve φ(t) in Mω0 as above, the first deriva-

tive of Iω0(φ(t)) is ddtIω0(φ(t)) = τφ(t)(φ(t)) = 1

VΩ

∫M φ(t) vg(t). Its second

derivative is then

d2

dt2Iω0(φ(t)) =

1

VΩ

∫M

(φ(t)− φ(t)∆g(t)φ(t)) vφ(t)

=1

VΩ

∫M

(φ(t)− |dφ(t)|2t ) vg(t).(4.6.2)

If φ(t) is a geodesic, Iω0(φ(t)) is then an affine function of t. In particular,

Iω0(φ(t)) ≡ 0, whenever φ(t) is a geodesic, φ(0) belongs to I−1ω0

(0) and φ(0)

belongs to Tφ(0)I−1ω0

(0). This confirms that I−1ω0

(0) is totally geodesic in Mω0 .

The equation of geodesics admits various alternative formulations. Oneof them relies on the following general construction. For any chosen base-point ω0 and any element ω = ω0 +ddcφ inMΩ let c(t) = ωt = ω+ddcφt beany curve in MΩ joining ω0 to ω = ω1, determined by the curve c(t) = φtin I−1

ω0(0) ⊂ Mω0 . Denote by Xt the family of vector fields defined by

XtgradXtgrad (4.6.3) Xt = −gradgt φ(t),

where gt stands for the riemannian metric associated to ωt. We denote by Φt

the corresponding “flow”, i.e. the 1-parameter familly of diffeomorphismsof M determined by

d

dtΦt(x) = Xt(Φt(x)),

Φ0 = Id,flowXtflowXt (4.6.4)

for each t in [0, 1] and each x in M (where Id stands for the identity map). IfZ := Xt+∂/∂t is viewed as a vector field on the product M×[0, 1], and if ΦZ

denotes the flow of Z, then Φt is related to ΦZ by: ΦZt ((x, 0)) = (Φt(x), t).

Notice that Φt is not a flow in the usual sense; in particular, Φt1+t2 6=Φt1 Φt2 and Φ−t 6= Φ−1

t in general. We have instead

lemmaflow Lemma 4.6.1. Let Φt the flow of Xt defined by (4.6.4). Then:

(i) Φ−1t is the flow of the 1-parameter vector field −Φ−1

t ·Xt, where, foreach t, Φ−1

t ·Xt denotes the direct image of the vector field Xt, cf. Section1.4.

(ii) For any tensor field T and for any fixed t, the Lie derivative LXtTof T along the vector field Xt is given by:

fakeliederivative1fakeliederivative1 (4.6.5) LXtT = Φt ·d

dt(Φ−1

t · T ).

Proof. (i) From ddt(Φ

−1t Φt) = 0, we infer: d

dtΦ−1t (Φt(x)) = −(Φ−1

t )∗(Xt(Φt(x))) =

−(Φ−1t ·Xt)(x), for any x in M . By substituting Φ−1

t (x), we get ddtΦ−1t (x) =

−(Φ−1t ·Xt)(Φ

−1t (x)), as required.

(ii) The derivative ddt(Φ

−1t ·T ) at t is better written as as d

ds |s=0Φ−1t+s ·T .

From (i), we infer dds |s=0

(Φt Φ−1t+s) = −Xt for each t. From the definition

(1.4.1) of the Lie derivative, we then get: dds |s=0

Φ−1t+s · T = d

ds |s=0(Φ−1

t Φt Φ−1t+s) · T = Φ−1

t · dds |s=0(Φt Φ−1

t+s) · T = Φ−1t · LXtT , which is (4.6.5).

4.6. THE EQUATION OF GEODESICS 101

Since Φ∗tωt = Φ−1t · ωt, from (4.6.5), we infer

d

dt(Φ∗tωt) = Φ∗t (LXtωt + ωt)

= Φ∗t (d(Xtyωt) + ωt)

= Φ∗t (−ddcφt + ddcφt) = 0.

flowtflowt (4.6.6)

We thus get

mosermoser (4.6.7) Φ∗tωt = ω0,

for each value of t in [0, 1]. This can be viewed as a special and simple case ofthe Moser lemma, cf. [74]. Accordingly, the curve t 7→ Φt will be called theMoser lift determined by c in the identity component Diff0M of the groupof diffeomorphisms of M (more on the Moser lift in Section 4.7.)

By using the Moser lift, we get the following characterization of geodesicsin MΩ or Mω0 :

propgeodesic2 Proposition 4.6.2. For any curve ωt = ω0 + ddcφt, t ∈ [0, 1], in MΩ,denote by Φt the corresponding Moser lift in Diff0(M). Then, ωt is a geodesicif and only if

geodesic2geodesic2 (4.6.8) Φ∗t φt = φ0,

for each t in [0, 1]. Equivalently,

geodesic3geodesic3 (4.6.9) φt =

∫ t

0(φ0 Φ−1

s ) ds.

Proof. Assume that ωt is a geodesic in MΩ. Then:

moserphimoserphi (4.6.10)d

dt(Φ∗t φt) = Φ∗t (LXt φt + φt) = Φ∗t (−(dφt, dφt)gt + φt) = 0.

We then get (4.6.8), hence (4.6.9). Conversely, by differentiating both sides

of (4.6.8) we get 0 = Φ∗t (φt + LXt φt), with Xt defined by (4.6.3), hence

φt − (dφt, dφt)gt = 0, where gt denotes the metric associated to ωt.

propmohsen1 Proposition 4.6.3. (i) Let X be any (real ) holomorphic vector field inhred of the form X = gradg0

fXg0, for some metric g0 in MΩ, of Kahler form

ω0, and for some real function fXg0satisfying

∫M fXg0

vg0 = 0. Denote by ΦXt

the flow of X. Then, the curve (gt, ωt) inMΩ defined by (gt = (ΦXt )∗g0, ωt =

(ΦXt )∗ω0) is a geodesic. Moreover, if ωt is written as ωt = ω0 + ddcφt, with

Iω0(φt) = 0, then X = gradgt φt, so that φt is a Killing potential with respect

to gt for any t, and Φt = ΦX−t is then the Moser lift of the geodesic (gt, ωt).

(ii) Conversely, let (gt, ωt = ω0 + ddcφt) be a (smooth) curve in MΩ

starting from ω0, such that φt is a Killing potential with respect to gt foreach t. Then this curve is contained in the H0(M,J)-orbit of ω0. Moreover,this curve is a geodesic in MΩ if and only if the 1-parameter (real) vector

field gradgt φt is independent of t. If so, gradgt φt is a well-defined element

X in hred(M,J) and we have ωt = (ΦXt )∗ω0, where ΦX

t denotes the flow ofX.

Proof. (i) First notice that ωt = Φ∗tω0 lives in MΩ for each t, asΦt belongs to Hred(M,J), hence preserves the complex structure J and


the Kahler class Ω. We then have ωt = ω0 + ddcφt, where, we assume,Iω0(φt) = 0, hence ωt = ddcφt, with

∫M φt vgt = 0. From ωt = Φ∗tω0,

we infer ωt = Φ∗t (LXω0) = Φ∗tddcfXg0

= ddc(Φ∗t f

Xg0

), with

∫M Φ∗t f

Xg0vgt =∫

M Φ∗t(fXg0

vg0

)=∫M fXg0

vg0 = 0. We thus get: φt = Φ∗t fXg0

for any t. On

the other hand, X = Φ−1t ·X = Φ−1

t · gradg0fXg0

= gradgt(Φ∗t f

X0 ), for any t;

we infer fXgt = Φ∗t fXg0

= φt, so that X = gradgt φt for each t; this means that

Φ−1t is the Moser lift of the curve ωt, and Φ∗t φ0 = φt. By Proposition 4.6.2,

this implies that the curve ωt is a geodesic.(ii) Define Xt = −gradgt φt and let Φt be the “flow” of the 1-parameter

vector field Xt, as defined above, so that Φ∗tωt = ω0 for any t, see (4.6.7).

Since φt is a Killing potential, Xt is a (real) holomorphic vector field in hred,by Proposition 2.2.1. By using (4.6.5), we infer that Φ−1

t ·J is constant alongthe curve, hence that Φ−1

t · J = J , meaning that Φ−1t belongs to H(M) for

any t. Moreover, since ddtΦt Φ−1

t = Xt belongs to hred, we conclude thatΦt actually belongs to Hred(M) for each t.

By using (3.3.4) in the proof of Proposition 3.3.1, we get Xt = ˙gradgt φt+

gradgt φt =(JXtyddcφt

)]gt +gradgt φt =(LJXtdcφt

)]gt −gradgt(JXtydcφt)+

gradgt φ. Since Xt, hence also JXt, belongs to hred, we have LJXtdcφt =

dc(dφt(JXt)

)= 0, as dφt(JXt) = (dφt, Jdφt)gt = 0. We thus eventually

get: Xt = gradgt(− (dφt, dφt)gt + φt

), meaning that Xt = 0 if and only

if the curve ωt = ω0 + ddcφt is a geodesic in MΩ. If so, X = gradgt φtis a well-defined vector field in hred(M,J), whose flow is ΦX

t = Φ−1t , and

ωt = (ΦXt )∗ω0 for each t.

Remark 4.6.2. Assume that (M,J) is a (compact) Riemann surface,i.e. that m = 1. Fix any Kahler form ω0, corresponding to a Kahler metricg0 (note that ω0 coincides with the volume form vg), and denote by Ω = [ω0]its Kahler class. Any curve in MΩ can be written as:

(4.6.11) ωt = ω0 + ddcφt = (1−∆φt)ω0.

On the other hand, for any (real) function φ, we have that |dφ|2gtωt =

|dφ|2g0ω0 = dφ ∧ dcφ, and similarly ddcφ = −∆gtφωt = −∆g0φω0, for each

value of t. The geodesic equation can then be written in the following form:

geodesicm1geodesicm1 (4.6.12) φt (1−∆g0φt)− |dφt|2g0= 0,

where the laplacian and the norm are relative to the fixed chosen metric g0.

4.7. The Moser lift as a horizontal liftssmoser

As observed by S. Donaldson in [74], part of the discussion in Section 4.6can be reinterpreted in terms of a connection on a Sp0(M,ω0)-principal bun-dle over MΩ as follows, where Sp0(M,ω0) denotes the identity componentof the group of symplectomorphisms of (M,ω0).

Denote by Q the space of pairs (Φ, ω) in Diff0(M)×MΩ such that

QQ (4.7.1) ω = Φ · ω0.

4.7. THE MOSER LIFT AS A HORIZONTAL LIFT 103

The natural projection to MΩ then makes Q into a Sp0(M,ω0)-principalbundle overMΩ, via the (right) action (Φ, ω) · γ = (Φ γ, ω), for any (Φ, ω)in Q and any γ in Sp0(M,ω0).

Fix (Φ, ω) in Q and let (Φt, ω(t)) be any curve in Q, with Φ0 = Φand ω(0) = ω. Then, ω(t) = Φt · ω0 = ΦtΦ

−1 · ω. It follows that ω(0) =−LXω = −d(ιXω), whereas ω(0) = ddcf , whith

∫M f vg = 0, where, as

usual, g = ω(·, J ·): f is then ω(0) as an element of TgMΩ according to ourusual identification. The tangent space T(Φ,ω)Q is then naturally identifiedwith the space of pairs (X, f), where X is a (real) vector field and f a (real)function, with

∫M f vg = 0, such that the (real) 1-form ιXω + dcf is closed.

Recall that the Lie algebra sp(M,ω0) of Sp0(M,ω0) is the space of (real)vector fields Z on M such that LZω0 = 0; equivalently, the ω0-symplecticdual 1-form ιXω0 is closed.

Let Z be any element of sp(M,ω0), generated by a curve γt in Sp0(M,ω0),

with γ0 = 1 (e.g. γt = ΦZt , the flow of Z). The induced vertical vector Z(Φ,ω)

on Q at (Φ, ω), tangent to the curve (Φt = Φ γt, ω) at t = 0, is then iden-tified with the pair (X = Φ · Z, 0).

In view of this, we define a Sp0(M,ω0)-equivariant connection 1-form,η, on Q in the following way: for any (X, f) in T(Φ,ω)Q, η((X, f)) is theω0-symplectic dual vector field of the (closed) 1-form Φ∗(ιXω + dcf).

We readily check that η(Z(Φ,ω)) = Z for any Z in sp(M,ω0) and that ηis Sp0(M,ω0)-equivariant as required.

lemmalift Lemma 4.7.1. For any (g, ω) in MΩ, for any f in TgMω and for any(Φ, ω) in Q over (g, ω), the horizontal lift of f in T(Φ,ω)Q is (−gradgf, f).

Proof. Any lift of f in T(Φ,ω)Q is of the form (X, f), with ιXω +dcf closed. Such a lift is horizontal if η((X, f)) = 0. By the above thismeans that Φ∗(ιXω + dcf) = 0, hence ιXω + dcf = 0; equivalently, X =−gradgf .

Proposition 4.7.1. For any curve ωt = ω0 +ddcφt inMΩ starting fromω0, a curve Φt in Diff0(M), with Φ0 = Id, is the Moser curve of ωt if andonly if (Φt, ωt) is a horizontal lift of ωt in Q with respect to the connectiondefined by η.

Proof. Let (Φt, ωt) be any lift of ωt in Q, with Φ0 = Id. By definitionof Q, Φ∗tωt = ω0 for all t. Moreover, the tangent vector field of the curve

(Φt, ωt) is the pair (Xt, φt), where Xt is the time dependent vector field

defined by (4.6.4). By Lemma 4.7.1, (Xt, φt) is horizontal if and only if

Xt = −gradgt φt: this is the way the Moser lift was defined.

Devote by C∞(M,ω0) the space of (smooth) real functions on M suchthat

∫M f vg0 = 0 (C∞(M,ω0) is then the tangent spave Tω0MΩ). We

regard C∞(M,ω0) as a representation space of Sp0(M,ω0) and we considerthe induced vector bundle Q×Sp0(M,ω0) C

∞(M,ω0) over MΩ.

prop-identification Proposition 4.7.2. (i) We have a natural identification of vector bun-dles:

identification1identification1 (4.7.2) TMΩ = Q×Sp0(M,ω0) C∞(M,ω0)


given by

identification2identification2 (4.7.3) [(Φ, ω), f ] 7→ Φ · f.(ii) In this identification, the linear connection, Dη, on TMΩ induced

by η coincides with the Levi-Civita connection D of the Mabuchi metric.

Proof. (i) To any element of Q ×Sp0(M,ω0) C∞(M,ω0) represente by

((Φ, ω), f) in Q×C∞(M,ω0) we associate Φ ·f in TωMΩ (notice that∫M (Φ ·

f) vg =∫M Φ · (f vg0) =

∫M f vg0 = 0); if (Φ γ, ω), γ−1 · f) is another

representative, with γ in Sp(M,ω0), we clearly get the same element ofTωMΩ.

(ii) It suffices to check that Dηf1f2 = Df1 f2 is satisfied for any f1 in

TgMΩ and for any “constant” vector field f2 = f2 − 1VΩ

∫M f2vg. Without

loss of generality, we can choose g = g0. Via (4.7.2), f2 is representedby ((Φ, ω),Φ−1 · f2 − 1

VΩ

∫M f2

ωm

m! ), whereas f1 is determined by the curve

ωt = ω0 + ddcφt, with φ0 = f1. By choosing (Φt, ωt) the horizontal lift ofωt in Q starting from (Φ0 = Id, ω0), the covariant derivative with respect to

the connection η of f2 is simply given by

(4.7.4) Dηf1f2 = ((Id, ω0),

d

dt |t=0(Φ−1

t · f2 −1

VΩ

∫Mf2ωmtm!

).

By using dΦtdt |t=0

= −gradg0f1, given by Lemma 4.7.1, and (3.2.4), we thus

get(4.7.5)

Dηf1f2 = ((Id, ω0),−(gradg0

f1, gradg0f2) +

1

VΩ

∫M

(gradg0f1, gradg0

f2) vg0).

This is Df1 f2, as given by (4.4.1), in the identification (4.7.2).

Remark 4.7.1. Let ωt = ω0 + ddcφt be a curve in MΩ starting fromω0. Let Φt the corresponding Moser lift in Diff0(M), so that (Φt, ωt) is ahorizontal lift of ωt in Q by Lemma 4.7.1. The tangent vector field alongthe curve ωt is φt. By (4.7.3), this corresponds to the curve [(Φt, ωt),Φ

∗t φt]

in Q ×Sp0(M,ω0) C∞(M,ω0). Since (Φt, ωt) is horizontal in Q, this curve is

η-horizontal in Q ×Sp0(M,ω0) C∞(M,ω0) for the induced connection if and

only if Φ∗t φt is constant, equal to φ0. By using Proposition 4.7.2, we thusrecover the characterization of geodesics of the Mabuchi spaceMΩ given byProposition 4.6.2.

4.8. The geodesic equation as a Monge-Ampere equationsssemmes

As shown by S. Semmes in [174], the geodesic equation (4.6.1) can bewritten as a (degenerate) Monge-Ampere equation in the following way. Let

φt be a geodesic in Mω0 , where t runs from 0 to T . Then, φt can beviewed as a real function defined on the product M × [0, T ]. By introducinga new real variable s, we define a function Φ = Φ(x, t, s) on the product

M := M × [0, T ] × S1 by putting Φ(x, t, s) = φt(x), for any x in M . Theadditional factor Σ := [0, T ]×S1 is then made into a Riemann surface via thecomplex structure JΣ defined by JΣ∂/∂t = ∂/∂s, or dct = ds. By combining

JΣ with J we get a complex structure on M , whose twisted differential is

4.8. THE GEODESIC EQUATION AS A MONGE-AMPERE EQUATION 105

still denoted by dc. Similarly, ω0 is viewed a 2-form defined on M (but of

course ω0 is no longer a symplectic form on M , as ωm+10 = 0). We then have

[174]:

propsemmes Proposition 4.8.1. A curve φt in Mω0 satisfies the geodesic equation(4.6.1) if and only if Φ satisfies the following Monge-Ampere equation:

semmessemmes (4.8.1) (ω0 + ddcΦ)m+1 = 0,

on the complex manifold M .

Proof. We readily get dΦ = dφt + φtdt, hence dcΦ = dcφt + φtds and

(4.8.2) ddcΦ = ddcφt − dcφt ∧ dt+ dφt ∧ ds+ φtdt ∧ ds.

We then have ω0 + ddcΦ = ωt + ψt, with ωt = ω0 + ddcφt and ψt = −dcφt ∧dt+ dφt ∧ ds+ φtdt ∧ ds. It follows that

(ω0 + ddcΦ)m+1 = (m+ 1)(φt ωmt −mωm−1

t ∧ dφt ∧ dcφt) ∧ dt ∧ ds

= (m+ 1)(φt − (dφt, dφt)t)ωmt ∧ dt ∧ ds.

(4.8.3)

The above computation is performed for S1-invariant solutions Φ of(4.8.1). More generally, we have (cf. [74] Formula (29)):

propsemmes-2 Proposition 4.8.2. Let Φ be a (smooth) real function on M × [0, T ]×S1, viewed as a family ϕt,s of functions on M , parametrized by (t, s) in[0, T ]× S1. Assume that ϕt,s is the relative Kahler potential with respect toω0 of an element (gt,s, ωt,s) of MΩ for any (t, s) in [0, t]× S1. Then Φ is asolution to (4.8.1) if and only if the family ϕt,s satisfies:

semmes-2semmes-2 (4.8.4) ϕtt + ϕss − |dϕt − dcϕs|2gt,s = 0,

where ϕt, ϕtt denote the first and second derivative of ϕ = ϕt,s with respectto t, ϕs, ϕss the first and second derivative of ϕ = ϕt,s with respect to s.

Proof. As in the proof of Proposition 4.8.1, we have: dcΦ = dcϕs,t +ϕt ds − ϕs dt, hence ddcΦ = ddcϕt,s − dcϕt ∧ dt − dcϕs ∧ ds + ϕttdt ∧ ds −ϕssds ∧ dt+ dϕt ∧ ds− dϕs ∧ dt, so that: ω0 + ddcΦ = ωt,s + ψ with

ψ = (ϕtt + ϕss) dt ∧ ds− (dcφt + dφs) ∧ dt+ (dφt − dcφs) ∧ ds.

In particular,

ψ2 = 2(dcφt + dφs) ∧ (dφt − dcφs) ∧ dt ∧ ds

and ψ3 = 0. It follows that

(ω0 + ddcΦ)m+1 = (ωt,s + ψ)m+1 = (m+ 1)ωmt,s ∧ ψ +(m+ 1)m

2ωm−1t,s ∧ ψ2

= (m+ 1)(ϕtt + ϕss − |dϕt − dcϕs|2)ωmt,s ∧ dt ∧ ds.

The above Monge-Ampere approach has proved a effective way of study-ing geodesics in Mω0 and MΩ, in particular in [55], [47], [78], [57], [59],[60], [56], cf. Section 4.15 below.


4.9. The curvature of the Mabuchi metricsscurvature

propcurvature Proposition 4.9.1. The curvature, RD, of the Levi-Civita connectionD of the Mabuchi metric on Mω0 is given at φ by

mabuchi-curvaturemabuchi-curvature (4.9.1) RDf1,f2f3 = −f1, f2ω, f3ω,

for any f1, f2, f3 in TφMω0 = C∞(M,R), where ·, ·ω denotes the Poissonbracket with respect to ω = ω0 + ddcφ. The same formula holds for thecurvature of D on MΩ for any (g, ω) in MΩ and any f1, f2, f3 in TgMΩ =C∞0,g(M,R).

Proof. According to the general definition of the curvature, cf. (1.18.1),RD has the following expression:

curvature-tempcurvature-temp (4.9.2) RDf1,f2f3 = −Df1(Df2f3) +Df2(Df1f3),

for any f1, f2, f3 in TφMΩ, regarded as constant vector fields on MΩ in therhs (with the effect that D[f1,f2]f3 ≡ 0.) From (4.4.1), we readily infer

DDf1DDf1 (4.9.3) −Df1(Df2f3) = f1 · (df2, df3)g − (df1, d((df2, df3)g))g,

with g = π(φ). By writing (df2, df3)g = Λg(df2∧dcf3) and by using (3.2.10),

we get f1 · (df2, df3)g = Λg(f1)(df2 ∧ dcf3) = −(ddcf1, df2 ∧ dcf3). In termsof the Levi-Civita connection D = Dg of g, this can be rewritten

f1 · (df2, df3)g = −(Dgradgf2dcf1)(Jgradgf3)

+ (DJgradgf3dcf1)(gradgf2)

= −(Dgradgf2df1)(gradgf3)

+ (DJgradgf3dcf1)(gradgf2),

(4.9.4)

whereas (df1, d((df2, df3)g))g can be rewritten(4.9.5)

(df1, d((df2, df3)g))g = (Dgradgf1df2)(gradgf3) + (Dgradgf1df3)(gradgf2).

Taking into account the symmetry of the hessian Ddf , most terms in therhs of (4.9.2) cancel, and (4.9.2) then becomes:

(4.9.6) RDf1,f2f3 = DJgradgf3d

cf1(gradgf2)−DJgradgf3dcf2(gradgf1).

In terms of symplectic gradients and of Poisson brackets with respect to theKahler form ω = ω0 + ddcφ, cf. (1.2.10), this can be rewritten

RDf1,f2f3 = −DJgradgf3df1(Jgradgf2) +DJgradgf3df2(Jgradgf1)

= −DJgradgf2df1(Jgradgf3) +DJgradgf1df2(Jgradgf3)

= −ω(Dgradωf1gradωf2 −Dgradωf2gradωf1, gradωf3)

= −ω([gradωf1, gradωf2], gradωf3)

= −ω(gradω(f1, f2ω), gradωf3)

= −f1, f2ω, f3ω,

(4.9.7)

which is (4.9.1). The second assertion follows immediately (notice that therhs of (4.9.1) only depends of f1, f2, f3 modulo constants and is of meanvalue 0 with respect to g, as are all Poisson brackets).

4.10. THE MABUCHI K-ENERGY 107

In particular, the sectional curvature of gΩ and gΩ is everywhere non-positive, [142], [74].

4.10. The Mabuchi K-energysssigma

For any g in MΩ, the difference sg − s, where, as usual, sg denotes the

scalar curvature of g and s = SΩVΩ

its mean value (the same for all g’s in

MΩ), can be viewed as an element of TgMω and the assignment g 7→ sg − scan therefore be regarded as a vector field on MΩ, denoted by s. Similarly,the assigment,

(4.10.1) g 7→ sgvg

or, indifferently, g 7→ (sg − s) vg, can be viewed as a 1-form on MΩ. This1-form will be denoted by σ and called the scalar curvature 1-form.

The vector field s and the 1-form σ can be considered as a riemanniandual pair on MΩ in the following (weak) sense:

dual-sigmadual-sigma (4.10.2) σφ(f) = 〈s, f〉g,

for each g in MΩ and each f in C∞0,g(M,R) = TgMΩ. In particular, theCalabi functional can be rewritten

CbisCbis (4.10.3) C(φ) = σg(s) +S2

Ω

VΩ.

Both σ and s0 are H(M,J)-invariant, as sγ∗g = γ∗(sg) and vγ∗g = γ∗(vg)for each riemannian metric g on M and each diffeomorphism γ of M .

A basic observation is that the the covariant derivative of σ with respectto D essentially coincides with the fourth order Lichnerowicz operator Lintroduced in Section 1.23 in the following precise sense:

lemmaDsigma Lemma 4.10.1. The covariant derivative of s and of σ with respect to Dare given by

DshatDshat (4.10.4) Df s = −2L(f),

and by

DsigmaDsigma (4.10.5) Dfσ = −2L(f) vg

for any metric g inMΩ and any f in TgMΩ, where L = δδD−d denotes thefourth order Lichnerowicz operator relative to g. In particular, σ is closed.

Proof. From (4.4.1) and (3.2.6), we readily infer

Df s = sf − (df, dsg) = −2L(f),

for any g in MΩ and any f in TgMΩ, which is (4.10.4) (recall that s isconstant on MΩ). From (4.10.2), we then obtain

(Df1σ)(f2) = 〈Df1 s, f2〉g = −2〈L(f1), f2〉g,

for any f1, f2 in MΩ, hence (4.10.5). Since L is self-adjoint, we infer(Df1σ)f2 = (Df2σ)f1, meaning that σ is closed.


SinceMΩ is contractible, σ is actually exact and we define the MabuchiK-energy, E, by

EdsigmaEdsigma (4.10.6) dσ = −dE,

hence by

EderEder (4.10.7) (dEω0)ω(f) = −∫Mf sgvg,

for any (g, ω) in MΩ and any f in TgMΩ (identified with the space of realsmooth functions on M of mean value zero with respect to vg). The MabuchiK-energy E defined that way is only determined up to an additive constant.For any chosen ω0 in MΩ, we then define Eω0 as the unique determinationof E which ivanishes at ω0. We then have:

mumu (4.10.8) Eω0(ω) = −∫ 1

0σ(c)dt,

for any ω in MΩ and any curve c : [0, 1] → MΩ connecting ω0 = c(0) toω = c(1).

remE Remark 4.10.1. In (4.10.7), f is assumed to be of mean value zero withrespect to vg. If, instead, TgMΩ is identified with the quotient C∞(M,R)/R,

in (4.10.7), sg must be replaced by sg− s, where s = SΩVΩ

is the mean value of

sg (the same for all elements ofMΩ). In the sequel, Eω0 will be occasionallycalled the K-energy centered at ω0.

By choosing the path c between ω0 and ω = ω0 +ddcφ equal to the linearinterpolation ωt = ω0 + tddcφ, we get the following expression for Eω0 :

EE (4.10.9) Eω0(φ) = −∫Mφ(

∫ 1

0(st − s) vtdt).

Notice that we must here integrate φ along∫ 1

0 (st − s) vtdt, and not along∫ 10 stvt, as the segment t 7→ tφ in Mω0 is never contained in I−1

ω0(0), unless

φ ≡ 0 (cf,. Remark 4.10.1 above). It readily follows from (4.10.5) that EΩ

is convex, meaning that its hessian with respect to D is everywhere non-negative. More precisely, at every point g ofMΩ and every non-zero vectorf in TgMΩ, we have that

hessianEhessianE (4.10.10) (DfdEω0)(f) = 2〈D−df,D−df〉g ≥ 0,

with equality if and only if f is the momentum of a hamiltonian Killingvector field, [142].

Moreover, since dEω0 = −σ, the critical points of Eω0 are exactly themetric of constant scalar curvature in MΩ.

Remark 4.10.2. Since s is the riemannian dual of σ, we also have

(4.10.11) Df s = −2δδD−df,

for each f in C∞(M,R) = TφMω0 . From (4.10.3), we then recover

(4.10.12) dCφ = −4δδD−dsg vg.

4.11. THE FUTAKI-MABUCHI BILINEAR FORM 109

4.11. The Futaki-Mabuchi bilinear formssfutakimabuchi

Let (M, g, J, ω) be any (connected) compact Kahler manifold. Recall— cf. Chapter 2 — that any element X of the Lie algebre h of (real)holomorphic vector fields on M can be written as X = XH + grad fXg +

JgradhXg , where XH is the dual of a g-harmonic 1-form and where fXg , hXg

are real functions normalized by∫M fXg vg =

∫M hXg vg = 0, and that the

complex function FXg := fXg + ihXg has been called the complex potential

of X with respect to g (we then have F JXg = iFXg ). Also recall that thesubspace of h defined by the condition XH ≡ 0 is actually a Lie subalgebraof h denoted by hred, independent of g, cf. Section 2.4.

propmabuchi Proposition 4.11.1 (A. Futaki-T. Mabuchi [88]). Let X1, . . . , Xr be anyr elements of hred and denote by FX1

g , . . . , FXrg the corresponding complex po-

tentials with respect to some metric g of MΩ, normalized by∫M FXkg vg = 0,

k = 1, . . . , r. For each k = 1, . . . , r, let Φk be any holomorphic function de-fined on some open subset of C containing the range of FXkg , so that Φk(F

Xkg )

is a well-defined complex function on M . Then

(4.11.1) Φ(X1,Φ1),...,(Xr,Φr) :=

∫M

Φ1(FX1g ) . . .Φr(F

Xrg )vg

is independent of the choice of g in MΩ.

Proof. Let ωt = ω0 +ddcφt be any curve inMΩ. By Lemma 3.2.1 andLemma 4.5.1, the corresponding variation of Φt := Φ(X1,Φ1),...,(Xr,Φr)(gt) isgiven by

dΦt

dt=

∫M

r∑k=1

Φ1(FX1gt ) . . .Φ′k(F

Xkgt ) . . .Φr(F

Xrgt )FXkgt vgt

−∫M

Φ1(FX1gt ) . . .Φr(F

Xrgt )∆gt φt)vgt

= 2

∫M

r∑k=1

Φ1(FX1gt ) . . .Φ′k(F

Xkgt ) . . .Φr(F

Xrgt )L

X1,0kφtvgt

−∫M

Φ1(FX1gt ) . . .Φr(F

Xrgt )∆gt φt)vgt

= 2〈r∑

k=1

Φ1(FX1gt ) . . .Φ′k(F

Xkgt ) . . .Φr(F

Xrgt )∂FXkgt , ∂φt〉

− 〈Φ1(FX1gt ) . . .Φr(F

Xrgt ),∆gt φt〉

= 2〈∂(Φ1(FX1gt ) . . .Φr(F

Xrgt )), ∂φt〉 − 〈Φ1(FX1

gt ) . . .Φr(FXrgt ),∆gt φt〉

= 〈Φ1(FX1gt ) . . .Φr(F

Xrgt ), (2−∆)φt〉 = 0

by Proposition 1.15.1 (here, 〈·, ·〉 denotes the (global) hermitian inner prod-

uct and we have used the fact that the hermitian dual of X1,0k is equal to

∂FXkgt — cf. Remark 2.5.1 — so that LX1,0kφt = (∂FXkgt , ∂φt), where, again,

(·, ·) denotes the (pointwise) hermitian inner product, cf. Section 1.3).


defiFMbilinear Definition 6. The Futaki-Mabuchi bilinear form relative to Ω is theC-bilinear form BΩ defined on hred by

mabuchiform1mabuchiform1 (4.11.2) BΩ(X1, X2) =

∫MF1F2 vg,

for any two elements X1, X2 of hred, of complex potentials F1 and F2 re-spectively with respect to some metric g in MΩ (normalized by (3.1.3)).

By Proposition 4.11.1, this expression only depends on X1, X2 and Ω,not upon the choice of g in MΩ, cf. [87].

The real part of BΩ will be denoted by BΩ: we thus have

BΩ(X1, X2) =

∫M<(F1F2) vg

= 〈f1, f2〉 − 〈h1, h2〉.mabuchiform2mabuchiform2 (4.11.3)

Notice that

(4.11.4) BΩ(JX1, JX2) = −BΩ(X1, X2).

For any choice of g inMΩ, kham and Jk0 are BΩ-orthogonal, the restric-tion of BΩ to kham is negative definite and its restriction to Jk0 is positivedefinite; in particular, the restriction of BΩ to the sum kham ⊕ Jk0 is non-degenerate. Notice that BΩ is H(M) invariant:

(4.11.5) BΩ(γ∗X1, γ∗X2) = BΩ(X1, X2),

for any γ in H(M); this is because, if ξ = ξ0 + df + dch is the Hodgedecomposition of the dual of X with respect to some g in MΩ, then theHodge decomposition of γ∗ξ with respect to γ∗g — which still belongs toMΩ — is γ∗ξ = γ∗ξ0 + d(f γ) + dc(h γ). In particular, we have that:

(4.11.6) BΩ([X1, X3], X2) +BΩ(X1, [X2, X3]) = 0,

for any X1, X2 in h0 and any X3 in h.For more information concerning BΩ in particular its link with the

Killing form of h0, see [87].

Remark 4.11.1. For any X1, X2 in hred, BΩ(X1, X2) can be rewrittenas

futakimabuchihatfutakimabuchihat (4.11.7) BΩ(X1, X2) = (X1, X2)g − ( ˆJX1, ˆJX2)g,

for any g in MΩ, where, we recall, X1, X2 etc. denote the induced vectorfields onMΩ — given by (3.1.6) — and (X1, X2)g etc. denotes the Mabuchiinner product at g defined by (4.3.1). The fact that the rhs of (4.11.7)is actually independent of g — which has been previously deduced fromProposition 4.11.1 — is then a straightforward consequence of the expression(4.5.6) of the covariant derivative of X1 etc. Indeed, for any X in hred, (4.5.6)can be simply read as

(4.11.8) Df X = −δg(fdchXg ),

whereas the derivative of the rhs of (4.11.7) at g along any f in TgMΩ is

equal to 〈Df X1, X2〉 + 〈X1,Df X2〉 − 〈Df ˆJX1, ˆJX2〉 − 〈 ˆJX1,Df ˆJX2〉; thiscan then be rewritten as

−〈f, (dchX1g , dfX2

g ) + (dchX2g , dfX1

g ) + (dcfX1g , dhX2

g ) + (dcfX2g , dhX1

g )〉,

4.12. THE FUTAKI CHARACTER 111

which is evidently equal to zero.

4.12. The Futaki characterssfutaki

Recall that for any fixed X in h, the assignment g 7→ fXg , where fXg maybe considered as a vector field on the Frechet manifold MΩ, and that thisvector field is equal to −X, where X denotes the vector field correspondingto the infinitesimal action of H(M,J) onMΩ, see Section 3.1. For any suchfixed vector field X in h, we then define FX :MΩ → R by

futakifutaki (4.12.1) FX(g) =

∫MfXg sgvg,

where sg denotes the scalar curvature, vg the volume form of g.By considering the 1-form σ onMΩ introduced in Section 4.10, this can

be rewritten as

futaki2futaki2 (4.12.2) FX(g) = σg(X).

propfutaki Proposition 4.12.1. For any X in h, FX is constant on MΩ.

Proof. We denote by a dot the directional derivative at g along φ ofFX and of all quantities appearing in the expression of FX ; sg and vg are

given by lemma 3.2.1, whereas, by lemma 4.5.1, fXg = LX φ; we thus obtain

FX =

∫M

(fXg sg + fXg s− fXg sg∆φ)vg

=

∫M

(sgLX φ− 2fXg δδD−dφ+ fXg (dφ, dsg)− fXg sg∆φ)vg,

where we have used (1.23.14). By integrating by parts, we get:∫M (fXg (dφ, dsg)−

fXg sg∆φ)vg = −∫M sg (dfXg , dφ)vg; we thus obtain:

FX =

∫M

(−2fXg δδD−dφ+ sg (dφ, ξ − df))vg,

or else, since ξ − df = ξH + dch is co-closed and L = δδD−d is self-adjoint:

(4.12.3) FX = −2〈φ,L(fXg ) +1

2(dsg, ξ − dfXg )〉 = 0,

by (2.6.4).

remjpb Remark 4.12.1. An alternative proof of Proposition 4.12.1, due to J.-P.Bourguignon [39] — cf. also [22] — directly relies on the geometry of MΩ

and the expression (4.12.2) of FX(g) in terms of the 1-form σ described inSection 4.10. Since σ is H(M,J)-invariant, we have that LXσ = 0, for each

X in h. By Cartan formula, LXσ = Xydσ + d(σ(X)). Since σ is closed we

get d(σ(X)) = 0, which is exactly what Proposition 4.12.1 says.

In view of Proposition 4.12.1, the Futaki functional will be simply de-noted by FΩ; we then have: FΩ(X) := FX(g), for any X in h and any g

in MΩ. We will sometimes prefer the normalized Futaki functional, FΩ,defined by

normalizedfutakinormalizedfutaki (4.12.4) FΩ =FΩ

VΩ,


where, we recall, VΩ denotes the total volume of M with respect to anymetric in MΩ.

The Futaki functional is H(M,J)-invariant:

futaki-invariancefutaki-invariance (4.12.5) FΩ(γ∗X) = FΩ(X),

for any γ in H(M,J) and any X in h (same argument as for BΩ). It followsthat FΩ is a character of the Lie algebra h, i.e. satisfies

futaki-characterfutaki-character (4.12.6) FΩ([X1, X2]) = 0,

for any pair X1, X2 in h. Accordingly, FΩ is called the Futaki character of(M,J).

One of the main application of the Futaki character has been to providean obstruction to the existence of metric of constant scalar curvature withina given Kahler class. More precisely:

thmfutaki-calabi Theorem 4.12.1 (A. Futaki [86], E. Calabi [46]). IfMΩ contains a met-ric g with constant scalar curvature, in particular a Kahler-Einstein metric,then FΩ = 0.

Conversely, if FΩ = 0, any extremal metric in MΩ — if any — is ofconstant scalar curvature.

If Ω is a multiple of the first Chern class of (M,J)) and FΩ = 0, thenany extremal metric in MΩ — if any — is Kahler-Einstein.

Proof. The first assertion readily follows from the definition (4.12.1)of FΩ (recall that the real potential fXg is normalized by

∫M fXg vg = 0).

The second assertion is also an easy consequence of (4.12.1). Indeed, if g isextremal, its scalar curvature sg is a real potential for the (real) holomorphicvector field −JK = gradgsg, as well as sg − s, which integrates to 0. IfFΩ ≡ 0, we have

0 = FΩ(−JK) = F−JK(g) =

∫M

(sg − s) sg vg =

∫M

(sg − s)2 vg.

It follows that sg = s, i.e. that sg is constant. If, moreover, Ω = k c1(M,J),for some (non-zero) real number k, then the Ricci form ρ is harmonic, as sgis constant — cf. Section 1.19 — and de Rham cohomologous to k ω: thisyields ρ = k ω.

Remark 4.12.2. The third assertion in the above theorem has been firstobserved by S. Kobayashi in [118].

Remark 4.12.3. It readily follows from Theorem 4.12.1 that if MΩ

admits two extremal metrics, then both are of constant scalar curvature orboth are of non-constant scalar curvature, according as FΩ = 0 or FΩ 6= 0.

rembasic Remark 4.12.4. It is a direct consequence of (4.12.2) that FΩ = 0 if and

only if σ(X) = 0 for each X in h. Since σ is H(M,J0)-invariant, the lattercondition just means that σ is basic relatively to the action of H(M,J0),i.e. is the pull-back of a 1-form defined on the quotient MΩ/H(M,J0).Equivalently, the Futaki character FΩ is zero if and only if the Mabuchifunctional is constant on each orbit of H(M,J0).

The Futaki character admits the following alternative formulation, dueto E. Calabi [46]:

4.13. THE EXTREMAL VECTOR FIELD 113

propfutakialternative Proposition 4.12.2. The Futaki character can be alternatively definedby:

futakialternativefutakialternative (4.12.7) FΩ(X) =

∫M

(X · pg) vg,

for any X in h, where pg denotes the Ricci potential defined in Section 1.19.

Proof. This follows from the following simple comutation:∫M

(X · pg) vg =

∫M

(dpg, ξ) vg =

∫M

(dpg, df) vg =

∫Mf ∆pg vg

=

∫Mf(sg − s)vg =

∫Mfsg vg = FΩ(X).

Remark 4.12.5. The space h∗ can be equivalently interpreted as the R-dual of h, viewed as a real vector space — this is the viewpoint developpedhere — or as the C-dual of h, viewed as a complex vector space (this amountsto identifying a complex linear form to its real part, i.e. to identifying α, aR-linear map from h to R, to 1

2(α − iα J) and vice versa). In the latteracceptation the (complex) Futaki character is defined by

futakicomplexfutakicomplex (4.12.8) FΩ(X) =

∫MsgF

Xg vg =

∫M

(X1,0 · pg)vg,

where FXg = fXg + ihXg is the complex potential of X. In the sequel, unlessotherwise specified, FΩ will always denote the (real) Futaki character definedby (4.12.1). Note that, according to this convention, FΩ(X) = 0, wheneverX is Killing with respect to some metric g inMΩ, as fXg is then identicallyzero.

More information concerning the Futaki character can be found in [86]and references therein.

4.13. The extremal vector field of a Kahler classssenergy

In this section, we consider metrics in MΩ which are invariant undersome maximal connected compact subgroup of Hred(M,J), where, we re-call, Hred(M,J) denotes the reduced automorphism group of (M,J) — seeSection 2.4 — and we denote by Mmax

Ω the space of these metrics.For any chosen maximal connected compact subgroup, sayG, of Hred(M,J),

we denote by MGΩ the space of G-invariant elements of MΩ. Then

(4.13.1) MmaxΩ =

⋃h∈Hred(M,J)/G

MhGh−1

Ω .

Notice that by Theorem 3.5.1 any extremal metric — if any — belongs toMmax

Ω , hence to the Hred(M,J)-orbit of an extremal metric in MGΩ .

We henceforth restrict our attention to MGΩ , for some chosen maximal

connected compact subgroup G of Hred(M,J). We denote by g ⊂ hred theLie algebra of G: g is then the space of hamiltonian Killing vector fields forall metrics g in MG

Ω .The space MG

Ω is connected and D-totally geodesic in MΩ as the set offixed points by an isometry group.


For any chosen metric g in MGΩ we denote by PGg the space of Killing

potentials relative to g, i.e. the kernel of the fourth order Lichnerowiczoperator L = δδD−d. cf. Section 2.2: PGg is then the direct sum of R, thespace of constant real functions on M , and of the space of real potentialsof elements of Jg ⊂ hred. Moreover, PGg is stable for the Poisson bracketdetermined by the Kahler form ω of g and, as a Lie algebra, is isomorphicto Lie algebra direct sum R⊕ g.

We denote by ΠGg the orthogonal projector with respect to the global

inner product 〈·, ·〉 of the (real) Hilbert L2(M,R) of real L2-functions onM onto PGg (recall that L2(M,R) is defined as the completion of the spaceC∞(M,R) of smooth real functions on M for the inner product 〈·, ·〉).

Notice that PGg is isomorphic to R⊕g for any g inMGΩ but, as a subspace

of C∞(M,R), depends on g, as does ΠGg .

For any real function f , ΠGg (f) will be called the Killing part of f with

respect to g. In particular, for any metric g in MΩ, ΠGg (sg) is called the

Killing part of the scalar curvature of g and we thus get we get the followingL2-orthogonal decomposition of sg:

sgsplitsgsplit (4.13.2) sg = sGg + ΠGg (sg),

where sGg — call it the reduced scalar curvature with respect to G — is

L2-orthogonal to PGg . Observe that g is extremal if and only if the reduced

scalar curvature sGg is identically zero.

We now define a vector field ZGΩ in Jg by

defZdefZ (4.13.3) FΩ(X) = BΩ(X,ZGΩ),

for any X in Jg, where, we recall, FΩ and BΩ denote the Futaki characterand the Futaki-Mabuchi bilinear form of Ω.

Notice that ZGΩ is well-defined, as the restriction of BΩ to Jg is positivedefinite. Moreover, ZGΩ and is G-invariant, so that JZGΩ belongs to the centerof g.

By its very definition, ZGΩ only depends on the Kahler class Ω and of thechoice of G. Following [87], we call it the extremal vector field of Ω relativeto G.

Any other maximal compact subgroup, G′, of Hred(M,J) is conjugate toG by an element, γ say, of Hred(M,J); since FΩ and BΩ are both H(M,J)-invariant, we infer:

invZinvZ (4.13.4) ZG′

Ω = γ · ZGΩ .

We then have the following proposition, cf. [107]:

propZ Proposition 4.13.1. For any metric g inMGΩ, the extremal vector field

ZGΩ is the gradient of the Killing part of the scalar curvature:

gradZgradZ (4.13.5) ZGΩ = gradg(ΠGg (sg)).

Moreover, the L2-square norm of ΠGg (sg) is given by

killingskillings (4.13.6)

∫M

(ΠGg (sg))

2 vg = EΩ,


by setting

AomegaAomega (4.13.7) E(Ω) =S2

Ω

VΩ+ FΩ(ZGΩ).

In particular,∫M (ΠG

g (sg))2 vg is independent of g in MG

Ω and of G inHred(M,J) and provides the following lower bound for the restriction of theCalabi functional C to Mmax

Ω :

hwanghwang (4.13.8) C(g) ≥ E(Ω),

with equality if and only if g is extremal.

Proof. By its very definition, for any g in MGΩ , ZGΩ = gradgf

ZGΩg for

some real function fZGΩg , determined, up to an additive constant, by

BΩ(ZGΩ , X) =

∫MfZGΩg fXg vg = FΩ(X) =

∫Msg f

Xg vg,

for any X in Jg, of real potential fXg (of mean value 0). It follows that

fZGΩg = ΠG

g (sg) up to an additive constant. We thus get (4.13.5). Notice that

the potential of ZGΩ , as defined in Section 2.1, is ΠGg (sg)− s; it follows that

(4.13.9) FΩ(ZGΩ) =

∫Msg(Π

Gg (sg)− s) vg =

∫M|ΠG

g (sg)|2 vg −S2

Ω

VΩ.

This gives (4.13.6), where E(Ω) is defined by (4.13.7). By using (4.13.4)and (4.12.5), we infer that EΩ is actually independent of G as well. From(4.13.2), we derive:

(4.13.10) C(g) =

∫M

(sGg )2 vg +

∫M|ΠG

g (sg)|2 vg ≥∫M|ΠG

g (sg)|2 vg = E(Ω)

for any g in MGΩ , with equality if and only if sGg ≡ 0, hence if and only

if g is extremal. Since any g in MmaxΩ belongs to MG

Ω for some maximalcompact subgroup of Hred(M,J), the last statement of Proposition 4.13.1follows.

Remark 4.13.1. All extremal metrics in MΩ belong to MmaxΩ by The-

orem 3.5.1. It then follows from Proposition 4.13.1 that

(4.13.11) C(g) = E(Ω)

for any extremal metric g inMΩ. It has been recently shown by X. X. Chenthat E(Ω) is a lower bound for C on the whole space MΩ [56]. The samelower bound for C on MΩ have been obtained by S. Donaldson, togetherwith an improved lower bounds in the case when Ω admits no extremalmetrics, in the polarized case, i.e. when Ω

2π is an integral class, hence theChern class of an ample holomorphic line bundle [79].

remcompact Remark 4.13.2. For any (connected) compact subgroupG of Hred(M,J)— not necesarily maximal — the space,MG

Ω , of G-invariant metrics inMΩ

is still a totally geodesic submanifold of MΩ and the Lie algebra g of G isstill a Lie algebra of hamiltonian Killing vector fields for all metrics g inMG

Ω , although some of them may now have more hamiltonian Killing vectorfields than the ones in g. For any metric g in MG

Ω , denote by PGg the space


of Killing potentials, including constants, of elements of g with respect to g— PGg is still a Lie algebra for the Poisson bracket, isomorphic to R⊕ g —

and by ΠGg the corresponding orthogonal projector. In particular, ΠG

g (sg)is called the Killing part of the scalar curvature sg with respect to G andwe still have the orthogonal decomposition (4.13.2) of sg as the sum of itsKilling part ΠG

g (sg) and its reduced part sGg .

We then defined an extremal vector field relative to G, denoted by ZGΩ , ei-

ther by (4.13.3), or by ZGΩ = gradgΠ

Gg (sg) for any g inMG

Ω . The equivalence

of the two definitions, in particular the fact that gradgΠGg (sg) is independent

of g in MGΩ , is shown as in the proof of the first part of Proposition 4.13.1.

Moreover, sg is G-invariant, as well as ΠGg (sg), so that JZG

Ω belongs tothe center of g.

If G = 1, the corresponding extremal vector field is identically zero,whereas, if G is a maximal compact subgroup of Hred(M,J), we simplyget the extremal vector field previously defined in this section. It may behowever of some interest to consider extremal vector field attached to non-maximal compact subgroups of H 0(M,J), in particular in the toric case —cf. e.g. Donaldson’s paper [77] — due in particular to the following simpleobservation:

propT Proposition 4.13.2. Let G be any maximal compact subgroup of Hred(M,J)and let T be any maximal torus in G. Then,

(4.13.12) ZTΩ = ZG

Ω .

Proof. Let g be any metric in MGΩ . Then, g belongs to MT

Ω as welland, as already observed, JZG

Ω = JgradgΠGg (sg) belongs to the center of g,

hence to t = Lie(T ). It follows that ΠGg (sg) belongs to P Tg , hence is equal

to ΠTg (sg). We thus have:

(4.13.13) ZGΩ = gradgΠ

Gg (sg) = gradgΠ

Tg (sg) = ZT

Ω .

Notice that in Proposition 4.13.2, T could have been replaced by any(connected) closed subgroup of G containing the identity component of thecenter of G.

4.14. Relative Mabuchi K-energy and relative Futaki characterssguan

In this section we fix a (connected) compact subgroup, G, of the reducedautomorphism group Hred(M,J) and we define a K-energy relative to G,denoted EGΩ , for a chosen Kahler class Ω. When G = 1, this is simply theMabuchi K-energy defined in Section 4.10, whereas in the case when G is amaximal compact subgroup of Hred(M,J), EG is the relative, or modified,K-energy introduced independently by D. Guan in [96] and by S. Simancain [175], which plays the same role for the search of a general extremalmetric within Ω as the Mabuchi K-energy for Kahler metrics of constantscalar curvature, cf. . We similarly define a Futaki character relative to,G, whose vanishing is a necessary condition that Ω can possibly admit anextremal metric if G is a maximal compact subgroup Gmax of Hred(M,J) ora maximal torus of Gmax.

4.14. RELATIVE MABUCHI K-ENERGY AND RELATIVE FUTAKI CHARACTER 117

We already observed — cf. Remark 4.13.2 — that an extremal vec-tor field ZGΩ can be defined for any (connected) compact subgroup G ofHred(M,J) and that JZGΩ then belongs to the center of g. The induced vec-

tor field ZG onMΩ is then tangent toMGΩ . Recall that ZG has the following

expression:

(4.14.1) ZG(g) = ΠGg (sg),

for any g in MGΩ . We denote by ζG the corresponding dual 1-form on MG

Ω :

(4.14.2) ζG(g) = ΠGg (sg) vg,

for any g in MGΩ .

lemmaextremalparallel Lemma 4.14.1. The restriction of ZG toMGΩ and its dual 1-form ζG are

D-parallel. In particular, ζG is closed.

Proof. JZG is a hamiltonian Killing vector field for any g in MGΩ ; by

Proposition 4.5.1, the restriction of ZG toMGΩ is then D-parallel. The dual

1-form ζG is D-parallel as well, hence closed.

We now define σ on MGΩ by

tildesigmatildesigma (4.14.3) σG = σ|MGΩ− ζG,

where, we recall, σ denotes the 1-form on MΩ determined by the scalarcurvature, namely the 1-form g 7→ sgvg.

Then, σG is a closed 1-form onMGΩ and its covariant derivative coincides

with the restriction to MGΩ of the covariant derivative of σ.

Moreover, we have that

(4.14.4) σG : g 7→ sGg vg,

where, we recall, sGg denotes the reduced scalar curvature appearing in(4.13.2).

In particular, σGg = 0 if and only if g is extremal.

The relative Mabuchi K-energy, EG, is defined as the opposite of theprimitive of σ:

(4.14.5) σ = −dEG.The relative Mabuchi K-energy is defined up to an additive constant. Forany chosen base-point ω0 inMG

Ω , denote by EGω0the relative K-energy which

vanishes at ω0; we then have:

(4.14.6) EGω0(ω) = −

∫ 1

0σ(c)dt,

for any path c(t) from ω0 = c(0) to ω = c(1). Since σ is closed and MGΩ is

contractible, EGω0(ω) is independent of the chosen path inMG

Ω . In particular,by chosing ct = tφ, corresponding to the curve ωt = ω0 + tddcφ in MΩ, weget

(4.14.7) EGω0(ω) = −

∫Mφ (

∫ 1

0sGt vt dt),

where st and vt are relative to the metric gt determined by ωt.


propguanconvex Proposition 4.14.1. At any point g of Mω0, the Hessian of the relativeMabuchi K-energy EG is given by

hessiantildeEhessiantildeE (4.14.8) (DfdEG)f = 2〈D−df,D−df〉gfor any G-invariant function f , with

∫M fvg = 0, viewed as an element of

TgMω0. In particular, DdEGΩ is everywhere non-negative and its kernel at gis the space of G-invariant momenta of hamiltonian g-Killing vector fields ing. Moreover, the critical points of EGΩ are exactly the G-invariant extremalmetrics in MΩ.

Proof. Since σG differs from σ|MGΩ

by a D-parallel 1-form, the hessian

of EG relatively to D is the restriction to MGΩ of the hessian of E: (4.14.8)

is then a direct consequence of (4.10.10), by only considering G-invariantfunctions f . The critical points of E are the metrics in MG

Ω at which σvanishes, hence the G-invariant extremal metrics, as already observed.

For any (connected) compact subgroup G of Hred(M,J), we denote byHG(M,J) the identity component of the normalizer of G in Hred(M,J) andby HG(M,J) the identity component of its centralizer. We similarly denoteby hG the Lie algebra of HG(M,J) and by hG the Lie algebra of HG(M,J).Hence HG(M,J) is a Lie subgroup of HG(M,J) and both act on the spaceMG

Ω of G-invariant metrics in MΩ. More precisely, the action of HG(M,J)on MG

Ω factors through the quotient HG(M,J)/G, whereas the action ofHG(M,J)factors through the quotient HG(M,J)/Z(G), where Z(G) denotesthe intersection of HG(M,J) with G (which is contained in the center of G).The two actions can then be considered as a same action, due to the followinglemma:

lemmaidaction Lemma 4.14.2. For any connected compact subgroup G of Hred(M,J)we have that

idactionidaction (4.14.9) HG(M,J)/G = HG(M,J)/Z(G).

Proof. We first check that

normalizercentralizernormalizercentralizer (4.14.10) hG = hG + g,

where g denotes the Lie algebra of G. Let Z = ZH + gradgfZ + Jgradgh

Z

be any element of h, where g is any fixed metric in MGΩ . Then, Z belongs

to hG if and only if [X,Z] = LXZ belongs to g for any X = JgradgfX

in g. Now, ZH is the dual vector field of a harmonic 1-form, say ζH , sothat LXζH = 0, cf. the proof of Propositions 2.2.1 and 2.3.2. Since, Xis a Killing vector field, this implies that LXZH = 0, hence that ZH isG-invariant. Since, X preserves G and J , we eventually get: [X,Z] =gradg(LXfZ) + Jgradg(LXhZ). This vector field clearly belongs to g for

all X in g if and only if LXfZ is constant, hence zero, for all X in g,meaning that fZ is G-invariant, and LXhZ belongs to PGg for all X in g.

This implies that γ∗hZ − hZ belongs to PGg for any γ in G. By integratingover G with respect to the bi-invariant measure of total volume 1 — thiscan be realized by using the volume-form vG of any bi-invariant riemannianmetric on G, as G is a compact Lie group — we get that hZ = hZ + f ,where hZ :=

∫G g∗hZ vG is G-invariant and f belongs to PGg . We thus

4.14. RELATIVE MABUCHI K-ENERGY AND RELATIVE FUTAKI CHARACTER 119

end up with Z = (ZH + gradgfZ + Jgradgh

Z) + Jgradgf , where Jgradgf

belongs to g, whereas ZH+gradgfZ+Jgradgh

Z = Z−Jgradf is G-invariant

in hG, hence belongs to hG. This completes the proof of (4.14.10). From(4.14.10), we deduce that HG(M,J)/G and its Lie subgroup HG(M,J)/Z(G)have the same Lie algbra, hence coincide (a more precise formulation of thisisomorphism will appears in Section 9.9).

defirealtivefutaki Definition 7. The Futaki character relative to G, denoted by FGΩ , isthe R-linear form on hG/g = hG/z(G) defined by

modfutakimodfutaki (4.14.11) FGΩ (X) =

∫MfXg s

Gg vg,

for any g in MGΩ , where fXg denotes the real potential of X with respect to

g and, we recall, sGg denotes the reduced scalar curvature of g with respectto G.

This definition relies on the following proposition:

propmodfutaki Proposition 4.14.2. The relative Futaki character FGΩ defined by (4.14.11)is independent of the choice of g in MG

Ω. It is equal to zero if and only ifthe relative K-energy EG is constant on each orbit in MG

Ω under the actionof HG(M,J)/G = HG(M,J)/Z(G).

Proof. For any X in hG, denote by X the induced vector field onMGΩ .

Then, X is given by X(g) = fXg for any g in MGΩ , where fXg is G-invariant.

Now, the closed 1-form σ is HG(M,J)-invariant, so that σ(X) =∫M fXg sg vg

is constant onMGΩ (cf. Remark 4.12.1). Moreover, since σ(X) = −dEG(X),

FGΩ ≡ 0 if and only EG is constant on each HG(M,J)-orbit in MGΩ .

In the case when G is a maximal compact subgroup of Hred(M,J) or amaximal torus, we get

Proposition 4.14.3. For any maximal compact subgroup Gmax of H0(M,J),

the Futaki character FGmaxΩ relative to Gmax and the Futaki character FTΩ

relative to any maximal torus T of Gmax — cf. Remark 4.13.2 — is identi-cally zero whenever there exists an extremal metric within the Kahler classΩ.

Proof. If there exists an extremal Kahler metric in MΩ, there existsone, g say, in MGmax

Ω by Calabi’s theorem 3.5.1. Then, sg is a Killing

potential, hence is equal to ΠGg (sg), in fact in ΠT

g (sg), cf. the proof of

Proposition 4.13.2. It follows that the reduced scalar curvature sGmaxg =

sg − ΠGmaxg (sg) = sg − ΠT

g (sg) = sTg is identically zero. The rhs of (4.14.11)is then equal to 0 for any X in hG, for G = Gmax and G = T and, moregenrally, for any subgroup G of Gmax which contains the center of Gmax, cf.Remark 4.13.2.


Remark 4.14.1. Since sGg = sg − ΠGg (sg), the Futaki character relative

to any (connected) compact subgroup G of Hred(M,J) is also given by

FGΩ (X) = FΩ(X)−BΩ(X,ZGΩ )

= FΩ(X)−BΩ(X,ZG

Ω )

BΩ(ZGΩ ,Z

GΩ )FΩ(ZG

Ω )gaborgabor (4.14.12)

(recall that FΩ(ZGΩ ) = BΩ(ZG

Ω ,ZGΩ ).) The relative Futaki character as

defined above then fits with the relative Futaki character introduced by G.Szekelyhidi in [180].

4.15. Uniqueness of extremal Kahler metricsssuniqueness

The uniqueness issue for Kahler-Einstein metrics has been understoodfor a long time. The uniqueness, up to scaling, of Kahler-Einstein metricswith negative (constant) scalar curvature on a compact complex manifold ofnegative first Chern class is a part of Theorem 1.21.1, whereas uniqueness ofKahler-Einstein metrics with zero scalar curvature within any given Kahlerclass on a compact complex manifold of zero first Chern class directly followsfrom the Calabi-Yau theorem, cf. Section 1.20. In contrast with the negativeand the zero case, no existence nor uniqueness theorem for Kahler-Einsteinmetrics of positive scalar curvature can be deduced from the Monge-Ampereequation (1.21.4). As a matter of fact, existence of Kahler metric of positivetype on a Fano manifold is not granted in general — cf. Section 6.6 —whereas the uniqueness issue was solved by S. Bando and T. Mabuchi [23]in the following form:

thmBM Theorem 4.15.1. Let (M, J) be a connected, compact manifold with pos-itive first Chern class c1(M,J). For any positive real number k, considerthe Kahler class Ω = 2π

k c1(M,J) and assume that the space, E, of Kahler-Einstein metrics in MΩ is non-empty. Then,

(i) E coincides with the orbit of any of its elements under the action ofH(M).

(ii) For any ω0 in E and any ω in MΩ, we have

(4.15.1) Eω0(ω) ≥ 0,

with equality if and only if ω belongs to E.

Proof. The proof of this theorem goes beyond the scope of these notes:the reader is referred to the original paper [23], cf. also [86].

The beautiful argument used by S. Bando and T. Mabuchi in [23] heav-ily relies on the Kahler-Einstein assumption, hence does not generalize toKahler metrics of constant scalar curvature, a fortiori to general extremalKahler metrics. On the other hand, as first observed by S. Donaldson in[74], uniqueness issues for extremal Kahler metrics could be easily solved ifthe existence of geodesics between any two points in MΩ were granted, asevidenced by the following proposition:

propuniquedonaldson Proposition 4.15.1 ([74] Proposition 10). Let (M,J) be a compact com-plex manifold and fix Ω a Kahler class on M . Assume that MΩ contains a

4.15. UNIQUENESS OF EXTREMAL KAHLER METRICS 121

Kahler metric of constant scalar curvature and denote by Eω0 the MabuchiK-energy which vanishes at ω0. Then:

(i) Any metric of constant scalar curvature in MΩ connected to ω0 by ageodesic belongs to the orbit of ω0 under the action of the reduced automor-phism group Hred(M,J).

(ii) For any ω in MΩ connected to ω0 by a geodesic , we have:

(4.15.2) Eω0(ω) ≥ 0,

with equality if and only if ω is of constant scalar curvature.

Proof. Let ω be any element ofMΩ liked to ω0 by a geodesic ωt = ω0+ddcφt, with Iω0(φt) = 0 for all t in the interval [0, T ]. Let T be the tangent

vector field along this geodesic, so that Tωt = φt in C∞0,gt(M,R) = TωtMΩ.

From (4.10.5), we infer

d

dtσφt(T ) = (DTσ)(T ) + σ(DTT )

= (DTσ)(T )

= −2

∫MφtδδD

−dφt vt

= −2

∫M|D−dφt|2gtvt

prop10prop10 (4.15.3)

(see [74], Proposition 10). Suppose now that ω = ω1 is of constant scalarcurvature. We then have: σω0(T ) = σω1(T ) = 0. Since the rhs of (4.15.3) is

non-positive for each t, we conclude that σωt(T ) = 0 and D−dφt = 0 for all

t. This means that Xt = −gradgt(φt) is a (real) holomorphic vector field foreach value of t in [0, 1]. It then follows from Proposition 4.6.3 that Xt = X0

for all t and that ω = (ΦX0t1

)∗ω0, where ΦX0t1

is the value at t1 of the flow ofX0, in particular belongs to Hred(M,J). This completes the proof of (i). Inorder to prove (ii), we observe that the identity (4.15.3) can be rewritten as

d2Ed2E (4.15.4)d2Eω0(ωt)

dt2= 2

∫M|D−dφt|2gtvt.

as σ = −dEω0 . Now, Eω0(ω0) = 0 by definition and, since ω0 is of con-

stant scalar curvature, we also havedEω0 (ωt)

dt |t=0= −σω0 = 0, whereas,

by (4.15.4), the second derivative of Eω0(ωt) is non-negative for all valuesof t in [0, 1]; Eω0(ωt is then non-negative along the geodesic, in particularEω0(ω) = Eω0(ω1 is non-negative. Moreover, Eω0(ω) = 0 if and only ifEω0(ωt) = 0 for each t; as we already saw, this happens if and only if ωbelongs to the Hred(M,J)-orbit of ω0, if and only if ω is of constant scalarcurvature.

By considering the relative K-energy EGω0, D. Guan adapted Donaldson’s

argument in the following manner:

propuniqueguan Proposition 4.15.2 ([96]). Let (M,J) be a compact complex manifoldand fix Ω a Kahler class on M . Choose a maximal compact subgroup, G, ofthe reduced automorphism group Hred(M,J) and assume that MG

Ω containsan extremal metric of Kahler form ω0. Denote by EGω0

the relative K-energyrelative to G which vanishes at ω0. Then:


(i) Any extremal metric in MGΩ connected to ω0 by a geodesic belongs to

the Hred(M,J)G-orbit of ω0.(ii) For any ω in MG

Ω connected to ω0 by a geodesic, we have:

(4.15.5) EGω0(ω) ≥ 0,

with equality if and only if ω is the Kahler form of an extremal metric.

Proof. The argument is quite similar to the argument for Proposition4.15.1. Let ω be any element of MG

Ω connected to ω0 by a geodesic ωt =ω0 + ddcφt, with I(φt) = 0 for all t in the interval [0, 1] (recall that MG

Ωis totally geodesic in MΩ). Let T be the tangent vector field along this

geodesic, so that Tωt = φt in C∞0,gt(M,R) = TωtMΩ. From (4.10.5), we infer

d

dtσφt(T ) = (DT σ)(T ) + σ(DTT ) = (DT σ)(T )

= (DTσ)(T )− (DT ζG)(T ).prop10bisprop10bis (4.15.6)

By Lemma 4.14.1, ζG is D-parallel: the above can then be rewritten:

(4.15.7) − d

dtσφt(T ) =

d2EGω0(ωt)

dt2= −(DTσ)(T ) = 2

∫M|D−dφt|2gtvt,

cf. the proof of Proposition 4.15.1. The end of the argument is as in Propo-sition 4.15.1.

In terms of the Monge-Ampere equation (4.8.1) in Section 4.8, Proposi-tion 4.15.2 can be extended as follows (the notations are as in Proposition4.8.2):

prophem Proposition 4.15.3 ([60], Theorem 6.1). Let Φ be a G-invariant (smooth)real function on M × Σ, with Σ = [0, T ] × S1. As in Proposition 4.8.2, Φis viewed as a family φt,s of G-invariant fonctions on M , parametrizedby (t, s) in [0, T ]× S1. Assume that for any (t, s) in [0, T ]× S1, φt,s is therelative Kahler potential with respect to ω0 of an element (gt,s, ωt,s) of MG

Ω.We then have:

hemhem (4.15.8)∂2EG

∂t2+∂2EG

∂s2= 2

∫M|D−(dφt − dcφs)|2gt,s vgt,s .

In particular, ∂2EG

∂t2+ ∂2EG

∂s2≥ 0, and equality holds if and only if Z :=

gradgt,s φt− Jgradgt,s φs is (real) holomorphic, hence belongs to hred, for any

t, s in [0, T ]× S1.

Proof. From the very definition of the relative MabuchiK-energy given

in Section 4.14, we infer ∂EG

∂t = −∫M φt s

Ggt,s vgt,s , hence

∂2EG

∂t2= −

∫Mφtts

Ggt,svgt,s −

∫Mφt

∂

∂t(sGgt,svgt,s).

By using (5.2.26) in Section 5.2 below, this can be re-written:

ddEtddEt (4.15.9)∂2EG

∂t2=

∫M

(−φtt+(dφt, dφt)gt,s) sGgt,s vgt,s +2

∫M|D−dφt|2gt,s vgt,s .

4.15. UNIQUENESS OF EXTREMAL KAHLER METRICS 123

Similarly,ddEsddEs (4.15.10)

∂2EG

∂s2=

∫M

(−φss + (dφs, dφs)gt,s) sGgt,s vgt,s + 2

∫M|D−dφs|2gt,s vgt,s .

By adding (4.15.9) and (4.15.10) and by using (4.8.4), we get

∂2EG

∂t2+∂2EG

∂s2= 2

∫M

(|D−dφt|2gt,s + |D−dφs|2gt,s) vgt,s

+ 2

∫M

(dφt, dcφs)gt,s s

Ggt,s vgt,s .

laplaceEprovlaplaceEprov (4.15.11)

From (1.23.15), we infer(4.15.12)∫M

(D−dφt, D−dcφs)gt,s vgt,s =

∫Mφt δδD

−dcφs vgt,s = −1

2

∫Mφt LKφs vgt,s .

In the last term, we can replace LKφs, where K = Jgradsgt,s — cf. Section

2.5 — with LKG φs, where KG := Jgradgt,ssGgt,s , as φs is G-invariant. It

follows that∫M

(D−dφt, D−dcφs)gt,s vgt,s = −1

2〈dcsGgt,s , φtdφs〉gt,s

=1

2〈δgt,s(sGgt,s ωt,s), φtdφs〉gt,s

=1

2〈sGgt,sωt,s, dφt ∧ dφs〉gt,s

= −1

2

∫M

(dφt, dcφs)gt,ss

Ggt,s vgt,s .

(4.15.13)

By substituting into (4.15.11), we get (4.15.8). The last statement followsfrom Lemma 1.23.2.

The space MΩ is contractible and of non-positive sectional curvature,hence an (infinite-dimensional) Hadamard space. In view of the expres-sion (4.9.1) of the curvature of the Levi Civita connection of the Mabuchimetric, it can be regarded as an (infinite dimensional) “symmetric spaceof non-compact type”, namely the “dual” of the group Hamω of hamilton-ian symplectomorphisms of (M,ω), hence the quotient of a (non existing)complexification of Hamω by Hamω (cf. [74] and Section 9 below for pre-cise statements related to this viewpoint). In spite of that, the issue ofexistence/non-existence of geodesics between any two metrics in MΩ — oreven between any two extremal metrics — is by no means granted a priori,cf. [133], [62].

As explained in Section 4.8, the search of geodesics between two pointsofMΩ essentially amounts to solving a degenerate Monge-Ampere equationon the manifold M × [0, 1]×S1, whith fixed boundary conditions. The maindifficulty is the lack of regularity of solutions, due to the non-ellipticity ofthe degenerate Monge-Ampere equation. Until recently, the main progressin this direction had been accomplished by X. X. Chen [55], X. X. Chen andE. Calabi [47], who proved a general weak existence theorem for geodesics

in the space of potentials Mω0 and a strong existence theorem for geodesics


in MΩ in the case that the first Chern class of (M,J) is negative, implyingthe uniqueness of extremal metrics in this case.

By using Proposition 4.15.3 and an improved regulraty theorem for su-lutions of the Monge-Ampere equation (4.8.1), the issue of uniqueness ofextremal metrics within a given Kahler class — up to the action of the re-duced automorphism group Hred(M,J) — as been completely solved in thissetting by X. X. Chen and G. Tian, who also proved that extremal Kahlermetrics — if any — realize the minimum value of the relative K-energy onMG

Ω , [60], [57], [59]. For the purpose of these notes, we here account forthese results via the following statement, taken from [6]:

thmchentian Theorem 4.15.2 (X.X. Chen-G. Tian [60], [57], [59]). For any compactcomplex manifold (M,J) and for any Kahler class Ω on M , extremal metricsin MΩ sit in a same orbit of H(M,J). Moreover, for any maximal com-pact subgroup G of Hred(M,J), extremal metrics in MG

Ω realize the absoluteminimum of the relative K-energy EGΩ . In particular, if MG

Ω contains anextremal metric, then EGΩ is bounded from below.

For the proof — which goes far beyond the scope of these notes — thereader is referred to the original papers.

CHAPTER 5

The extremal Kahler cone

scone

The extremal Kahler cone of a complex manifold (M,J) is defined as theset of Kahler classes which contain an extremal Kahler metric. This chapteris mainly devoted to the LeBrun-Simanca openness theorem, according towhich the extremal Kahler cone is open in the Kahler cone. We also in-troduced the so-called strongly extremal Kahler metrics, which are definedas the critical points of the (normalized) Calabi functional, when the lat-ter is defined on the space of all Kahler metrics on (M,J). When M is of(real) dimension 4, strongly extremal Kahler metrics are characterized bythe vanishing of the Bach tensor, which is the subjet of Section 5.5.

5.1. LeBrun-Simanca openness theoremssopenness

In this chapter, (M,J) denotes a (connected) compact complex manifoldof (complex) dimension m. Let H1,1(M) be the subspace of H2

dR(M,R) ofthose classes which can be represented by (closed) J-invariant real 2-forms.The Kahler cone, K(M,J), of (M,J) is then an open set — possibly empty— of H1,1(M). A Kahler class Ω in K(M,J) is called extremal if it can berepresented by the Kahler form of an extremal Kahler metric. The set ofextremal Kahler classes in K(M,J) is clearly a subcone of K(M,J), calledthe extremal Kahler cone of (M,J).

thmstability Theorem 5.1.1 (C. LeBrun-S. Simanca [131] Theorem A). For any com-pact, connected complex manifold (M,J), the extremal Kahler cone is openin K(M,J).

The proof of this theorem occupies the next two sections of this chapter.We essentially reproduce LeBrun-Simanca’s argument. Missing details canbe found in the original papers [130] and [131] and also in [85, Section 6].

Fix an extremal Kahler class Ω0 in K(M,J) and an extremal Kahlermetric g0 of Kahler class ω0, with [ω0] = Ω0. Without loss of generality,we can assume that g0 belongs to MG

Ω , where G is a maximal connectedcompact subgroup of the reduces automorphism group Hred(M,J), like inSection 4.13.

Denote byMK the space of all Kahler metrics and byMGK the space of

G-invariant Kahler metrics on (M,J): a typical element ofMK orMGK will

be indifferently viewed as a metric g or as the corresponding Kahler formω, whose de Rham class can be any element in the Kahler cone K(M,J).

Denote by H1,1G the space of J-invariant, g0-harmonic real 2-forms which

are also G-invariant and by C∞G the space of those real, G-invariant func-tions which are orthogonal to the space PGg0

introduced in Section 4.13 (in

particular, the elements of C∞G are of zero mean value).

125

126 5. THE EXTREMAL KAHLER CONE

Following LeBrun-Simanca, we consider the map

(5.1.1) Ψ :MGK → H

1,1G × C

∞G ,

defined by

LSmapLSmap (5.1.2) Ψ(g) = (α, (I −ΠGg0

)(sGg )),

for any g inMGK , where α denotes the g0-harmonic representative of [ω−ω0],

I the identity operator, ΠGg0

the orthogonal projector of L2G(M,R) to PGg0

with respect to g0, and, we recall, sGg := sg − ΠGg (sg) denotes the reduced

scalar curvature of g, cf. Section 4.13. The map Ψ will be referred to as theLeBrun-Simanca map.

Recall — cf. Section 4.13 — that g is extremal if and only if sGg isidentically zero. Since g0 is extremal, we thus have that Ψ(g0) = (0, 0).Moreover, if g is close enough to g0 so that PGg0

∩ (PGg )⊥ = 0, and if

(I −ΠGg0

)(sGg ) = 0, i.e. if Ψ(g) = (α, 0), then sGg = 0, i.e. g is extremal.

The derivative, Tg0Ψ of Ψ at g0 is a linear map from Tg0MG = H1,1G ⊕C∞G

to T(0,0)H1,1G × C∞G = H1,1

G ⊕ C∞G , hence an endomorphism of H1,1G ⊕ C∞G .

The key step of the argument consists in showing that this derivativeis an isomorphism in a sense which willbe made precise in Section 5.3, cf.Proposition 5.3.1.

The actual computation of Tg0Ψ requires some additional general varia-tional formulae which are furnished in the next Section 5.2. The end of theproof is given in Section 5.3.

5.2. A few variational formulaessvariation

In view of the foregoing, we need to know the derivatives of certainquantities, in particular of the reduced scalar curvature sGg , when the met-

ric g moves in MK or MGK . Since derivatives corresponding to variations

whithin a fixed Kahler class have already been computed, it is sufficient toconsider variations of the metric which are transverse to the Kahler cone,e.g. variations of the form

varharmonicvarharmonic (5.2.1) ω = α,

where α is harmonic with respect to ω. To avoid trivial variations of theform ω = k ω, we moreover demand that α be orthogonal to ω, i.e. that

orthoortho (5.2.2) (α, ω) = 0

(recall — cf. Section 4.12 — that the inner product of any harmonic 2-formα with the Kahler form ω is constant).

Variations as (5.2.1), with (α, ω) = 0, will be called trace-free harmonicvariations of the metric.

lemmavarvharmonic Lemma 5.2.1. For any Kahler metric (g, ω) and any trace-free harmonicvariation (5.2.1) of the metric in MK , the corresponding derivative at g ofthe volume form vg is trivial:

varvharmonicvarvharmonic (5.2.3) vg = 0.

Proof. From vg = ωm

m! we get vg = α∧ωm−1

(m−1)! = (α, ω) vg, which is equal

to zero by the orthogonality condition (5.2.2).

5.2. A FEW VARIATIONAL FORMULAE 127

lemmavarrhoharmonic Lemma 5.2.2. For any Kahler metric (g, ω) and any trace-free harmonicvariation (5.2.1) of the metric in MK , the corresponding derivative at g ofthe Ricci form ρ is trivial:

varrhoharmonicvarrhoharmonic (5.2.4) ρ = 0.

Proof. Since ρ only depends upon the volume form and the complexstructure — cf. Section 1.19 — Proposition 5.2.4 readily follows from Propo-sition 5.2.3.

lemmasharmonic Lemma 5.2.3. For any Kahler metric (g, ω) and any trace-free harmonicvariation (5.2.1) of the metric in MK , the corresponding derivative at g ofthe scalar curvature sg is given by:

varsharmonicvarsharmonic (5.2.5) sg = −2(α, ρ),

where ρ stands for the Ricci form of g.

Proof. Easy consequence of s = 2Λρ, (5.2.4) and (3.2.11).

lemmavarfXharmonic Lemma 5.2.4. For any Kahler metric (g, ω), for any fixed (real) holo-morphic vector field X in h and for any trace-free harmonic variation (5.2.1)of the metric in MK , the corresponding derivative at g of its real potentialfXg is given by

varfXharmonicvarfXharmonic (5.2.6) fXg = −G(δ(ιJXα)),

where G stands for the Green operator, acting on the space of functions ofmean value zero; equivalently, fXg is of mean value zero and

varfXharmonicbisvarfXharmonicbis (5.2.7) ∆fXg = −δ(ιJXα).

Proof. From (2.1.5, we infer

(5.2.8) LXα = ddcfXg ,

hence, by using the Kahler identity (1.14.1),

∆fXg = −ΛddcfXg = −Λ(LXα) = −Λd(ιXα)

= −[Λ, d]ιXα = δc(ιXα) = −δ(ιJXα).

lemmavarfutaki Lemma 5.2.5. For any Kahler metric (g, ω) and any orthogonal har-monic variation (5.2.1) of the metric in MK , the corresponding derivativeat g of the Futaki character is given by

varfutakivarfutaki (5.2.9) FΩ(X) = −2〈fXg , (ρH , α)〉,

for any X in h of real potential fXg , where ρH stands for the g-harmonicpart of the Ricci form ρ.

Proof. From (4.12.1), we get

(5.2.10) FΩ(X) =

∫MfXg vg +

∫MfXg sg vg = 〈fXg , sg〉+ 〈fXg , sg〉


(recall that vg = 0, cf. Lemma 5.2.1)). From (5.2.6), we infer

〈fXg , sg〉 = −〈Gδ(ιJXα, sg〉= −〈δ(ιJXα),G(sg − s)〉= −〈ιJXα, d(G(sg − s))〉= −〈α, Jξ ∧ d(G(sg − s))〉= −〈α, dfXg ∧ d(G(sg − s)〉= −〈α, fXg ddc(G(sg − s))〉

for any X in h, of g-dual ξ = ξH + dfXg + dchXg , cf. Lemma 2.1.1 (we severaltimes used the fact that α is harmonic). On the other hand, we have thatρ = ρH − 1

2ddc(G(sg − s), cf. Section 1.19. We then obtain

〈fXg , sg〉 = 2〈fXg , (ρ, α)〉 − 2〈fXg , (ρH , α)〉.

By using (5.2.5), we get (5.2.9).

lemmavarmabuchi Lemma 5.2.6. For any Kahler metric (g, ω) in MGK and for any trace-

free harmonic variation (5.2.1) of the metric in MGK , the corresponding de-

rivative at g of the restriction of the Futaki-Mabuchi bilinear form BΩ to Jgis given by

(5.2.11) BΩ(X1, X2) = −〈α, fX1g ddc(G(fX2

g )) + (G(fX1g )) ddcfX2

g 〉,

for any two (fixed) elements X1 = gradgfX1g , X2 = gradgf

X2g of Jg.

Proof. FromBΩ(X1, X2) = 〈fX1g , fX2

g 〉, we get BΩ(X1, X2) = 〈fX1g , fX2

g 〉+〈fX1g , fX2

g 〉 (recall that the volume form vg is preserved by this type of vari-ations). By Lemma 5.2.4, this becomes

BΩ(X1, X2) = −〈G(δ(ιJX1α)), fX2g 〉 − 〈fX1

g ,G(δ(ιJX1α))〉= −〈ιJX1α, d(G(fX2

g )〉 − 〈ιJX2α, d(G(fX1g )〉

= −〈α, dcfX1g ∧ d(G(fX2

g ))− d(G(fX1g )) ∧ dcfX2

g 〉

= −〈α, fX1f ddc(G(fX2

g )) + (G(fX1g )) ddcfX2

g 〉.

In this computation we several times used the fact that α is harmonic, henced- and δc-closed, and, for further use, we deliberately broke the symmetrybetween X1, X2 in the final expression.

lemmavarextremal Lemma 5.2.7. For any extremal Kahler metric (g, ω) inMGK and for any

trace-free harmonic variation (5.2.1) of the metric inMGK , the corresponding

derivative at g of the extremal vector field ZGΩ is given by

varextremalvarextremal (5.2.12) ZGΩ = gradg(ΠGg (G((α, ddcsg))− 2(ρ, α))).

Proof. For the first part of the argument, we only assume that g be-longs to MG

Ω . Since ZGΩ belongs to the fixed space Jg — in fact to Jz0,

where z0 denotes the center of g — ZGΩ still belongs to Jg, hence is of the

form ZGΩ = gradg(aGg ) for some function aGg in PGg . From (4.13.3) and by

5.2. A FEW VARIATIONAL FORMULAE 129

using Lemmas 5.2.5 and 5.2.6 for any X = gradgfXg we then get:

BΩ(X, ZGΩ) = 〈fXg , aGg 〉

= FΩ(X)− BΩ(X,ZGΩ)

= 〈fXg , (α,−2ρH + ddc(G(ΠGg (s)g)))〉+ 〈fXg ,G(ddc ΠG

g (sg), α)〉.

This holds for any Kahler metric (g, ω) in MGΩ . If moreover, g is extremal,

then ΠGg (sg) = sg and the above becomes

(5.2.13) 〈fXg , aGg 〉 = 〈fXg ,G(α, ddcsg)− 2(α, ρ)〉,

for any X in Jg. We readily infer

(5.2.14) aGg = ΠGg (G(α, ddcsg)− 2(α, ρ)).

For convenience, we set

derzderz (5.2.15) zGg = ΠGg (sg)

for the Killing part of sg with respect to G. We then have:

lemmavarz Lemma 5.2.8. For any extremal Kahler metric (g, ω) inMGΩ and for any

trace-free harmonic variation (5.2.1) of the metric inMGK , the corresponding

derivative of zGg := ΠGg (sg) is given by

varzvarz (5.2.16) zGg = −2ΠGg ((α, ρ)) + (ΠG

g − I)(G(α, ddcsg)).

Proof. From ZGΩ = gradg zGg , we infer

zZzZ (5.2.17) ZGΩ = ˙gradgzGg + gradg z

Gg .

We already know ZGΩ by Lemma 5.2.7 and it is easy to check that

(5.2.18) ˙gradgf = (ιJgradgfα)],

for any function f (e.g. compute the variation of the identity ιgradgfω = Jdf ,

where only gradg depends on g). Morover, from (5.2.17), we infer that˙gradgz

Gg is itself a gradient, i.e. ˙gradgz

Gg = gradgb

Gg for the function bGg of

mean value zero defined by

(5.2.19) bGg = G(δg(ιJZGΩα)) = G((α, ddczGg ))

(the second equality follows from the adjunction formula (1.23.16)). Bycombining this equality with (5.2.12), by using the fact that g is extremal,so that zGg = sg and by noting that

∫M zGg vg =

∫M sgvg, so that

∫M zGg vg =∫

M sgvg = −2〈α, ρ〉 — cf. (5.2.3)-(5.2.5) — we eventually get:

(5.2.20) zGg (α) = −2ΠGg ((α, ρ)) + ΠG

g (G(α, ddcsg))−G(α, ddcsg).

In view of Theorem 5.1.1, the main result of this section is the following


lemmavartildes Lemma 5.2.9. Let (g, ω) be any extremal Kahler metric in MGK . Then,

for any trace-free harmonic variation (5.2.1) of the metric in MGK , the cor-

responding derivative of the reduced scalar curvature sGg at g is given by

varstildeharmonicvarstildeharmonic (5.2.21) sGg (α) = (I −ΠGg )(G((α, ddcsg))− 2 (α, ρ)

).

For any variations of the metric in MGΩ, of the form ω = ddcf , where f is

a G-invariant real function of mean value zero, the derivative of sGg at g isgiven by

varstildefvarstildef (5.2.22) sGg (f) = −2δδD−df.

The derivative at g of the LeBrun-Simanca map Ψ is then given by

LSderivativeLSderivative (5.2.23) TgΨ(α, f) = (α, (I −ΠGg )(G((α, ddcsg))− 2 (α, ρ))− 2δδD−df),

for any (α, f) in TgMGK = H1,1

G ⊕ C∞G .

Proof. The reduced scalar curvature, sGg , of any G-invariant Kahler

metric g is defined by sGg = sg − zGg , cf. (4.13.2): (5.2.21) then readilyfollows from (5.2.16) and (5.2.5). The variations of the scalar curvaturewithin MΩ is given by (3.2.6). In order to compute the variation of zGgwithin MG

Ω , we may apply Lemma 4.5.1, as ZGΩ is now a fixed element of h:we thus get

varzfvarzf (5.2.24) zGg (f) = LZGΩf = (df, dzGg ) = (df, dsg),

as g is extremal. By combining (3.2.6) and (5.2.24), we get (5.2.22). Inview of (5.1.2), (5.2.23) follows (5.2.21) and (5.2.22) (notice that δδD−df isorthogonal to PGg , so that δδD−df = (I −ΠG

g ) δδD−df).

remdotreduced Remark 5.2.1. Form the proof of Lemma 5.2.9, we easily extract thefollowing:

lemmadotreduced Lemma 5.2.10. The first variation of the reduced scalar curvature sG inMG

Ω is given by

dotreduceddotreduced (5.2.25) sGg (f) = −2δδD−f + (dsG, df),

whereas

dotstildevdotstildev (5.2.26) ˙(sGg vg)(f) = δ(−2δD−df − sGg df) vg,

for any g in MGΩ and any f in TgMG

Ω.

Proof. We have: sGg = sg − zGg , whereas, by the proof of Lemma 5.2.9,

(5.2.27) ˙zGg(f) = (dzGg , df).

We then readily deduce (5.2.25) and (5.2.26) from (3.2.6) and (3.2.7).

5.3. THE END OF THE PROOF 131

5.3. The end of the proof of LeBrun-Simanca openness theoremssproof

The space C∞G , with its natural structure of Frechet space, is not suitablefor analysis; we thus substitute appropriate Sobolev spaces, namely, for anynon-negative integer k, the space W k

G defined as the completions in L2G(M,R)

of C∞G for the Sobolev norm involving derivatives up to order k (a precisedescription of the Sobolev machinery and the elliptic theory which is usedbelow would go beyond the limits of these notes; for this we refer the readerto any one of the numerous excellent textbooks on the subject, e.g. [126]

or [193]). All these spaces are Banach spaces and we denote by MG,(k)K the

corresponding Banach manifolds.

lemmaelliptic Lemma 5.3.1. For any non-negative integer k, the operator δδD−d ex-tends to a continuous linear operator from W k+4

G to W kG, which is an iso-

morphism.

Proof. Recall that δδD−d is a (formally) self-adjoint, G-invariant, el-liptic linear fourth-order differential operator acting on C∞G (M,R). Accord-ing to the standard elliptic theory, C∞G (M,R) then splits as C∞G (M,R) =ker (δδD−d)⊕ im (δδD−d), where each summand is closed in C∞G (M,R) —for the Frechet topology — and the direct sum is orthogonal with respect tothe (global) inner product 〈·, ·〉 relative to g0. Since ker(δδD−d) = (R⊕PGg0

)

and C∞G (M,R) has been defined as the L2-ortogonal complement of (R⊕PGg0)

in C∞G (M,R), this decomposition is the same as C∞G (M,R) = (R ⊕ PGg0) ⊕

C∞G (M,R). In particular, the restriction of δδD−d to C∞G (M,R) is an iso-

morphism of C∞G (M,R) to itself. It follows that the natural extensions of

δδD−d to W k+4 are isomorphisms from W k+4 to W k, for any non-negativeinteger k.

We denote by Ψ(k) the extended map of Banach manifolds fromMG,(k+4)G

to H1,1 × W kG. We then have:

propLSiso Proposition 5.3.1. For any non-negative integer k, the derivative Tg0Ψ(k)

is an isomorphism from H1,1 ⊕ W k+4G to H1,1 ⊕ W k

G.

Proof. It readily follows from (5.2.23) that the equation Tg0Ψ(k)(α, f) =0 is solved by the space of those pairs (0, f) such that δδD−df = 0, when

f belongs to W k+4G . Since δδD−d is elliptic, f is smooth, hence belongs

to PGg0. Since, moreover, W k+4

G is the completion of C∞G (M,R), we infer

that f ≡ 0. This proves the injectivity of Tg0Ψ(k). In order to check the

surjectivity of Tg0Ψ(k), choose any (β, h) in H1,1× W kG, and try to solve the

equation Tg0Ψ(k)(α, f) = (β, h). This is clearly equivalent to the system: (i)α = β and (ii) 2δδD−f = −h + (I − ΠG

g0)(G((α, ddcsg0)) − 2 (α, ρ)). Since

h belongs to W kG, whereas G((α, ddcsg0)) and (α, ρ)) are smooth, the rhs of

equation (ii) belongs to W kG. We then conclude by using Lemma 5.3.1.

The end of the proof of Theorem 5.1.1 goes as follows. Choose k large

enough so thatMG,(k+4)K be contained in the space of Kahler metrics of reg-

ularity al least C4. This means that with respect to any smooth coordinatesystem, in particular with respect to any smooth coordinate system arising


from a holomorphic chart determined by the complex structure J , the coeffi-cients of the metric belong to the class C4. Since Tg0Ψ(k) is an isomorphism,

it follows from the inverse function theorem for Banach manifolds that Ψ(k)

determines an isomorphism from an open neigbourhood of g0 in MG,(k+4)K

to an open neigbourhood of (0, 0) = Ψ(k)(g0) in H1,1G × W k

G. In particular,

there exists ε > 0 such that for any α in H1,1G with 〈α, α〉 ≤ ε there exists g

inMG,(k+4)K with the property that Ψ(k)(g) = (α, 0). As observed in Section

5.1, if ε is small enough, g is then an extremal Kahler metric, of regular-ity at least C4. We conclude by using the following regularity result (thestatement and the argument are taken from [131]):

Proposition 5.3.2 ([131] Proposition 4). Any extremal Kahler metricof regularity at least C4 is smooth.

Proof. By definition, a Kahler metric g is extremal if the gradientgradgsg of its scalar curvature preserves g or, equivalently, if gradgsg pre-

serves J , i.e. is the real part of a holomorphic section of T 1,0M , cf. Sections1.8 and 1.23. Up to now, we only considered smooth metrics, bu this makessense if g is only assumed to be of regularity C4, since then gradgsg, hence

also (gradgsg)1,0, is of regularity C1. Being holomorphic, it is actually real

analytic. The dual dsg of gradgsg with respect to g is then of regularity

C4, and sg itself of regularity C5 when expressed in terms of a holomorphiccoordinate chart. Now, on any such holomorphic coordinate chart, sg hasthe following expression

(5.3.1) sg = 2Λρ = ∆g(logvgv0

),

where v0 denotes the volume form of the (local) flat Kahler metric deter-mined by the holomorphic coordinates — cf. Section 1.19 — and vg is ofregularity C4, as g itself. By a standard bootstrapping argument using theellipticity of ∆g — cf. [131] for more details and references — we concludethat g is smooth.

5.4. Strongly extremal Kahler metricssstrongly

In this chapter, we consider the (normalized) Calabi functional definedon the space of all Kahler metrics on a given compact complex manifold(M,J) of real dimension n = 2m > 0, with a special interest for the casem = 2, i.e. when (M,J) is a complex surface.

We denote: by M the space of all riemannian metrics on M ; by MH

the subspace of all J-hermitian (riemannian) metrics; by MK the subspaceof all Kahler J-hermitian metrics on M ; we then have

(5.4.1) MΩ ⊂MK ⊂MH ⊂M,

where, we recall,MΩ denotes the space of all J-Kahler metrics of M whoseKahler class belongs to some fixed Kahler class Ω.

The normalized Calabi functional C is defined on MK by:

newcalabinewcalabi (5.4.2) C(g) =

∫M s2

g vg

Vn−4n

g

,

5.4. STRONGLY EXTREMAL KAHLER METRICS 133

where Vg =∫M vg denotes the total volume of (M, g). The exponent of Vg in

the denominator has been chosen in such a way that the normalized Calabifunctional be scale invariant.

Following S. Simanca, a critical element of the (normalized) Calabi func-tional C (5.4.2) defined on the whole ofMK will be called strongly extremalKahler metric. Of course, a strongly extremal Kahler metric is extremal.More precisely:

thmcritic Theorem 5.4.1 (S. Simanca [176]). Let g be a Kahler metric on a com-pact Kahler manifold (M,J) of complex dimension m ≥ 2. Denote by ω theKahler form, by ρ the Ricci form, by ρ0 the trace-free — or primitive —part of ρ and by s the scalar curvature, so that ρ = ρ0 + s

2mω. Then, g isstrongly extremal, i.e. critical for the restriction of C to the spaceMK of allKahler metrics on (M,J) metrics, if and only the following two conditionsare satisfied

(i) g is extremal, i.e. Jgradgs is a Killing vector field, and(ii) the harmonic part of sρ0 is zero.

If m = 2, this happens if and and only if

(i) g is extremal, and(ii) the riemannian metric g := s−2g is Einstein on the open set U

where s 6= 0.

Proof. Since the normalized Calabi functional is scale invariant, a met-ric in MK is strongly extremal if and only if it is extremal and critical forany trace-free harmonic variation (5.2.1). The first statement of the theoremis then a direct consequence of (5.2.5).

In order to get an easier derivation of the second statement — and alsofor its own sake — we now provide an alternative derivation of the firstone by first considering the Calabi functional C defined on the whole of M;the gradient of C, denoted by gradC, is then a vector field on M, whosevalue at g is a symmetric bilinear form on M viewed as an element of thetangent space TgM. For simplicity, the scalar curvature and the Ricci tensorwill be denoted by s and r, without mention of g; similarly, the Levi-Civitaconnection of g is denoted by D, whereas ∆ = δd+dδ denotes the riemannianLaplace operator with respect to g. The total volume will be simply denotedby V . We then have:

Lemma 5.4.1. For any g in M,

gradgrad (5.4.3) (grad C)(g) = 2Dds+ 2∆s g − 2s r +s2

2g − (m− 2)

2mV(

∫Ms2 vg) g.

Proof. For any variation, g, of g within the space of all Riemannianmetrics on M , the corresponding variations, sg and vg, of the scalar curva-ture and of the volume form are given by

dotsgeneraldotsgeneral (5.4.4) sg = ∆(trg g) + δδg − (r, g),

and

dotvgeneraldotvgeneral (5.4.5) vg =1

2(trg g) vg,


where trg g is the trace of g with respect to g, see e.g. [28, Theorem 1.174]:(5.4.3) follows easily.

Remark 5.4.1. The Calabi functional on M is clearly invariant underthe natural action of the group of diffeomorphisms of M . At each point gof M, gradC of C(g) must then be a δg-closed bilinear form on M , cf. [28,Chapter 2]. This can also be checked directly:

δ(Dds+ ∆s g − s r +s2

4g) = δDds−∆(ds) + r(ds)

− s δr − 1

2s ds,

(5.4.6)

where each line of the RHS is zero: the first one because of the generalBochner identity for 1-forms: ∆α−δDα = r(α), here applied to α = ds, thesecond one because of the (contracted) differential identity: δr + 1

2ds = 0.

If g belongs toMK , (gradC)(g) splits as the sum of its J-invariant part,(gradCJ,+)(g), and of its J-anti-invariant part, (gradCJ,−)(g), with:

(gradCJ,+)(g) = 2(Dds)J,+ + 2∆s g − 2s r +s2

2g − (m− 2)

2mV(

∫Ms2 vg) g,

(gradCJ,−)(g) = 2D−ds.

(5.4.7)

At any g in MK , the J-invariant part gradCJ,+ can be identified withthe J-invariant 2-form γ = (gradCJ,+)(J ·, ·). Since g is Kahler, γ can bewritten as follows:

γ = ddcs+ 2∆s ω − 2s ρ+s2

2ω − (m− 2)

2mV(

∫Ms2 vg)ω

= ddcs+ 2∆s ω − 2sρ0 +(m− 2)

2m(s2 − 1

V

∫Ms2vg)ω

= ddcs+ 2∆s ω − 2sρ0 +(m− 2)

2m∆σ ω,

gammagamma (5.4.8)

where ρ0 denotes the trace-free part of ρ and σ is defined by σ = G(s2 −s2), where, we recall, G denotes the Green operator acting on the space of

functions of zero mean value — cf. Section 1.17 — and s2 := 1V

∫M s2vg

denotes the mean value of s2. By using the identity

deltafomegadeltafomega (5.4.9) (∆f)ω = ∆(f ω) = −ddcf + δ(df ∧ ω),

which holds for any function f on any Kahler manifold — we leave the easyverification as an exercice for the reader — we get

gammadefgammadef (5.4.10) γ = −2sρ0 − ddcs−(m− 2)

2mddcσ + δ((2ds+

(m− 2)

2mdσ) ∧ ω).

Notice that γ can be viewed as the orthogonal projection of (gradC)(g) onTgMH , when elements of MH are represented by their Kahler form. Wethen have:

lemmagamma Lemma 5.4.2. A Kahler metric g in MK is

(i) extremal if and only if γ is coclosed,(ii) strongly extremal, i.e. critical for C|MK

, if and only if γ is coexact.

5.4. STRONGLY EXTREMAL KAHLER METRICS 135

Proof. The 2-form γ is coclosed if and only if δgradCJ,+g = 0. Since

δgradCg = 0 for any metric g, this happens if and only if δgradCJ,−g =δD−ds = 0. Since M is compact, this is equivalent to D−ds = 0, i.e.K := Jgradgs is Killing, hence g is extremal. To prove the second assertion,it is convenient to represent metrics in MH by their Kahler form, so thatTgMH

∼= Ω1,1M , the space of (real) J-invariant 2-forms on M , and TgMK

is then identified with the space of closed elements of Ω1,1M . It follows thatγ can be regarded as the orthogonal projection of gradCg on TgMH and thatits orthogonal projection on TgMK

J is zero, so that g is critical for C|MK, if

and only if γ belongs to Im δ.

From (5.4.10), we readily infer that the harmonic part of γ is equal tothe harmonic part of −sρ0. The first assertion of Theorem 5.4.1 then readilyfollows from the above lemma. This assertion can actually be formulatedin a slightly more informative way as follows: an extremal Kahler metric iscritical for C|MK

if and only if the following condition is satisfied:

(5.4.11) sρ0 ≡ −1

2ddcs− (m− 2)

4mddcσ mod Im δ

or, equivalently,

srho0srho0 (5.4.12) sρ0 ≡ −m

2(m− 1)(ddcs)0 −

(m− 2)

4(m− 1)(ddcσ)0 mod Im δ.

This easily follows from (5.4.10) and from Lemma 5.4.2. When m = 2,(5.4.12) reduces to

cond2cond2 (5.4.13) sρ0 + (ddcs)0 ≡ 0 mod Im δ.

But sρ0 + (ddcs)0 is a trace-free J-invariant, hence antiselfdual 2-form, andany co-closed antiselfdual 2-form is clearly closed, hence harmonic. Condi-tion (5.4.13) then simply reduces to

cond2biscond2bis (5.4.14) sρ0 + (ddcs)0 = 0,

so that γ = −32 ∗ dd

cs. Equivalently, since we already know that Hesss isJ-invariant, as g is extremal,

cond2tercond2ter (5.4.15) sr0 + 2 Hess0s = 0,

where Hess0 denotes the trace-free part of the hessian. In general, forany two riemannian metrics g, g conformally related by g = φ−2 g on a n-dimensional manifold, where φ is positive function, the corresponding trace-free Ricci tensors, r, r, are related by

rconformalrconformal (5.4.16) r0 = r0 + (n− 2)Hess0(φ)

φ,

cf. e.g. [28, Chapter 2]. Condition (5.4.15) then means that the conformallyrelated metric g := s−2g is Einstein wherever it is defined, i.e. on the openset U where s has no zero. This completes the proof of the second assertionin Theorem 5.4.1.


5.5. The four-dimensional case: the Bach tensorssbach

When n = 4, the Calabi functional C : g 7→∫M s2

gvg is already scale in-variant and C|MK

is actually the restriction toMK of a conformally invariant

functional defined on M, namely the functional g → 24∫M |W

+g |2vg, where

the selfdual Weyl tensor W+ is relative to the orientation induced by J , forwhich the Kahler form of any metric inMH is selfdual. This directly followsfrom (A.4.3) in Appendix A. Moreover, by (A.4.10), up to additive and mul-tiplicative universal constants

∫M |W

+g |2vg can be replaced by

∫M |W

−g |2vg =∫

M |W−g |2vg − 12π2τ or by

∫M |Wg|2vg =

∫M |W

+g |2vg +

∫M |W

−g |2vg, where

W−g denotes the anti-selfdual Weyl tensor and Wg = W+g + W−g the whole

Weyl tensor.For any oriented, four-dimensional manifold M , the Bach tensor, Bg, of

a riemannian metric g on M is defined as the opposite of the gradient at gof the functional

(5.5.1) g 7→ W(g) :=

∫M|Wg|2 vg,

defined on the space M of all riemannian metrics on M , where Wg denotesthe Weyl tensor of g, cf. [21]. We then have:

(5.5.2) dWg(h) =dW(g + th)

dt |t=0= −〈Bg, h〉g,

for symmetric bilinear form h in TgM. SinceW is both conformally invariantand invariant under the action of the group Diff(M) of diffeomorphisms ofM , the Bach tensor is a co-closed, trace-free symmetric bilinear form, which

is conformally covariant of weight −2: Bf−2g = f2Bg, for any positivefunction f .

In order to express the Bach tensor, first in the general case, then in theKahler case, we first note that the whole curvature R can written as

decRSdecRS (5.5.3) R = S ∧ I +W,

which is an alternative way of writing (A.1.7): here I denotes the identityand S the renormalized Ricci tensor, defined by S = 1

2(r − s6 g). In (5.5.3),

S is viewed as a T ∗M -valued 1-form, the identity I as a TM -valued 1-formand the wedge product S ∧ I is the curvature-like tensor defined by

(5.5.4) (S ∧ I)X,Y = S(X) ∧ Y +X ∧ S(Y ).

We then introduce the so-called Cotton-York tensor, C, defined by

(5.5.5) CX,Y,Z = −(DXS)(Y, Z) + (DY S)(X,Z).

The Cotton-York tensor is a three-linear form which can be viewed either asa T ∗M -valued 2-form or as a Λ2M -valued 1-form. As a T ∗M -valued 2-form,C we have that

CYCY (5.5.6) C = −dDS,where S is regarded as a T ∗M -valued 1-form and, we recall, dD denotes theformal exterior differential with respect to the Levi-Civita connection, cf.Section 1.6. As a Λ2M -valued 1-form, C is related to the Weyl tensor W by

CWCW (5.5.7) C = δDW,

5.5. THE FOUR-DIMENSIONAL CASE: THE BACH TENSOR 137

where δD denotes the formal codifferential with respect to D, defined as themetric formal adjoint of dD (this easily follows from the differential Binachiidentity, expressed as dDR = 0). We denote by C+, resp. C−, the selfdual,resp. anti-selfdual, part of C, and we then have

CW+CW+ (5.5.8) C+ = δDW+, C− = δDW−.

We also recall that, on any n-dimensional riemannian manifold, the curva-ture R, as well as the Weyl tensor W , act on the space of symmetric bilinearforms by

(5.5.9) R(h)(X,Y ) =

n∑=1

(RX,ejY, h(ej)),

for any symmetric bilinear form h, also viewed as a symmetric operator, andfor any auxiliary orthonormal frame e1, . . . , en; R(h)(X,Y ) is then the traceof the product of h by the operator Z 7→ RX,ZY ; in particular, R(g) = r.We define W (h) and W±(h) in a similar way. In particular, R(h) and W (h)are related by

RWRW (5.5.10) R(h) = W (h) + trhS + tr(h S) g − S h− h S.

We then have

propbach Proposition 5.5.1. The Bach tensor, B, of any 4-dimensional orientedmanifold has the following expressions

B = δDC +W (S)

= 2(δDC+ +W+(S))

= 2(δDC− +W−(S)).

bachbach (5.5.11)

Proof. We first prove the first identity in (5.5.11). For that we recall

that the expression∫M ( s

2

6 −|r|22 +|W |2) vg is equal to the Euler characteristic

of M , up to some universal positive multiplicative factor, hence is the samefor all metrics on M . The derivative of W is then readily deduced fromthe derivative of the functional g 7→ C(g) :=

∫M s2 vg and g 7→ R(g) :=∫

M |r|2 vg, whose gradients are given by

(5.5.12) gradgC = 2Dds+ 2∆s g − 2s r +s2

2g,

and

(5.5.13) gradgR = δDr +Dds+∆s

2g +|r|2

2g − 2R(r),

cf. [28], Chapter 4. We then get

Bg = −gradgW =1

6gradgC −

1

2gradR

= −1

2δDr − 1

6Dds− 1

3s r +R(r)

+∆s

12g +

s2

12g − |r|

2

4g.

BprovBprov (5.5.14)


On the other hand, we have

(δDC)X,Y = −4∑j=1

(DejC)ej ,X,Y =4∑j=1

((D2ej ,ejS)X,Y − (D2

ej ,XS)ej ,Y )

= −(δDDS)X,Y −4∑j=1

(D2Y,ejS)ej ,Y +

4∑j=1

(Rej ,XS)ej ,Y

− (δDDS)X,Y + (DX(δDS))Y +

4∑j=1

(Rej ,XS)ej ,Y

= −(δDDS)X,Y −1

6(Dds)X,Y +

4∑j=1

(Rej ,XS)ej ,Y .

(5.5.15)

Now,

4∑j=1

(Rej ,XS)ej ,Y = −4∑j=1

(S(Rej ,Xej , Y ) + S(ej , Rej ,XY ))

= (−S r +R(S))X,Y

=1

2(−r r +R(r))X,Y .

(5.5.16)

We thus get

(5.5.17) δDC = −1

2δDDr − 1

6Dds+

δDs

12g − 1

2r r +

1

2R(r).

By comparing with (5.5.14) and by using (5.5.10) for h = S = r2 −

s12 g, we

obtain

Bg = δDC +1

2R(r) +

1

2r r − 1

3s r +

s2

12g − |r|

2

4g

= δDC +R(S)− 1

4s r +

1

2r r +

s2

12g − |r|

2

4g

= δDC +W (S)

(5.5.18)

In order to get the next two identities, it suffices to check the identity

(5.5.19) δD(C+ − C−) = −(W+ −W−)(S) = −(∗W )(S).

This follows from δD = − ∗ dD∗, so that

δD(C+ − C−) = − ∗ dDC= ∗(dD dD)S

= − ∗ (R ∧ S)

= − ∗ (W ∧ S) = −(∗W )(S).

(5.5.20)

In the above computation we used the definition (5.5.6) of C, as well asthe expression (1.6.7) of the curvature, operating on C as a T ∗M -valued 2-form; the wedge product R∧S then denotes the TM -valued 3-form definedby (R ∧ S)X,Y,Z = SX,Y,ZRX,Y S(Z) and it is easy to check that the Ricci-part S ∧ I contributes for nothing in this expression, as ((S ∧ I)∧S)X,Y,Z =


SX,Y,Z((S(X), S(Y ))Z − S(Z, Y )S(X) + S(Z,X)S(Y ) − (S(Y ), S(Z))X),which is clearly 0; finally, it is an easy exercice to check that (∗W )(h) =∗(W ∧ h), for any symmetric bilinear form h.

corbach1 Corollary 5.5.1. The Bach tensor vanishes identically whenever oneof the following three conditions is satisfied:

(1) g is conformally Einsten, or(2) g is selfdual, i.e. W− ≡ 0, or(3) g is anti-selfdual, i.e. W+ ≡ 0.

Proof. If g is Einstein, then r, hence also S, are parallel and C ≡ 0;a fortiori, δDC ≡ 0, whereas W (S) = s

12W (g) ≡ 0; we thus get Bg ≡ 0 byusing the first identity in (5.5.11). If W± ≡ 0, then C± ≡ 0 by (5.5.8) andwe conclude by using the second and third identities in (5.5.11).

corbach2 Corollary 5.5.2 ([67] Lemma 5, [8] Lemma 3). The Bach tensor of aKahler surface (M, g, J, ω) has the following expression

KbachKbach (5.5.21) B =1

6((2D+ds+ s r)0 −D−ds),

where s denotes the scalar curvature and (D+ds− s2 r)0 stands for the trace-

free part of D+ds − s2 r (recall that D+ds, resp. D−ds, denotes the J-

invariant, resp. J-anti-invariant, part of the hessian Dds, cf. Section 1.23).

Proof. It is here understood that M is oriented in such a way that theKahler form ω is selfdual. Then, Corollary 5.5.2 relies on the fact that theselfdual Weyl tensor W+ is given by (A.1.8), meaning that W+ acts on theKahler form ω by W+(ω) = s

6 ω and on the orthogonal complement of ω in

Λ+M by W+|ω⊥ = − s

12 I|ω⊥ ; in tensorial form, W+ can then be written as

W+tensW+tens (5.5.22) W+ =s

8ω ⊗ ω − s

12Π+,

where Π+ denotes the orthogonal projection from Λ2M to Λ+M . Now,Λ+M splits as Λ+M = Rω ⊕ ΛJ,−M , where ΛJ,−M denotes the bundle ofJ-anti-invariant 2-forms, so that Π+ is determined by

(5.5.23) Π+(X ∧ Y ) =1

2(ω(X,Y )ω +X ∧ Y − JX ∧ JY )

(we here identify vectors and covectors as we often tacitly do in these noteswhenever the metric is specified). We thus get the following explicit expres-sion of W+ in the Kahler case:

W+X,Y Z =

s

12ω(X,Y )ω

+s

24(−g(X,Z)Y + g(Y,Z)X + ω(X,Z)JY − ω(JY, Z)JX).

W+bisW+bis (5.5.24)

Recall that W+(S)X,Y is defined as the trace of the product of S with the

operator Z 7→W+X,ZY ; we thus get:

W+SW+S (5.5.25) W+(S) =1

6S0 =

s

12r0,


where S0, r0 denote the trace-free parts of S, r. From (5.5.24) we also easilyobtain:

(5.5.26) (DXW+)Y,Z =

ds(X)

24(2ω(Y,Z)ω − Y ∧ Z + JX ∧ JZ).

By using (5.5.7), we thus obtain the following expression of the selfdualCotton-York tensor C+, viewed as a TM -valued selfdual 2-form:

C+X,Y = − 1

12ω(X,Y )ω

+1

24(ds(X)Y − ds(Y )X + ds(JX)JY − ds(JY )JX).

(5.5.27)

We then easily infer

(5.5.28) (δC+)X,Y =1

24(DXds)(Y ) +

1

8(DJXds)(JY ) +

1

24∆s g(X,Y ),

or

(5.5.29) δC+ =1

6(D+ds+

1

4∆s g)− 1

12D−ds,

where D+ds + 14∆s g is the trace-free part of D+ds (note that D−ds is

already trace-free, as a J-anti-invariant bilinear form). By using (5.5.25)and the second identity in (5.5.11), we finally get (5.5.21).

We conclude this section by rephrasing the second part of Theorem 5.4.1in the following way:

thmcritic4 Theorem 5.5.1. Let (M,J) be a compact complex surface. Then, aKahler metric (g, J, ω) on (M,J) is

(1) extremal, i.e. critical for the restriction of the Calabi functional toM[ω], if and only if the Bach tensor Bg is J-invariant;

(2) strongly extremal, i.e. critical for the restriction of the Calabi func-tional to MK , if and only if the Bach tensor Bg is identically zero.

Proof. From (5.5.21) we see that the J-anti-invariant part of the Bachtensor Bg is −1

6D−ds: it follows that Bg is J-invariant if and only if D−ds =

0; this condition in turn just means that s is the momentum of a hamiltonianKilling vector field, cf. Proposition 2.6.1, i.e. that the metric is extremal.By (5.5.21) again, we infer that Bg ≡ 0 if and only if g is extremal and(s r + 2D+ds)0 ≡ 0; since D−ds ≡ 0, the latter condtion can be written as(s r+ 2Dds)0 ≡ 0, which is exactly the condition that the metric g := s−2 gbe Einstein, whenever it is well-defined, i.e. on the open set, say M0, wheres 6= 0. We then conclude by using the second part of Theorem 5.4.1.

Remark 5.5.1. As observed in the beginning of this section, on anycompact complex surface (M,J), the Calabi functional C can be written as

(5.5.30) C(g) =1

24

∫M|W+|2 vg =

π2

2τ +

1

24

∫M|W−|2 vg,

for any g in MK , where, we recall, τ stands for the signature of M , whenM is oriented by the complex structure. It follows that any selfdual Kahlermetric, if any, is not only critical — selfdual Kahler surfaces are Bach-flat,cf. Corollary 5.5.1 — but a (global) minimum for C.


remlebrun Remark 5.5.2. Until recently, the only known example of a compactKahler surface whose Bach tensor is zero and whose scalar curvature s isnon-constant and positive everywhere (so that the Einstein metric g = s−2g

is globally defined) was the first Hirzebruch surface F1 = P2#P2, equippedwith one of the U(2)-invariant extremal Kahler metrics constructed by E.Calabi in [45]: the Einstein metric g then coincides with the celebratedPage metric constructed by D. Page in [158], cf. Section 10.5. Moreover,it was proved by C. LeBrun in [127] that the only compact complex sur-faces which possibly admit Kahler metrics satisfying (ii), with non-constant

s, are P2#kP2, with k = 1, 2 or 3. Notice that P2#P2 and P2#2P2 areknown to admit no Kahler-Einstein metric — cf. Section 6.6 — whereas theexistence of extremal Kahler metrics on P2#2P2 has recently been provedby C. Arezzo, F. Pacard and Michael Singer [15]. Similarly, P2#3P2 admitsKahler-Einstein metrics — cf. Theorem 6.6.1 — and also extremal Kahlermetrics of non-constant scalar curvature, cf. [15]. More recently, the exis-tence of Bach-flat Kahler metrics of non-constant, everywhere positive scalarcurvature — hence globally conformal to an Einstein metric, cf. above —was established by X. X. Chen, C. LeBrun and B. Weber in [58].

CHAPTER 6

The complex projective space

sprojective6.1. The complex projective space as a complex manifold

ssprojFor any complex vector space V of (complex) dimension m + 1, the

(complex) projective space P(V ) is defined as the space of complex vectorsubspaces of (complex) dimension 1, simply called complex lines, of V . Thestructure of complex manifold of P(V ) is determined by the set of affinecharts associated to non-zero elements of the (complex) dual V ∗. For anynon-zero α in V ∗, the domain of the corresponding affine chart is the openset P(α)(V ) := P(V )\P(ker(α)), i.e. the set of those complex lines x in P(V )which are generated by elements u in V such that α(u) 6= 0. This is naturallyidentified with the affine subspace, Vα, of V , defined by the condition α(u) =1, which is modeled on the kernel ker(α) of α. By chooosing any element wαof P(α)(V ) ∼= Vα as an origin, we thus get a natural bijection v ∈ ker(α) 7→x = C(wα + v) ∈ P(α)(V ) from ker(α) to P(α)(V ).

The set of affine charts defined that way makes P(V ) into a compactcomplex manifold of complex dimensionm, with respect to which the naturalprojection, π, from V \ 0 onto P(V ) is holomorphic.

In particular, any basis e0, . . . , em of V gives rise to a a full coveringof P(V ) by affine charts by successively choosing α equal to any element ofthe (algebraic) dual basis e∗0, . . . , e∗m of V ∗. Via the basis e0, . . . , em,each element x of P(V ) is represented by its homogeneous coordinates (u0 :

u1 : . . . : um) in Cm+1; if, e.g. we choose α = e∗0, P(α)(V ) is determined bythe condition u0 6= 0 and the affine space Vα is then the space of elementsof Cm+1 of the form (1, z1, . . . , zm), with zj =

uju0

, j = 1, . . . ,m, whereas

ker(α) is the subspace Cm ⊂ Cm+1 of elements of the form (0, z1, . . . , zm);

then, the z1, . . . , zm provide a holomorphic coordinate system on P(e∗0)(V ).We can similarly choose α = e∗j , for any j = 1, . . . ,m, and get a holomorphic

coordinate system on P(e∗j )(V ). Since, P(V ) =⋃mi=0 P(e∗i )(V ), we thus get a

holomorphic atlas for P(V ).

A (complex) 1-dimensional, resp. 2-dimensional, projective space is usu-ally called a (complex) projective line, resp. a (complex) projective plane.

The tautological line bundle, Λ, is the complex line bundle over P(V )whose fiber at x is x itself, viewed as a (vector) complex line of V . By itsvery definition, the tautological line bundle Λ is a subbundle of the productbundle P(V )× V and comes equipped with the induced holomorphic struc-ture: a (local) section of Λ over some open set U of P(V ) is holomorphic ifit is holomorphic a V -valued function on U . For any non-zero element α ofV ∗, the restriction of Λ to P(V )(α) is trivialized by the holomorphic section

σα defined as follows: for any x in P(V )(α), σα(x) is the unique generator

143

144 6. THE COMPLEX PROJECTIVE SPACE

of the complex line x such that α(σα(x)) = 1. If α = e∗0 as above, we thenhave σe∗0(u0 : u1 : . . . : um) = (1, u1

u0, . . . , umu0

) = (1, z1, . . . , zm).

For any k in Z, Λk will denote the k-th (complex) tensor power of Λ ifk is positive, the (−k)-th (complex) tensor power of the (complex) dual Λ∗

if k is negative, the trivial holomorphic line bundle if k = 0.

Remark 6.1.1. The tautological line bundle Λ will be occasionally de-noted by O(−1), as is usually done in the literature in algebraic geometry,where holomorphic vector bundles are commonly identified with the associ-ated sheaves of germs of holomorphic sections (in particular, O traditionallydenotes the sheaf of germs of holomorphic functions and the associated vec-tor bundle is them the trivial hiolomorphic line bundle). Accordingly, Λk

will be then denoted by O(−k). In particular, O(1) then stands for the dualtautological line bundle Λ∗. In view of Proposition 6.1.1, O(1) is also calledthe hyperplane line bundle of P(V ).

As a subbundle of the product bundle P(V ) × V , the tautological linebundle Λ fits into to exact sequence

exactexact (6.1.1) 0→ Λ→ P(V )× V → Q→ 0,

where Q is defined as the vector bundle over P(V ) whose fiber at x is thequotient V/x. On the other hand, at each point x of P(V ), the tangentspace TxP(V ) admits the following natural identification

TxhomTxhom (6.1.2) TxP(V ) = Hom(x, V/x) = x∗ ⊗ V/x,where Hom(x, V/x) denotes the space of C-linear homomorphisms from thecomplex line x to the quotient V/x. This can be shown as follows. For anyα in V ∗ such that α|x 6= 0, denote by Pα the (complex) hyperplane kerα in

V . We thus have V = x ⊕ Pα and P(α)(V ) is identified with the complex

vector space Hom(x, Pα) by representing any y in P(V )(α) by its graph withrespect to the above splitting of V , namely y = u + Ay(u) |u ∈ x, for

some well-defined Ay in Hom(x, Pα). We thus get: TxP(V ) = TxP(α)(V ) =Hom(x, Pα = Hom(x, V/x), via the identification Pα = V/x, deduced fromthe above splitting of V . It is easy to check that the resulting isomorphismTxP(V ) = Hom(x, V/x) is independent of the choice of α. Alternatively, anycurve xt in P(V ), with xt = x, can be lifted to a curve ut in V \ 0, withu0 = u any non-zero element of x; then x0 is entirely determined by u0 modx. It follows that TP(V ) = Λ∗ ⊗Q, hence fits into the exact sequence

eulereuler (6.1.3) 0→ 1→ Λ∗ ⊗ V → TP(V )→ 0

obtained by tensoring (8.5.1) by Λ∗; here, 1 = Λ∗ ⊗ Λ stands for the trivialbundle P(V ) × C and Λ∗ ⊗ V is (non canonically) isomorphic to (m + 1)copies of Λ∗.

remcanonicproj Remark 6.1.2. From (6.1.2), we easily infer the following isomorphism:

(6.1.4) K∗(P(V )) = O(m+ 1),

where, we recall — cf. Section 1.19 — K∗(P(V )) = Λm(TP(V )) denotesthe anti-canonical line bundle of P(V ). Indeed, from (6.1.2) we infer thatthe fiber at x of K∗(P(V )) is canonically identified with (x∗)m ⊗ Λm(V/x),

6.1. THE COMPLEX PROJECTIVE SPACE AS A COMPLEX MANIFOLD 145

whereas Λm(V/x) is canonically identified with x∗ ⊗ Λm+1V (equivalently,Λm+1V = x⊗ Λm(V/x) for any complex line x in V ).

propholsection Proposition 6.1.1. For any positive integer k, the space H0(P(V ),O(k))of holomorphic sections of O(k) is naturally isomorphic to the complex k-th

symmetric tensorial power⊙k V ∗, whereas H0(P(V ),O(−k)) = 0.

Proof. For simplicity we give a (sketchy) argument for k = 1 only.Any holomorphic section σ of Λ∗ determines a holomorphic function σ onV \ 0 defined by σ(u) = σx(u) where x = π(u). By Hartogs’ theorem —cf. e.g. [104], [155] — σ extends to a holomorphic function, still denotedby σ, defined on V , with σ(0) = 0. Moreover, σ satisfied the identityσ(λu) = λ σ(u), for any u in V , any λ in C: it thus coincides with itsderivative at 0 and is then a linear form on V . Conversely, any element α ofV ∗ determines a holomorphic section of Λ∗, say α, defined by α(x) = α|x,where α|x denotes the restriction of α to the complex line x. We then get

H0(P(V ),Λ−1) = V ∗. The argument for H0(P(V ),Λ−k) = kV ∗ is quitesimilar. The second assertion H0(P(V ),Λk) = 0 follows by the followinggeneral simple fact: A holomorphic line bundle L and its dual L∗ over acompact, connected, complex manifold M cannot both admit a non-trivialholomorphic section, unless they are both (holomorphically) trivial. Indeed,if σ is a non-trivial holomorphic section of L and α a holomorphic sectionof L∗, then α(σ) is a holomorphic function on M hence is constant, as M iscompact; if L is not trivial, σ has zeros, so that this constant is 0; since σis non-trivial it has no zeros on a dense open set of M , on which α vanishesnecessarily as α(σ) ≡ 0: α is then identically 0.

propaut Proposition 6.1.2. The group Aut(P(V )) of complex automorphismsof P(V ) is naturally isomorphic to the projectivized complex linear groupPG`(V ) = G`(V )/C∗.

Proof. Any element γ of G`(V ) tautologically induces a holomorphicautomorphism, say γ, of P(V ), by setting γ(x) = π(γ(u)), for any non-zero uin x; we thus get a homomorphism from G`(V ) to Aut(P(V )), whose kernelis clearly the subgroup C∗ I, where I denotes the identity of G`(V ). Wethus get an injective homomorphism from PG`(V ) to Aut(P(V )). In orderto chow that this homomorphism is surjective, i.e. that PG`(V ) is actuallyequal to Aut(P(V )), we use the fact that P(V ) admits no other holomorphicline bundle than the Λk already mentioned, for k ∈ Z. This fact in turnfollows from the fact that H2(P(V ),Z) = Z — so that (topological) complexline bundles are parametrized by their degree, equal to −k for Λk — andthat P(V ) is simply-connected — as a quotient of the sphere S2m+1 by a freeaction of the circle S1, cf. Section 6.2 — so that Λk has no other holomorphicstructure than the standard one (for a justification of these assertions werefer the reader to e.g. [61] or [93]). For any Φ in Aut(P(V )), Φ∗(Λ) isa holomorphic line bundle of degree ±1, as Φ is a diffeomorphism, in fact−1, as Φ∗(Λ), like Λ, has no non-zero global holomorphic section; it followsthat Φ∗(Λ) is isomorphic to Λ as a holomorphic line bundle. For any chosenholomorphic isomorphism of Φ∗(Λ) with Λ we then get a biholomorphic map,

say Φ, from V \ 0 to itself, which commutes with π; by Hartogs’ theorem


— cf. the proof of Proposition 6.1.1 — Φ extends to a biholomorphic mapfrom V to itself; moreover, since Φ satisfies Φ(λu) = λ Φ(u), for any u in V ,

any λ in C, we infer as in the proof of Proposition 6.1.1, that Φ is actuallya linear map, hence an element of G`(V ); since Aut(L) = C∗ — as P(V ) is

compact — for any other chosen isomorphism Φ∗(Λ) ∼= Λ, the resulting Φ isa multiple of the former one by an element of C∗: we thus get a well definedelement of PG`(V ), whose image in Aut(P(V )) is obviously Φ.

remholproj Remark 6.1.3. From Proposition 6.1.2 we infer that the Lie algebrah of (real) holomorphic vector fields on P(V ), which is the Lie algebra ofAut(P(V )), is naturally isomorphic to the Lie algebra gl(V )/CI, the quotientof the Lie algebra gl(V ) of C-linear endomorphisms of V by the the spacegenerated by the identity, I, of V , which is the center of gl(V ). For any ain gl(V ), the corresponding vector field Za is given by

vfprojvfproj (6.1.5) Za(x) = a|x modx,

for any x in P(V ) (we here use the identification TxP(V ) = Hom(x, V/x),cf. (6.1.2), and a|x denotes the restriction of a to the complex line x); wereadily check that Za ≡ 0 if a = I, whereas a can be recovered from Za,mod I.

Let e0, . . . , em be any basis of V , so that a has the following matricialexpression: a(ej) =

∑mk=0 akj ek and consider any affine chart determined

by any element e∗j of the dual basis of V ∗, as explained in the beginning of

this section; to simplify notation, choose e∗0: we then have P(e∗0)(V ) = (u0 :. . . : um) |u0 6= 0, which is identified with Cm = ker(e∗0) ⊂ V = Cm+1

by setting z1 = u1u0, . . . , zm = um

u0. Then, the restriction of Za to P(e∗0)(V ),

viewed as a vector field on Cm, has the following expression:

ZaffineZaffine (6.1.6) Za =m∑j=1

(aj0 +

m∑k=1

ajkzk − zj (a00 +m∑k=1

a0kzk))ej .

Indeed, any x in P(e∗0)(V ) = Cm, of affine holomorphic coordinates z1, . . . , zm,is generated by ux := e0 +

∑mj=1 zjej , whose image by a is a(ux) = a00e0 +∑m

k=1 ak0ek + zj(a0je0 +∑m

k=1 akjek); by projecting a(ux) on Cm = ker(e∗0)along x, we readily get the rhs of (6.1.6). Notice that the rhs of (6.1.6)can ve indifferently regarded as a (real) holomorphic vector field — ourusual viewpoint — or as a holomorphic (complex) vector field of type (1,0), when it is better written as Za =

∑mj=1

(aj0 +

∑mk=1 ajkzk − zj (a00 +∑m

k=1 a0kzk))∂/∂zj .

The (complex) Lie algebra h = sl(V ) is reductive, even semi-simple, andthen provides no obstruction to the existence of Kahler-Einstein metricson complex projective spaces. These actually exist and are the so-calledFubini-Study metrics described in the next section.

6.2. The Fubini-Study metricssFS

Any basis of V identifies V with Cm+1 and P(V ) with the standardprojective space Pm := P(Cm+1).

Via the standard hermitian inner product (·, ·) of Cm+1, Pm comesequipped with a 1-parameter family of canonical Kahler metrics defined

6.2. THE FUBINI-STUDY METRIC 147

as follows. Denote by S2m+1 the unit sphere in Cm+1 and by π the restric-tion of π to S2m+1. Then, π makes S2m+1 into a principal S1-bundle overPm, for the action of the S1 determined by the usual diagonal action of S1

on Cm+1. The standard euclidean inner product Re (·, ·) of Cm+1 inducesa riemannian metric on S2m+1, which is the standard metric of constantsectional curvature +1, denoted by g0.

If h denotes the fiberwise hermitian structure of the tautological linebundle Λ induced by the inner product of Cm+1, S2m+1 is then the associatedS1-principal bundle of “unit frames”. Any x in Pm is then represented bya unit element u in S2m+1, which is well-defined up to the action of S1:u→ eitu. Let T be the generator of the action of S1 on S2m+1 and denoteby η the dual 1-form with respect to g0. A vector in TuS

2m+1 is calledhorizontal if its belongs to the kernel of η. Then, for any u in S2m+1 over x,any vector X in TxPm is uniquely represented by a unique horizontal vectorX in TuS

2m+1 which projects to X by π∗. For any positive real number c,we then define an almost hermitian structure (gFS,c, JFS , ωFS,c) on Pm bysetting:

gFS,c(X,Y ) =4

cg0(X, Y ),

JFSX = π∗(iX),

ωFS,c(X,Y ) =4

cg0(iX, Y ).

FSFS (6.2.1)

It is easily checked that the above expressions are independent of the choiceof u in π−1(x) and that the almost-structure JFS coincides with the almost-structure induced by the natural complex structure of Pm defined via theaffine charts. In particular, JFS is integrable. Moreover we easily check that

quantizequantize (6.2.2) π∗ωFS,c =2

cdη.

It follows that ωFS,c is closed. By Proposition 1.1.1 the almost hermitianstructure defined by (6.2.1) is then a Kahler structure, called the Fubini-Study metric of P(V ) of parameter c induced by the basis s (later on in thissection, the parameter c will be indentified with the (constant) holomorphicsectional curvature of ωFS,c).

rembasis Remark 6.2.1. The Fubini-Study metric of parameter c induced by anybasis of V only depends on the hermitian inner product of V determined bythis basis; moreover, it remains unchanged when the basis is replaced by anyhomothetic basis. Denote by B(V ) the space of all basis of V and by B0(V )the quotient B(V )/C∗. Then, the natural (right) action of the special lineargroup Sl(m+ 1,C) on B0(V ) is transitive and, for any chosen base-point inB(V ), the space, FS(V ), of all Fubini-Study metrics of P(V ) of any chosenparameter c is then identified with the quotient Sl(m+1,C)/SU(s0), whereSU(s0) ∼= SU(m+ 1) stands for the special unitary group of the hermitianinner product determines by s0. Since SU(m + 1) is a maximal connectedcompact Lie subgroup of Sl(m + 1,C), FS(V ) ∼= Sl(m + 1,C)/SU(m + 1)has the structure of a riemannian symmetric space of non-compact type, ofdimension m(m+2). In particular, FS(V ) is diffeomorphic to Rm(m+2) (formore information on symmetric spaces we refer the reader to e.g. [98]).


In order to check that the Fubini-Study metric is Kahler-Einstein, thereis no need to compute the whole curvature: we can use instead the directdefinition of the Ricci form ρ as the curvature of the Chern connection of theanti-canonical line bundle K−1

Pm which we already identified with O(−(m +

1)) = Λ−(m−1), cf. Remark 6.1.2. Indeed, the 1-form η is nothing elsethan the connection 1-form on the principal bundle S2m+1 of the Chernconnection ∇Λ of Λ and (6.2.2) then simply means that the curvature formρΛ of ∇Λ, as defined in Section 1.19, is related to ωFS by

rhoFSrhoFS (6.2.3) ωFS,c = −2

cρΛ.

By (1.7.3), we infer that

pirhoFSpirhoFS (6.2.4) π∗ωFS,c =1

cddc log r2

where π denotes the projection of Λ := Λ \ Σ0 onto Pm.

rem-affinechart Remark 6.2.2. On the affine chart determined by e∗0, where the affineholomorphic coordinates on Pe∗0(Cm+1) are denoted by z1, . . . , zm, cf. Sec-tion 6.1, the map s : (z1, . . . , zm) 7→ (1, z1, . . . , zm) is a holomorphic trivial-ization of the tautological line bundle Λ over Pe∗0(Cm+1), whose square normis equal to 1 +

∑mj=1 |zj |2). It then follows from (1.7.2) that the restriction

of ωFS,c to Pe∗0(Cm+1) has the following expression:

FSpotentialFSpotential (6.2.5) ωFS,c =1

cddc log (1 +

m∑j=1

|zj |2).

The Ricci form ρFS is the curvature form of the anti-canonical line bun-dle K−1

Pm = Λ−(m+1), equipped with the induced hermitian structure; it isthen given by

rhoriccirhoricci (6.2.6) ρFS = −(m+ 1) ρΛ =c

2(m+ 1)ωFS .

It follows that the Fubini-Study metric is Kahler-Einstein, of (constant)scalar curvature given by

(6.2.7) sFS,c = cm(m+ 1).

On the other hand, it is easy to compute the whole curvature, RFS,c, ofgFS,c, and we then get:

propFScurvature Proposition 6.2.1. For any c > 0, the curvature RFS,c of the corre-sponding Fubini-Study metric is given by

RFS,cX,Y Z =c

4

((X,Z)Y − (Y,Z)X

+ (JX,Z)JY − (JY, Z)JX + 2 (JX, Y ) JZ),

FScurvatureFScurvature (6.2.8)

for any three vector fields X,Y, Z. For any tangent 2-plane P , generated byon orthonormal pair X,Y , the sectional curvature K(P ) of P is then givenby

(6.2.9) K(P ) =c

4(1 + 3 cos2 θ),

6.2. THE FUBINI-STUDY METRIC 149

where θ, called the holomorphy angle of P , is defined by cos θ = ωF,c(X,Y ).In particular, the holomorphic sectional curvature is constant, equal to c,and the Bochner tensor — see Section (1.19) — vanishes identically.

Proof. For any vector fields X,Y on Pm, represented by their horizon-

tal lifts X, Y , the horizontal lift, DXY , of the covariant derivative DXYrelative to the Levi-Civita connection of the Fubini-Study metric — thesame for all values of c — is equal the horizontal projection of DS

XY along

T , where DS denotes the Levi-Civita connection of the sphere (S2m+1, g0)(easy verification). We also easily check that DST is given by: (i) DS

TT = 0;

(ii) DSXT = iX, for any horizontal vector field X on S2m+1. Then, (6.2.8)

readily follows from these observations. The rest of the Proposition followsreadily (of course, (6.2.6) also readily follows from (6.2.8)).

propisomproj Proposition 6.2.2. The connected isometry group K(Pm, gFS) is thesubgroup PU(m+1) = U(m+1)/C∗ = SU(m+1)/µm+1 of PG`(m+1,C) =G`(m + 1,C)/C∗, where µm+1 = C∗ ∩ SU(m + 1) stands for the group of(m+ 1)-th roots of 1, realized as the center of SU(m+ 1).

Proof. By its very definition, the unitary group U(m+1) preserves thenatural hermitian inner product of Cm+1; the projectivized unitary groupPU(m + 1), as a subgroup of PG`(m = 1,C) = Aut(P(V )), is then asubgroup of K(Pm, gFS); in fact, the two groups coincide, as PU(m + 1) isa maximal compact subgroup of PG`(m+ 1,C).

Proposition 6.2.3. All Kahler-Einstein metrics on Pm are of (positive)constant holomorphic sectional curvature and are isomorphic to a Fubini-Study metric by the action of an element of Aut(Pm).

Proof. We first notice that Pm admits no Kahler-Einstein metric withnegative or zero scalar curvature, since its first Chern class is positive, cf.Section 1.21. By Theorem 3.6.2, the (identity component of the) isome-try group of any Kahler-Einstein metric with positive scalar curvature isa maximal connected compact subgroup of Aut(Pm) and is then conjugatein Aut(Pm) to SU(m + 1)/µm+1. The Kahler-Einstein metric is then theimage by an element of Aut(Pm) of a Kahler-Einstein metric whose isome-try group is SU(m + 1)/µm+1. The latter is then equal to a Fubini-Studymetric. Indeed, the Ricci form of any Kahler metric on a given complexmanifold is entirely determined by its volume form — cf. Section 1.19 —and, up to rescaling, there is only one SU(m + 1)/µm+1-invariant volumeform, as SU(m+ 1)/µm+1 acts transitively on Pm.

remFSs Remark 6.2.3. The volume of Pm with respect to the Fubini-Studymetric of holomorphic sectional curvature c is equal to 1

m! (4c π)m. This

follows readily from (6.2.3) and the fact that the Chern class of the dualtautological line bundle Λ∗ = O(1) is the generator of H2(Pm,Z), so that∫Pm(−ρ∇

2π )m = 1.

remhyperbolic Remark 6.2.4. By choosing any base-point x0, say x0 = (1 : 0 : . . . : 0),in Pm, the latter, equipped with a Fubini-Study Kahler structure of con-stant holomorphic sectional curvature c, can be identified with the quo-tient of SU(m + 1)/µm+1 by the isotropy group of x0 which is clearly the


group S(U(1) × U(m))/µm+1 := SU(m + 1) ∩ (U(1) × U(m))/µm+1. Wethus identify Pm with the quotient SU(m + 1)/S(U(1) × U(m)), which isa hermitian symmetric space of compact type, cf. e.g. [98]. The readeris invited to explore and retrieve the main properties of the Fubini- Studymetric in this setting. Any riemannian symmetric space of compact typehas an associated riemannian symmetric space of non-compact type [98].In the current case, this is the standard complex hyperbolic space CHm =SU(m, 1)/S(U(1) × U(m)), where SU(m, 1) stands for the unitary groupof C1,m := C ⊕ Cm, equipped with the Lorentian hermitian form, (·, ·)1,m

obtained by reversing the sign of the standard hermitian form on the factorCm. As a complex manifold, the complex hyperbolic space CHm can berealized as the space of timelike complex lines in C1,m, defined as the (vec-tor) complex lines on which (·, ·)m,1 is positive definite. The constructionof the dual Fubini-Study metric can then be done as before, by simply re-placing the former unit sphere S2m+1 by the Lorentzian unit sphere S1,2m inC1,m: this “unit sphere” S1,2m is no longer a topological sphere, but is stilla S1-principal bundle over CHm and comes equipped with a S1-invariantlorentzian metric of signature (2m, 1), induced by −〈·, ·〉1,m, which is posi-tive definite on the horizontal space determined by the S1-action. The dualFubini-Study metric is then the induced metric on CHm = S1,2m/S1, asdefined by (6.2.1). It is easily checked that the resulting Kahler metric is ofconstant (negative) holomorphic sectional curvature, equal to −c.

remvariant Remark 6.2.5. A variant of the construction of the Fubini-Study andthe dual Fubini-Study metrics relies on the natural isomorphism

(6.2.10) TxPm = Hom(x,Cm+1/x),

for any x in Pm (cf. Section 6.1). If Cm+1 is equipped with its standard(positive definite) hermitian product, then Cm+1/x is identified with the her-mitian orthogonal complement, x⊥ of x in Cm+1, which recovers a definitepositive hermitian product, as well as the complex line x itself. It followsthat Hom(x,Cm+1/x) = x∗ ⊗ x⊥ admits a positive definite hermitian innerproduct, which makes Pm into a hermitian complex manifold. It is easy tocheck that this hermitian structure coincides with the Fubini-Study Kahlerstructure defined by (6.2.1), for the value c = 4 of the holomorphic sectionalcurvature. The dual Fubini-Study can be defined similarly, when CHm isholomorphically embedded in Pm as the open set of timelike elements, whenCm+1 is now equipped with a lorentian hermitian inner product of signa-ture (1,m), cf. Remark 6.2.4. If now, for any x in CHm, x⊥ denotes theorthogonal complement of x in Cm+1 with respect to the lorentzian hermit-ian inner product, then the latter restrict to a negative definite hermitianinner product on x⊥ and to a positive definite hermitian inner product on xitself. By changing the sign, we eventually get a positive definite hermitianinner product on x∗ ⊗ x⊥ = TxCHm, hence a hermitian structure on CHm,which is the dual Fubini-Study Kahler structure of holomorphic sectionalcurvature −4.

6.3. THE COMPLEX PROJECTIVE SPACE AS A (CO)ADJOINT ORBIT 151

6.3. The complex projective space as a (co)adjoint orbitssorbitproj

We start this section by recalling some well-known general facts concern-ing hamiltonian actions and momentum maps. Let (M,ω) be a symplecticmanifold acted on — on the left — by a Lie group G and suppose that thisaction preserves the symplectic form ω: γ∗ω = ω for any γ in G. For any ain the Lie algebra g of G, the induced vector field a is the vector field on Mwhose flow is given by the action of the 1-parameter subgroup exp(ta) of Ggenerated by a, where exp stands for the exponential map from g to G; wethen have:

inducedinduced (6.3.1) a(x) =d

dt |t=0exp(ta) · x

for any x in M . Note that

hatabhatab (6.3.2) [a, b] = −[a, b],for any two elements a, b of g, as the action of G is a left action. From (6.3.1)and (1.4.1) we infer that

(6.3.3) Laω =d

dt |t=0exp(ta)∗ω = 0.

Each a is then a symplectic vector field, as defined in Section 1.5.The action of G is called hamiltonian if there exists a momentum map

(6.3.4) µ : M → g∗

from M to the dual g∗ of g, satisfying the following two properties:

(H1) For each a in g, µa := 〈µ, a〉 is a momentum of a:

mommommommom (6.3.5) dµa = −ιaω,and

(H2) µ is G-equivariant:

momequivmomequiv (6.3.6) µ(γ · x) = Ad∗γ(µ(x)),

for any x in M and any γ in G.

In (6.3.6), γ · x stands for the given action of G on M and Ad∗γ denotes thecoadjoint action of γ on g∗, defined by 〈Ad∗γζ, a〉 := 〈ζ,Adγ−1a〉, for any ζin g∗ and any a in g.

Proposition 6.3.1. If G is connected, the equivariance condition (6.3.6)can be equivalently formulated as

equivpoissonequivpoisson (6.3.7) µ[a,b] = −µa, µb,for any a, b in g.

Proof. The condition (6.3.6) can be rewritten as

momequivbismomequivbis (6.3.8) µAdγa(γ · x) = µa(x),

for any γ in G, any a in g and any x im M . By subtituting γt := exp(tb)

and by derivating in t at t = 0, we get µ[b,a](x)+dµax(b) = 0, hence, by using(6.3.5),

momequivtermomequivter (6.3.9) µ[a,b](x) = −ωx(a, b),


for any a, b in g and any x in M . By (1.2.10), ω(a, b) is the Poisson bracketµa, µb. We then get (6.3.7). Conversely, (6.3.9)-(6.3.7) implies (6.3.6)-(6.3.8) if G is connected.

Remark 6.3.1. The above proposition can be rephrased as follows:If G is connected, a momentum map for the action of G on M is the assign-ment of a momentum µa for each vector field a in such a way that the mapa 7→ µa be a Lie algebra anti-homomorphism from g to C∞(M,R), equippedwith the Poisson bracket. Note that the minus sign in the rhs of (6.3.7) isdue to the fact that the action of G on M has been chosen a left action,which is also the reason why a minus sign appears in the rhs of (6.3.2). Ifwe substitute a right action by putting Rγ(x) := γ−1 · x, the minus signdisappears in (6.3.2) as well as in (6.3.7) and the induced map from g toC∞(M,R) then becomes a Lie algebra homomorphism.

remmomentum Remark 6.3.2. (i) A momentum map for a hamiltonian action of G onM is uniquely defined up to the addition of a Ad∗-invariant element of g∗,i.e. an element ζ of g∗ such that 〈ζ, a〉 = 0, for any a in the derived Liealgebra [g, g]. If g = [g, g], in particular if g is semi-simple, the momentummap for a hamiltonian action is then uniquely defined.

(ii) If (M,ω) is compact and H1dR(M,R) = 0, any symplectic action

of a connected Lie group G on M is hamiltonian. Indeed, for any a in g,a is certainly hamiltonian, as H1

dR(M,R) = 0, and if we define µa as themomentum of a normalized

∫M µa ωm = 0, then (6.3.7) is certainly satisfied.

(iii) If g is semi simple, its Killing form is non-degenerate and thusidentifies g with g∗ in a G-equivariant way. The momentum map — if any— can then be viewed as an equivariant map from M to g.

We now apply the above general considerations to the action of theconnected isometry group K(Pm, gFS,c) = SU(m + 1)/µm+1 on Pm, whoseaction preserves the full Kahler structure, in particular the symplectic formωFS,c. Since M = Pm is compact and simply-connected we know that amomentum map exists, and since the Lie algebra g = su(m + 1) is semi-simple, this momentum map is uniquely defined. Moreover, since the actionof G is transitive, the momentum map is an immersion — direct consequenceof (6.3.5) — and its image is a unique coadjoint orbit of SU(m + 1)/µm+1

in su(m + 1). Since Pm is compact and simply-connected, the momentummap µFS then identifies Pm with a coadjoint orbit of SU(m + 1)/µm+1 insu(m+ 1).

The explicit form of the momentum map µFS,c : Pm → su(m + 1)∗ iseasy to determine. From (6.1.5) we infer that for any a in the Lie algebrasu(m+ 1) the induced vector field a on Pm is given by

(6.3.10) ax(u) = a(u)− (a(u), u)u,

for any unit generator u of the complex line x (we here identify Cm+1/xwith the (hermitian) orthogonal complement of x in Cm+1). From (6.2.1)and (6.3.5) we easily deduce the following expression of µaFS,c = 〈µ, a〉:

momentumPmmomentumPm (6.3.11) µaFS,c(x) = −2

ci (a(u), u),

6.4. THE BLOWING UP PROCESS 153

for any a in su(m+ 1) and for any x in Pm (here and henceforth, u standsfor any unit generator of the complex line x). Note that

balancedbalanced (6.3.12)

∫Pm

µaFS,c(x) vFS,c = 0

for any a in su(m + 1), as required. This follows from the fact that theelements of su(m+1) are trace-free and from the general following expressionfor the trace of any endomorphism, A, of Cm+1:

(6.3.13) trA =m+ 1

V(S2m+1)

∫S2m+1

(Au, u) vS2m+1 =m+ 1

V(Pm)

∫Pm

(Au, u) vFS .

Here, V(S2m+1) denotes the volume of the unit sphere S2m+1 and V(Pm) thevolume of the standard projective space for the Fubini-Study of holomorphic

sectional curvature c = 4, with V(S2m+1) = 2πV(Pm) = 2πm+1

m! , cf. Remark6.2.3.

By Remark 6.3.2, µFS,c can be viewed as a map from Pm to the Lie

algebra su(m + 1) itself, via the Killing form Ksu(m+1) of su(m + 1). Thelatter is easily checked to be given by

(6.3.14) Ksu(m+1)(a, b) = 2(m+ 1)tr(a b).As the Killing form of any semi-simple compact Lie group, it is negative def-inite, and is here conveniently replaced by the positive definite renormalizedKilling form Ksu(m+1)(a, b) = −tr(ab). By using Ksu(m+1), the momentummap µFS,c becomes a map from Pm to su(m+ 1) and has then the followingexpression:

FSmomentFSmoment (6.3.15) µFS,c(x) =2

ci (Πx −

1

(m+ 1)I),

where I denotes the identity of Cm+1 and Πx stands for the hermitian or-thogonal projector from Cm+1 to the complex line x. In terms of matrices,we thus have

FSmoment-matrixFSmoment-matrix (6.3.16) (µFS,c(x))ij =2i

c(

uiuj∑mk=0 |uk|2

− 1

(m+ 1)δij),

for any i, j = 0, . . . ,m, for any generator u = (u0, u1, . . . , uN ) of the complexline x.

6.4. The blowing up processssblowup

Let P(V ) be a complex projective space of (complex) dimension (m −1), m > 1; V is then a complex vector space of dimension m. As a m-dimensional complex manifold, the (total space of) the tautological linebundle Λ = O(−1) of P(V ) can be viewed as a submanifold of the productV × P(V ), namely the set of pairs (u, x) in V × P(V ) such that u belongsto x. Then, the second projection makes Λ into a holomorphic line bundleover P(V ), as we initially defined it, for which the zero section, say Σ0, isthe subset of pairs (0, x) and is then isomorphic to P(V ) itself. Denote by pthe map from Λ to V determined by the first projection: then, p−1(0) = Σ0,whereas p|Λ\Σ0

is an isomorphism from Λ \ Σ0 to V \ 0. In other words,p realizes Λ as the blow-up of V at the origin 0, which consists in replacingthe point 0 of V by the set of tangent complex directions at 0.


In terms of the natural (linear) holomorphic coordinates, say u0, . . . , um−1,of V determined by a basis e0, . . . , em−1 of V , the map p has the follow-ing expression. Denote by Vj the open set of V defined by the condition

uj 6= 0; then, Vj/C∗ is the affine subspace Pe∗j (V ) of P(V ), cf. Section

6.1. For simplicity, consider the case when j = 0; then, Pe∗0(V ) is iden-tified with Cm−1, by setting zj =

uju0

, j = 1, . . . ,m − 1, and the map

σe∗0 : (z1, . . . , zm−1) 7→ (1, z1, . . . , zm−1) is then a holomorphic section, with-

out zero, of Λ, over Pe∗0(V ) = Cm (cf. Section 6.1). On the open subsetπ−1(Pe∗0(V )) of Λ we then get a system of holomorphic coordinates, say(λ, z1, . . . , zm−1) by identifying (λ, z1, . . . , zm−1) with λσ((z1, . . . , zm−1)).On π−1(Pe∗0(V )), the map p has then the following analytical expression:

blowupexpressionblowupexpression (6.4.1) p : (λ, z1, . . . , zm−1) 7→ (u0 = λ, u1 = λz1, . . . , um−1 = λzm−1).

The operation of blowing up can then be performed for anym-dimensionalcomplex manifold M , m ≥ 1, and for any point x0 of M . The proceduregoes a follows: (i) choose a holomorphic chart domain U around x0, in sucha way that U be identified with a ball B in V centered as 0; (ii) replace Uby ΛU := p−1(B) ⊂ Λ; more precisely, U \ x0 appears as an open set inΛU as well as in M \ x0 and we glue together ΛU and M \ x0 alongU \ x0, cf. e. g. [123] for details.

The resulting object, call it Mx0 or simply M if x0 is understood, is thendescribed by

(6.4.2) Mx0 = (M \ x0)⊔

U\x0=p−1(B)\Σ0

p−1(B)

meaning that Mx0 is obtained from the disjoint union M \x0⊔p−1(B) by

gluing these two pieces via the common identification of U \x0 ⊂M \x0and p−1(B)\Σ0 with B \0. This can be easily made into a m-dimensionalcomplex manifold and comes equipped with a natural holomorphic map, sayp again, from Mx0 to M defined as follows: (i) if y belongs to M \ x0,then p(y) = y; (ii) if y belongs to Σ0 in p−1(B), then p(y) = x0. This

map is then an isomorphism from M \ p−1(x0) onto M \ x0, whereas theexceptional fiber — or exceptional divisor 1 — x0 := p−1(x0) is isomorphicto the complex projective space P(Tx0M).

1 For any (smooth) complex manifold M , the group of divisors, D of M is the freedivisorabelian group whose generators are the connected, irreducible subvarieties of M of codi-mension 1. A non-zero element of D is then a formal sum D =

∑Ni=1 miDi, where each

Di is a connected, irreducible, complex analytic subvariety of M and the coefficients mi

are integers, which we can assume non-zero.For any holomorphic line bundle, L, over M , each meromorphic section, s, of L

determines a divisor, where each Di is a zero or a pole of s with multiplicity mi, positiveif Di is a zero, negative if Di is a pole. Conversely, any divisor D is the divisor of ameromorphic section sD of a holomorphic line bundle LD over M , uniquely defined up toisomorphism. If D is a (smooth, connected) submanifold of codimension 1, LD is trivialoutside some open neighbourhood of D in M and its restriction to D is isomorphic to thenormal vector bundle ν(D) of D in M ; moreover, the first Chern class of D is the Poincaredual of the homology class of D.

Let ID denote the defining sheaf of D in M , i.e. the sheaf of germs of holomorphicfunctions which vanish on D; we then have: ID = L∗D, meaning that ID is isomorphicto the sheaf of germs of holomorphic sections of the dual complex line bundle L∗D (this

6.4. THE BLOWING UP PROCESS 155

The blowing up operation at x0 depends upon the choice of a localholomorphic chart around x0, but any two blow-up’s M , M ′ of M at thesame point x0 are isomorphic via complex isomorphisms which can be sochosen as to commute with the projections p, p′ and to induce the identity ofM outside an open neighbourhood of x0. On the other hand, blow-up’s of asame complex manifold at two different points are in general non-isomorphiccomplex manifolds.

In general, for any (closed) complex submanifold, say M0, of a com-plex manifold M , the normal bundle, ν(M), of M0 in M is the holomorphicvector bundle over M defined by ν(M) = TM|M/TM . By the very con-

struction of Mx0 , the exceptional divisor x0 admits an open neighbourhood

in Mx0 isomorphic to p−1(B) ⊂ Λ. The normal bundle of x0 in Mx0 is thenisomorphic to Λ = O(−1).

propblowup Proposition 6.4.1. For any compact complex manifold M and any pointx0 on M , denote by M the blow-up of M at x0. Then, H(M) is naturallyrealized as the subgroup of elements of H(M) which fix x0.

Proof. We first show that any (real) holomorphic vector field, X, on

M descends to a (real) holomorphic vector field, X, on M , with X(x0) = 0.

Indeed, since p is a biholomorphism from M \ x0 to M \ x0, X certainlydescends to a holomorphic vector field on M \ x0. On the other hand,

the restriction of X on the exceptional divisor x0 determines a holomorphicsection of the normal bundle N = TM|x0

/T x0, which, as we already observe,is isomorphic to O(−1), hence has no non trival global holomorphic section

(cf. Prop 6.1.1); the restriction of X is then tangent to x0. It follows that

X descends to a continuous vector field, X, on M , which is holomorphicoutside x0 and equal to 0 at x0. By Hartogs’ theorem, X is holomorphiceverywhere on M , hence belongs to the Lie subalgebra, hx0(M), of elements

of h(M) which vanish at x0. Moreover, the resulting map from h(M) tohx0(M) is a Lie algebra morphism. This map is clearly injective and wenow show that its is surjective, hence a Lie algebra isomorphism. Indeed,any element X of hx0 lifts to a (real) holomorphic vector field, say X, on

M \ x0, via the isomorphism p from M \ x0 to M \ x0; we then complete

X into a (real) vector field, still denoted by X, on M by defining X on x0

as the (real) holomorphic vector field determined by the complex derivativeof X at x0, viewed as an endomorphism of Tx0M , cf. Remark 6.1.3 (sinceX(x0) = 0, the complex derivative of X at x0, say a, is well defined; forexample, a = ∇X for any C-linear connection on TM). We now have to

check that X is well defined as an element of h(M). Since X is holomorphic

on M \ x0 we only have to worry about the behaviour of X near points of the

exceptional divisor x0 and since, x0 has a neighbourhood in M isomorphicto a neighbourhood of Σ0 in Λ, we can use the local expression of p givenby (6.4.1), where u0, . . . , um are the chosen holomorphic coordinates usedto perform the blow-up. Relatively to these coordinates, X — which will behere conveniently regarded as a holomorphic (complex) vector field of type

holds, more generally, for any effective divisors, i.e. divisors D =∑Nj=1 mjDj where all

mj positive). cf. e.g. [100, Chapter IV] for details.


(1, 0) — is written as: X =∑m−1

j=1 Xj∂/∂uj , where the Xj are holomorphicfunctions of the u0, . . . , um−1; in terms of the coordinates λ, z1, . . . , zm, Xhas then the following expression:

XblowupXblowup (6.4.3) X = X0∂/∂λ+m−1∑j=1

(Xj

λ− zjX0

λ) ∂/∂zj .

This expression only makes sense a priori on the open set where λ 6= 0. Onthe other hand, near the origin, X0 and the Xj , j = 1, . . . ,m − 1, can be

written as X0 = a00u0 +∑m−1

k=1 a0kuk, Xj = aj0u0 +∑m−1

k=1 ajkuk, mod termsof higher orders in the u0, . . . , um−1, where a is the matrix of the complexderivative of X at 0 in terms of the coordinates u0, . . . , um−1. By plugginginto (6.4.3), we thus get:

(6.4.4) X = X0∂/∂λ+m−1∑j=1

(m−1∑k=1

(aj0 + ajkzk − a00zj − a0kzjzk))∂/∂zj

mod terms which are of order at least 1 in λ. This expression is now welldefined, even for λ = 0, and, for λ = 0, reduces to (6.1.6). This show that

X is well defined on the whole of M as an element of h(M) and that the

restriction of X to x0 coincides, as was claimed, with the holomorphic vectorfield on P(Tx0M) by the complex derivative of X at x0.

Let γ be an element of H(M) and assume that γ belongs to the image

of the exponential map from h(M) to H(M), i.e. is the value at t = 1 of

the flow ΦXt of some element X of h(M); then, X descends to a well defined

element X of hx0 , whose flow, ΦXt , belongs to H(M,J) and fixes x0 for each

t; moreover, ΦXt = ΦX

t p for each t. It follows that γ globally preserves theexceptional divisor x0, hence descends to a homeomorphism of M , say γ,wich fixes x0 and is holomorphic on M \ x0; by Hartogs’ theorem again,γ is holomorphic everywhere, hence belongs to Hx0(M), if Hx0(M) denotesthe (closed) Lie subgroup of elements of H(M,J) which fix x0. In general,

not any element of H(M) belongs to image the exponential map, but any γ

in H(M) is the product of a finite number of such elements: it follows that

any γ in H(M) (globally) preserves x0, hence descends to an eleemnt, γ, of

H(M,J) which fixes x0. The resulting homomorphism, say p, from H(M)

to Hx0(M), covers the above Lie algebra ismomorphism p from h(M) tohx0(M); it is then a covering map; on the other hand, p is clearly injective

(if γ in H(M) is the identity on M \ x0, it certainly restricts to the identityon x0); p is then an Lie group isomorphism. Observe that γ can be recovered

from γ in the following manner: (i) the action of γ on M \ x0 coincides withthe action of γ on M \ x0; (ii) the action of γ on the exceptional divisorx0 = P(Tx0M) coincides with the induced linear action of γ on Tx0M .

Proposition 6.4.2. Denote by Lx0 the holomorphic line bundle on Mdetermined by the exceptional divisor x0, cf. footnote of page 154. Then,the canonical line bundles, K(M) and K(M), of M and M are related by:

canonicblowupcanonicblowup (6.4.5) K(M) = p∗K(M)⊗ Lm−1x0

.

6.5. BLOW-UP OF PROJECTIVE SPACES AND HIRZEBRUCH SURFACES 157

In particular, the first Chern classes of M and M are related by

chernblowupchernblowup (6.4.6) c1(M) = p∗c1(M)− (m− 1)c1(Lx0).

Proof. Outside the exceptional divisor x0, we certainly have K(M) =

p∗K(M), whereas a neighbourhood of x0 in M is covered by m holomorphic

charts, U (0), . . . , U (m−1), with holomorphic coordinates (λ = λ(0), z1 = z(0)1 ,

. . ., zm−1 = z(0)m−1), · · · , (λ(m−1), z

(m−1)1 , . . . , z

(m−1)m−1 ), in such a way that p be

locally expressed in each U (0) as in (6.4.1), and on each U (j), j > 0, by

p : (λ(j), z(j)1 , . . . , z

(j)m−1) 7→ (λ(j)z

(j)1 , . . . , λ(j)z

(j)j−1, λ

(j), λ(j)z(j)j+1, . . . , λ

(j)z(j)m−1).

On each U (j), λ(j) is a equation of the exeptional divisor x0, meaning that,if Ix0 denotes the defining sheaf of x0 — whic, we recall, is isomorphic to

L∗x0, cf. footnote of page 154 — Ix0(U (j)) is freely generatedby λ(j) over

O(U (j) (the space of holomorphic functions defined on U (j)). From (6.4.1),we get p∗(du0 ∧ du1 ∧ . . . ∧ dum−1) = λm−1dλ ∧ dz − 1 ∧ . . . ∧ dzm−1 on

U (0), and we get similar expression on the other open sets U (j). We thusget p∗K(M) = K(M) ⊗ (L∗x0

)m−1, which readily implies (6.4.5), as well as(6.4.6) (recall that, for any comple manifold M , c1(M) is defined by as thefirst Chern class of the anti-canonical line bundle, i.e. c1(M) = c1(K−1

M ) =−c1(KM )).

6.5. Blow-up of projective spaces and Hirzebruch surfacesssblowupproj

The blow-up at some point x0 of a complex projective space P(V ) ,with V ∼= Cm+1 and m ≥ 1, can be described in the following way (sinceH(P(V )) = PG`(V ) acts transitively on P(V ), the blow-up, in this case, isessentially independent of the choice of x0). Denote by Qx0 the space of

projective lines in P(V ) which pass through x02 and define P(V )x0

as theset of pairs (x,D) in P(V )×Qx0 such that x belongs to D:

(6.5.1) P(V )x0= (x,D) ∈ P(V )×Qx0 | x ∈ D.

Denote by p the projection from P(V )x0to P(V ) determined by the obvious

projection from Qx0 × P(V ) to P(V ). Then, p is clearly an isomorphismoutside p−1(x0), whereas p−1(x0) is the whole of Qx0 , which is naturallyidentified with P(Tx0P(V )).

In order to get a more concrete grasp on P(V )x0it is convenient to choose

a complex vector subspace, V0 say, of V such that V = x0 ⊕ V0. P(V0) isthen a projective hyperplane of P(V ) which does not meet x0. It followsthat each projective line D of P(V ) in Qx0 meets P(V0) in a unique point, sothat Qx0 is naturally identified with P(V0). Denote by ΛV0 the tautologicalline bundle of P(V0).

2A projective line in P(V ) is a complex projective line D = P(P ), where P is anycomplex 2-plane, i.e. any complex vector subspace of dimension 2, of V ; the points ofD are then the complex lines of V which are contained in P ; Qx0 is then the space ofcomplex vector 2-planes of V which contain the complex line x0.


prophatproj Proposition 6.5.1. For any choice of a complement V0 of x0 in V , wehave a natural identification of complex manifolds:

hatprojhatproj (6.5.2) P(V )x0= P(1x0 ⊕ ΛV0).

Here 1x0 stands for the trivial line bundle P(V0) × x0 and P(1x0 ⊕ ΛV0)denotes the holomorphic projective line bundle over P(V0) whose fiber at yis the projective line P(x0 ⊕ y), when y is viewed as a complex line of V0.

Proof. The identification (6.5.2) is essentially tautological. To any

pair (x,D) in P(V )x0we first associate the complex line y := D ∩ P(V0) of

V0, viewed as a point of P(V0). Then, D = P(x0 ⊕ y) and x can then beregarded as a point of the fiber at y of P(1x0 ⊕ ΛV0). Conversely, any pointx of P(1x0⊕ΛV0) over y is a complex line in the 2-plane x0⊕y of V , hence a

point of P(V ). The corresponding element of P(V )x0is then the pair (x,D),

where D = P(x0 ⊕ y) is viewed as a projective line in P(V ).

It is convenient to identify the chosen complex line x0 with C, by chosinga generator of x0, and P(1x0⊕ΛV0) is then identified with the projective linebundle P(1⊕ ΛV0), where 1 stands for the trivial line bundle P(V )× C.

The projective line bundle P(1x0 ⊕ΛV0) is now defined independently ofthe projective space P(V ) and Proposition 6.5.1 can then be reformulatedas follows: For any complex projective space P(V0), we have a canonicalidentification

hatproj-bishatproj-bis (6.5.3) P(1⊕ ΛV0) = P(C⊕ V0)C,

where the rhs denotes the blow-up of the complex projective space P(C⊕ V0)at the point C as defined above, i.e. as the incidence submanifold of P(V0)×P(C⊕ V0).

In the sequel, the natural projection from P(1 ⊕ ΛV0) to P(V0) will bedenoted π.

The tautological line bundle ΛV0 is naturally embedded as an open set— actually an open subbundle — of P(1⊕ΛV0) by identifying any element uof ΛV0 , over some point y of P(V0), with the complex line, x, of the 2-planeC⊕ y generated by (1, u). For any y in P(V0) we then have

(6.5.4) ΛV0y = π−1(y) \ y,

where, in the rhs, y is viewed as the complex line y in the 2-plane C ⊕ y,hence as a point in π−1(y).

The projective line bundle P(1⊕ΛV0) thus appears as a compactificationof ΛV0 obtained by adding a point at infinity to each fiber ΛV0

y , namely

the complex line y itself as a distinguished point of π−1(y). The reunionof the points at infinity is denoted by Σ∞; Σ∞ is then the image of the(holomorphic) infinity section σ∞ : y 7→ y ⊂ C⊕ y. For simplicity, Σ∞ itselfwill be called the infinity section of P(1⊕ ΛV0).

Via the embedding of ΛV0 in P(1 ⊕ ΛV0), the zero section of ΛV0 canve viewed as a complex submanifold, Σ0, of P(1 ⊕ ΛV0), namely the imageof the (holomorphic) zero section σ0 : y 7→ C ⊂ C ⊕ y of P(1 ⊕ ΛV0). Forsimplicity again, Σ0 itself will be called the zero section of P(1⊕ΛV0). BothΣ0 and Σ∞ are isomorphic to P(V0) via the projection π.

6.5. BLOW-UP OF PROJECTIVE SPACES AND HIRZEBRUCH SURFACES 159

Via the isomorphism (6.5.3), the blow-up map p, viewed as a map fromP(1⊕ΛV0) to P(V ), V = C⊕V0, can be described in the following way: anyx in π−1(y) is defined as an element of P(x0⊕ y), i.e. as a complex line of Vcontained in the complex 2-plane x0 ⊕ y of V , and we then have p(x) = x,where x is simply regarded as a complex line in V , i.e. as a point of P(V ),with no more reference to the 2-plane x0⊕ y (p then appears as a forgettingmap). It follows that p−1(x0) is the whole zero section, Σ0, whereas forany x 6= x0, p−1(x) is simply x itself, viewed as an element of π−1(y), if ydenotes the intersection of P(V0) with the complex projective line in P(V )determined by x and x0. Observe that p maps Σ∞ biholomorphically toP(V0), whereas Σ0 is mapped to x0.

remcompactification Remark 6.5.1. In general, for any complex manifold M and any holo-morphic line bundle, L, over M , the associated projective line bundle π :P(1⊕ L)→M can similarly be viewed as a compactification of L obtainedby adding an infinity section Σ∞, image of the section σ∞ : y 7→ Ly ⊂ C⊕Lyof π, whereas the zero section, Σ0, is the image of the section σ0 : y 7→ C ⊂C⊕ Ly.

The first Hirzebruch surface, F1, is the (compact) complex surface P(1⊕Λ), when Λ denotes the tautological line bundle of the complex projectiveline P1. In view of (6.5.3), F1 is isomorphic to the blow-up of the complexprojective plane P2 = P(C⊕ C2) at C.

For any positive integer ` > 1, the k-th Hirzebruch surface, F`, is definedsimilarly, by substituting Λ` to Λ, i.e. F` = P(1 ⊕ Λ`). It turns out thatthe zero section Σ0 can again be blown down, but the resulting object isno longer a smooth complex manifold but a projective variety with orbilfoldsingularities, namely the weighted projective plane of weight (1, 1, `), cf. [71].

remhirzebruch Remark 6.5.2. It readily follows from the above discussion that thefirst Hirzebruch surface F1 can be realized as the complex submanifold ofthe product P1 × P2 defined by the equation

H1H1 (6.5.5) x1y2 − x2y1 = 0,

for x = (x0 : x1 : x2) in P2 and y = (y1 : y2) in P1. Here, P2 is understood asthe standard complex projective plane P(C3, where C3 comes equipped withits natural basis e0, e1, e2, and P1 is then P(C2), where C2 is understood asthe complex 2-plane generated by e1, e2. The identification goes as follows:to any pair (x = (x0 : x1 : x2), y = (y1 : y2) satisfying (6.5.5), we associatethe complex line in the complex 2-plane C e0 ⊕ y — the fiber at h of thevector bundle 1 ⊕ Λ over P1 — generated by x0 e0 + λ (y1 e1 + y2 e2), withλ = x1

y1= x2

y2. In this identification, the projection to the second factor P1

determines the natural projection from F1 to P1 whereas the projection tothe first factor P2 realizes F1 as the blow-up op P2 at the point (1 : 0 : 0).Moreover, the zero section Σ0 is identified with the set of pairs (x = (0 : 0 :1), y = (y0 : y1), whereas the infinity section Σ∞ is identified with the set ofpairs (x = (0 : y1 : y2), y = (y1 : y2) for all (y0 : y1) in P1.

More geenrally, any Hirzebruch surface can be realized as the set ofsolutions of the equation

HellHell (6.5.6) x1y`2 − x2y

`1 = 0,


in P2×P1. With the above notation, we simply identify each pair (x = (x0 :x1 : x2), y = (y1 : y2) satisfying (6.5.6) with the element of the fiber at y ofP(1⊕ Λ`) over generated by x0 e0 + λ (y1 e1 + y2 e1)⊗`), with λ = x1

y`1= x2

y`2.

For any `, the Hirzebruch surface F` can be realized as a blow-up by sub-stituting the weighted projective plane P2

`,1,1 to P2. In general, for any set

(a0, . . . , am) of positive integers, with gcd (a0, . . . , am) = 1, the correspond-ing weighted (complex) projective space P2

a0,...,am is defined as the quotient

of Cm+1 \ 0 by the weighted action of C∗ defined by

(6.5.7) ζ · (z0, . . . , zm) = (ζa0z0, . . . , ζamzm),

for any ζ in C∗ and any (z0, . . . , zm) in Cm+1. We here consider the casewhen m = 2 and (a0, a1, a2) = (`, 1, 1). The resulting weighted projectiveplane is smooth except at the point (1 : 0 : 0) where it has an orbifoldsingularity of type C2/Z`, where Z` stands for the group of `-roots of 1.The weighted projective plane P2

`,1,1 admits a canonical map to the standard

complex projective space P2 — viewed as the weighted projective space P21,1,1

— determined by the map

(6.5.8) q` : (z0, z1, z2) 7→ (x0 = z0, x1 = z`1, x2 = z`2)

from C3 to itself (this map is clearly equivariant for the appropriate C∗-actions, of weight (`, 1, 1) in the lhs, of weight (1, 1, 1) in the rhs, hencedetermines a well-defined (holomorphic) map from P2

`,1,1 to P2). We now

show that F` can be realized in the product P2`,1,1 × P1 by the equation

H1-bisH1-bis (6.5.9) z1y2 − z2y1 = 0.

Denote by F` the subvariety of P2`,1,1×P1 defined by (6.5.9) and consider the

map, q`, from P2`,1,1×P1 to P2×P1 determined by q` on the first factor and

the identity on the second factor. From (6.5.9) we deduce z`1y`2 − z`2y`1 = 0,

hence x1y`2 − x2y

`1 = 0: this shows that q` maps F` onto F` in P2 × P1.

Moreover, the restriction of q` to F` is injective. Indeed, for any pair (x =

(x0 : x1 : x2), y = (y1 : y2)) in F`, the elements of F` which map to (x, y)are the pairs (z = (z0 : z1 : z2), y = (y1 : y2)) such that z0 = x0, z`1 = x1,z`2 = x2 and z1y2 = z2y1. Since y1, y2 cannot ve both zero, we may assumein the following argument that y1 6= 0, so that z2 is determined by z1. Thetriple (z0, z1, z2) is then entirely determined by the choice of z1 as a `-th rootof y1. Moreover, if z1 is replaced by ζ z1, for ζ any `-th root of 1, then thecorresponding new triple is (z0, ζ z1, ζ z2) = (ζ` z0, ζ z1, ζ z2), which is equalto ζ · (z0, z1, z2), for the C∗-action of weight (`, 1, 1), hence determines thesame element of P2

`,1,1.

6.6. Fano complex surfacesssfano

A compact complex manifold (M,J) whose first Chern class c1(M,J)is positive, i.e. belongs to the Kahler cone, is called a Fano manifold. Byits very definition any Fano manifold admits Kahler metrics polarized by

6.6. FANO COMPLEX SURFACES 161

its anti-canonical line bundle K−1M , cf. Section 1.19. Any compact Kahler-

Einstein manifold with positive (constant) scalar curvature is a Fano mani-fold but not any Fano manifold admits a Kahler-Einstein metric, cf. Section1.20.

Fano complex surfaces are well understood [196], [101]: these are P2,P1 × P1 and the blow-ups of P2 at k distinct points p1, . . . , pk in generalposition, with k ≤ 8. Here in general position means that: (i) if k ≥ 3, nothree of them sit on a same projective line; (ii) if k ≥ 6, no 6 of them sit ona same conic; (iii) if k = 8, there is no singular cubic passing through all ofthem and having one of them as a singular (double) point.

The complex projective plane P2 admits a Kahler-Einstein metric, theFubini-Study metric, which is actually of constant holomorphic sectionalcurvature and is unique up to rescaling and the action of H(P2), cf. Section6.2.

Similarly, the product P1 × P1 admits a Kahler-Einstein metric, namelythe product of two Fubini-Study metrics with the same value of the constantholomorphic sectional curvature. Again this metric is unique up to rescalingand the action of Aut(P1 × P1). Note that the product of two Fubini-Studymetrics with different (constant) holomorphic sectional curvatures is still ofconstant scalar curvature, but no longer Kahler-Einstein.

propM1M2 Proposition 6.6.1. The blow-up’s of P2 at one point or at any two dis-tinct points are Fano but admit no Kahler-Einstein metric, not even Kahlermetrics of constant scalar curvature.

Proof. This is a simple application of the non-existence criterion givenby Lichnerowicz-Matsushima theorems 3.6.1, 3.6.2. For any k, denote byMk any blow-up of P2 at k distinct points p1, . . . , pk in general position asabove. By Proposition 6.4.1, H(Mk) is naturally identified with the subgroupof elements of H(P2) which preserve each point p1, . . . , pk. It remains tocheck that H(Mk) is not reductive whenever k = 1 or k = 2. Recall — cf.Proposition 6.1.2 — that H(P2) = PG`(3,C). Since H(P2) acts transitivelyon P2 and on pairs of (distnct) points of P2, we can assume without loss ofgenerality that M1 is the blow-up of P2 at x1 = (1 : 0 : 0) and that M2 isthe blow-up of P2 at x1 = (1 : 0 : 0) and x2 = (0 : 1 : 0). A typical elementof H(M1) can then be written as

autmatrix1autmatrix1 (6.6.1) γ =

1 u v0 a b0 c d

where u, v stand for any two complex numbers and

(a bc d

)for any element

of G`(2,C), whereas a typical element of H(M2) is written as

autmatrix2autmatrix2 (6.6.2) γ =

1 0 v0 a b0 0 d

where v stands for any complex number and

(a b0 d

)for element of the

group T+(2,C) of invertible upper 2 × 2 complex triangular matrix. In


other words, we have that

(6.6.3) H(M1) ≡ G`(2,C) nC2,

the semi-direct product of the full complex linear group G`(2,C) with C2

for the natural action of G`(2,C) on C2, whereas

(6.6.4) H(M2) ≡ T+(2,C) nC,semi-direct product of T+(2,C) and C for the action of T+(2,C) on C given

by

(a b0 d

)· v = d v. At the level of Lie algebras, we then have

(6.6.5) h(M1) = gl(2,C)⊕ C2,

with bracket given by [(A, x), (B, y)] = ([A,B], Ax − By), for any A,B ingl(2,C), any x, y in C2, and

(6.6.6) h(M2) = t+(2,C)⊕ C,with bracket similarly given by: [(A, x), (B, y)] = ([A,B], Ay−Bx), for anyA,B in t+(2,C), any x, y in C. We easily check that the center of h(M1)and the center of h(M2) are both reduced to 0, whereas h(M1) and h(M2)are clearly ly not semi-simple: they are then both non-reductive 3 . It thenfollows from Theorems 3.6.1, 3.6.2 that M1 and M2 admit no Kahler metricwith constant scalar curvature, a fortiori no Einstein-Kahler metric.

Since H(P2) acts transitively on all triple of distinct points of P2, anyM3 is isomorphic to the blow-up of P2 at the points x1 = (1 : 0 : 0),x2 = (0 : 1 : 0) and x3 = (0 : 0 : 1), A typical element of H(M3) is thenwritten as

(6.6.8) γ =

1 0 00 a 00 0 b

with a, b any two complex numbers such that ab 6= 0. We then have

(6.6.9) H(M3) = C∗ × C∗,

3 Let G, H be any two Lie groups and suppose that G acts on H by group automor-semidirectphisms. The semi-direct product, GnH, of G by H relative to this action is then definedas the Lie goup whose underlying manifold is the product G×H, with the multiplicationlaw: (g1, h1)(g2, h2) = (g1g2, (g

−12 · h1)h2), for any g1, g2 in G and any h1, h2 in H; then,

the maps g 7→ (g, 1) and h 7→ (1, h) realize G and H as subgroups of GnH, H is normalin GnH, and the initial action of G on H simply becomes the natural action of G on Hby inner automorphisms inside GnH.

The semi-direct products considered in this section are of the type GnρV , where G is aLie group and ρ : G→ Aut(V ) is a linear representation of G: each elements of GnρV is apair (g, v) in G×V , and the product is defined by: (g1, v1)(g2, v2) = (g1g2, ρ(g−1

2 )(v1)+v2).Similarly, the elements of Lie(G nρ V ) are pairs (a, u) in g × V , and the bracket is then:[(a1, u1), (a2, u2)] = ([a1, a2], dρ(a1)(u2) − dρ(a2)(u1)), where dρ : g → End(V ) denotesthe induced representation of the Lie algebra g of G, i.e. derivative at the origin of ρ. Inparticular, it is easy to check that the Killing form, say BGnρV , is then given by

(6.6.7) BGnρV ((a1, u1), (a2, u2)) = BG(a1, a2) + tr(dρ(a1) dρ(a2)).

In particular, V , as an (abelian) Lie subalgebra of Lie(G nρ V ), is always contained inthe kernel of BGnρV and Lie(G nρ V ) is then never semisimple, unless ρ is a trivialrepresentation and g is semisimple.

6.6. FANO COMPLEX SURFACES 163

which is abelian, hence reductive. It is easy to check that Aut0(Mk) = 1if k ≥ 4. It follows that Aut0(Mk) is reductive for all k ≥ 3 and theLichnerowicz-Matsushima criterion does not apply in these cases.

It may be asked whether a Fano manifold with reductive group of au-tomorphisms or, more generally, a Fano manifold for which Futaki criterionis fulfilled — cf. Theorem 4.12.1 below — admits Kahler-Einstein metrics.This long standing “folkloric conjecture” was supported by the following:

thmtiansurface Theorem 6.6.1 (G. Tian [181]). All (smooth) Fano complex surfaceswhose automorphism group is reductive, i.e. all Fano complex surfaces apartfrom the blow-up of P2 at one or at two points, admit a Kahler-Einsteinmetric.

For the proof of this result, we refer to Tian’s original paper [181], cf.also [191], [157] (it must be observed that the assumption smooth is hereessential and that the conclusion of Theorem 6.6.1 does not hold for Fanocomplex surfaces with orbifold singularities, [69]).

The above “folkloric conjecture” proved however to be false when G.Tian constructed the first known example of a Fano manifold of complexdimension 3, which has no non-trivial holomorphic vector field and whichadmits no Kahler-Einstein metric, cf. [186]: in this paper, G. Tian estab-lishes the non-existence of a Kahler-Einstein metric bu showing that themanifold contradicts the so-called weak K-stability condition introduced inthe same paper, and formulates the conjecture that a Fano complex manifoldadmits Kahler-Einstein metric if and only if it is weakly K-stable (K-stableif it has no non-trivial holomorphic vector field), [186, Conjecture 5], cf.also [186, Acknowledgement] for comments about this conjecture.

Remark 6.6.1. By Proposition 6.6.1 the blow-up of P2 at one point— which is also the first Hirzebruch surface F1, cf. Section 6.4 — admitsno Kahler metric with constant scalar curvature, but it does admit extremalKahler metrics as shown by E. Calabi in [45]. Calabi construction on F1 andthe other Hirzebruch surfaces, as well as its natural extension to Hirzebruch-like ruled surfaces of any genus is the main subject of Chapter 10.

On the other hand, the issue of the existence/non-existence of extremalKahler metrics on the blow-up of P2 at two points has long remained anopen question until the recent works of C. Arezzo-F. Pacard [13], [14] andof C. Arezzo-F. Pacard-M. Singer [15].

CHAPTER 7

Kahler-Einstein metrics on Fano manifolds

sfano

Recall that a Fano manifold is a (connected) compact complex manifold(M,J) whose (real) first Chern class c1(M,J) in H2(M,R) is a Kahler class,cf. Section 1.20.

The existence issue for Kahler-Einstein metrics on Fano manifolds wascompletely solved by G. Tian in the cas when m = 2, cf. Theorem 6.6.1,but has remained a most challenging open question when m > 2.

7.1. The Ricci potential of a Fano manifoldssRPfano

Throughout this Chapter — unless otherwise specified — (M,J) willdenote a Fano manifold, of complex dimension m ≥ 2, equipped with theKahler class

Omega-fanoOmega-fano (7.1.1) Ω =2π

kc1(M,J),

for some k > 0. We also fix a base-point (g0, ω0) in the space MΩ ofKahler metrics of Kahler class Ω, which will be most often identified withI−1ω0

(0) ⊂ Mω0 , where Iω0 denotes the Donaldson gauge functional, cf. Sec-tion 4.1. Unless otherwise specified, any element ofMΩ will then be writtenunambiguously as ωφ = ω0 + ddcφ, with Iω0(φ) = 0.

The total volume∫M vg and the mean curvature

∫M sgvg∫M vg

with respect to

any metric g in MΩ are denoted by V and s respectively.

lemmapgphi-fano Lemma 7.1.1. For any (gφ, ωφ = ω0+ddcφ), the mean curvature is givenby

mean-fanomean-fano (7.1.2) s = 2km,

and the Ricci form by

rho-fanorho-fano (7.1.3) ρgφ = kωφ −1

2ddcpgφ ,

where the Ricci potential pg, normalized by (1.19.11), has the following ex-pression

pg-fanopg-fano (7.1.4) pgφ = pg0 + logvgφvg0

+ 2k(φ− Iω0(φ))− 1

VEω0(ωφ),

where Eω0 denotes the Mabuchi K-energy centered at ω0, as defined in Sec-tion 4.13.

Proof. By Proposition 1.19.1, the class of ρgφ inH2(M,R) is 2π c1(M,J).Since 2π c1(M,J) = kΩ, the harmonic part of ρgφ with respect to gφ is thenk ωφ. In particular, s = 2Λωφ(kωφ) = 2km, whereas ρφ is given by (7.1.3),

165

166 7. KAHLER-EINSTEIN METRICS ON FANO MANIFOLDS

where the Ricci potential pgφ , as defined by (1.19.10)-(1.19.11), is given by(1.21.3), up to an additive constant, hence by

pg-provpg-prov (7.1.5) pgφ = pg0 + 2k φ+ logvgφvg0

− C(φ)

V,

where C(φ) is constant on M and smoothly depends on φ. By integratingalong vgφ , this yields

(7.1.6) C(φ) =

∫Mpg0 vgφ + 2k

∫Mφ vgφ +

∫M

logvgφvg0

vgφ .

We now regard C as a function defined on Mω0 and we compute its derivativeat φ — corresponding to the metric (gφ, ωφ) in MΩ — along any f in

TφMω0 = C∞(M,R). By using (3.2.4), we obtain

(dC)φ(f) =−∫Mpg0 ∆gφf vgφ + 2k

∫Mf vgφ − 2k

∫Mφ∆gφf vgφ −

∫M

∆gφf vgφ

−∫M

logvgφvg0

∆gφf vgφ

= −∫Mf∆gφ

(pg0 + 2kφ+ log

vgφvg0

)vgφ + 2k

∫Mf vgφ .

(7.1.7)

By using (7.1.5) and (1.15.6) — as well as (1.19.2) and (7.1.2) — we thenget

(dC)φ(f) =

∫Mf Λωφdd

cpgφ vgφ + 2k

∫Mf vgφ

=

∫M

Λωφ(2kωφ − 2ρgφ)f vgφ + 2k

∫Mf vgφ

= −∫M

(sgφ − s) f vgφ + 2k

∫Mf vgφ

= d(Eω0 + 2kV Iω0)φ(f).

(7.1.8)

Since C(0) = 0, we deduce C(φ) = Eω0(φ) + 2kVΩ Iω0(φ).

Remark 7.1.1. By integrating both sides of (7.1.4) along vgφ , we getthe following expression of the Mabuchi K-energy on a Fano manifold

E-expressionE-expression (7.1.9) Eω0(ωφ) =

∫M

logvgφvg0

vgφ + 2k

∫Mφ vφ +

∫Mpg0 vgφ ,

for any ωφ = ω0 +ddcφ, with Iω0(φ) = 0. By Proposition 4.2.2,∫M φ vgφ ≤ 0;

we thus get the following majoration of Eω0 :

E-majorationE-majoration (7.1.10)1

VEω0(ωφ) ≤ 1

V

∫M

logvgφvg0

vgφ + maxM

pg0 .

Lemma 7.1.2. The first variation of the Ricci potential pg in MΩ isgiven by

pg-varpg-var (7.1.11) pg = −∆gf + 2k f +1

V

∫Mpg ∆gf vg,

for any g in MΩ = I−1ω0

(0) and any f in TgMΩ (so that∫M f vg = 0).

7.2. HOLOMORPHIC VECTOR FIELDS AND POINCARE INEQUALITIES 167

Proof. Direct consequence of (7.1.4), of (4.10.7) and of (1.19.12).

Notice that (7.1.11) implies∫M

(pg vg + pg vg

)= 0, in accordance with

the chosen normalization∫M pg vg = 0 of the Ricci potential.

By the very definition of pg, a Kahler metric g inMΩ is Kahler-Einsteinif and only if pg = 0. More generally

defi-soliton Definition 8. A Kahler metric in MΩ is a Kahler-Ricci soliton if theRicci potential pg is a Killing potential, i.e. if Xf := gradωpg = Jgradgpgis a Killing vector field with respect to g. Equivalently, g is a Kahler-Riccisoliton if

solitonsoliton (7.1.12) ρ = k ω +1

2LJXω,

for some hamiltonian Killing (real) vector field X.

7.2. Holomorphic vector fields and Poincare inequalitiessspoincarelemma-crux Lemma 7.2.1. For any complex 1-form α such that dα is J-invariant

and for any metric g in MΩ, we have the following identity

id-cruxid-crux (7.2.1) epg δ(e−pg D−α

)=

1

2∆gα− k α+

1

2d 〈α, dpg〉 −

1

2dc 〈Jα, dpg〉.

Proof. In (7.2.1) and throughout this paragraph, 〈·, ·〉 denotes the her-mitian inner product determined by g, which is C-linear in the first argumentand C-anti-linear in the second one. From (1.23.10), with ρ = kω− 1

2ddcpg,

we infer

δD−α =1

2∆gα− k α−

1

2Jα]yddcpg

=1

2∆gα− k α−

1

2LJα]dcpg +

1

2d(Jα]ydcpg

)=

1

2∆gα− k α+

1

2d(α, dpg)−

1

2dc(Jα, dpg) +

1

2dpg LJα]J

for any 1-form α, where α] denotes the C-linear dual of α, defined byg(α], Z) = α(Z) for any (complex) vector field Z, and g is C-bilinearly ex-tended to complex vector fields. The rhs of (7.2.1) is then equal to δD−α−12dpg LJα]J , whereas, by (1.23.7), −1

2dpg LJα]J = D−α(·, gradgpg). Now,

dα is J-invariant if and only ifD−α is symmetric (easy verification). We thusget: δD−α− 1

2dpgLJα]J = δD−α+D−α(gradgpg, ·) = epgδ(e−pgD−α

).

lemma-crux-bis Lemma 7.2.2. For any Kahler metric g in MΩ, of Ricci potential pg,and for any complex-valued function F , we have

id-crux-bisid-crux-bis (7.2.2) epg δ(e−pg D−∂F

)= ∂

(D+g F − k F

),

where D+g denote the Futaki twisted Laplacian defined by:

futaki-laplacianfutaki-laplacian (7.2.3) D+g = epg ∂∗ (e−pg ∂F ),

cf. [86, page 40].


Proof. Direct application of (7.2.1), with α = ∂F , via the alternativeformulation of D+

g :

futaki-laplacian-bisfutaki-laplacian-bis (7.2.4) D+g F = gF + 〈∂F, dpg〉 =

1

2∆gF + 〈∂F, dpg〉.

prop-poincare Proposition 7.2.1 ([86] Theorem 2.4.3,[190] Lemma 3.1). Let g be anyKahler metric in MΩ, and denote by pg its Ricci potential. Then, for anycomplex function F such that

∫M F e−pg vg = 0, we have:

poincarepoincare (7.2.5)

∫M|∂F |2g e−pg vg − k

∫M|F |2 e−pg vg ≥ 0,

with equality if and only if F is a holomorphic potential with respect to g, cf.Definition 2. In particular, for any real function f such that

∫M fe−pg vg =

0, we have

poincare-realpoincare-real (7.2.6)

∫M|df |2g e−pg vg − 2k

∫Mf2 e−pg vg ≥ 0,

with equality if and only if f is a Killing potential with respect to g.

Proof. The Futaki twisted laplacian D+g is a linear elliptic second or-

der differential operator, defined on the space C∞(M,C) of complex-valuedfunctions. From (7.2.3), we readily infer that it is self-adjoint with respectto the volume form e−pg vg, i.e. D+

g satisfies:

(7.2.7)

∫M〈D+

g F1, F2〉 e−pg vg =

∫M〈F1,D+

g F2〉 e−pg vg,

for any complex functions F1, F2, and that

pre-Poincarepre-Poincare (7.2.8)

∫M〈D+

g F, F 〉 e−pg vg =

∫M|∂F |2 e−pg vg,

which implies that D+g is semi-positive and that its kernel is the space of

holomorphic functions, hence the space C of constant functions, since M iscompact. We then have

futaki-decfutaki-dec (7.2.9) C∞(M,C) ⊂ L2(M,C; e−pg vg) =⊕∞

r=0Eλr ,

where 0 = λ0 < λ1 < . . . < λr < . . . denotes the eigenvalues of D+g , whereas

the sum is a Hilbert sum with respect to the hermitian product

futaki-innerfutaki-inner (7.2.10) 〈F1, F2〉pg :=

∫M〈F1, F2〉 e−pg vg,

with E0 = C. In particular

D0 := C∞(M,C) ∩⊕∞

r=1Eλr

= F ∈ C∞(M,C) |∫MF e−pg vg = 0.

D0D0 (7.2.11)

From (7.2.2), we infer that

(7.2.12)

∫M|D−∂F |2 e−pg vg = λr(λr − k)

∫M|F |2 e−pg vg,

7.2. HOLOMORPHIC VECTOR FIELDS AND POINCARE INEQUALITIES 169

for any eigenvalue λr and any (complex) eigenfunction F of D+g relative

to λr. This implies that any positive eigenvalue λr of D+g satisfies λr ≥

k and that k is actually an eigenvalue of D+g if and only if the space of

holomorphic potentials F with respect to g, normalized by∫M F e−pgvg = 0,

is not reduced to 0; this is then identified with the k-eigenspace of D+g ,

cf. Section 2.5. It follows that the restriction of D+g − k Id to D0 is semi-

positive and that its kernel is either 0 or Eλ1 , i.e. the space of holomorphicpotentials F normalized by

∫M F e−pg vg, if the latter is not reduced to 0,

i.e. if λ1 = k. From (7.2.8), we then infer(7.2.13)∫

M〈(D+g F − k F

), F 〉 e−pg vg =

∫M|∂F |2 e−pg vg − k

∫M|F |2 e−pg vg ≥ 0,

for any complex function F such that∫M F e−pg vg = 0, with equality if and

only if F is holomorphic potential. We thus get the first part of Proposition7.2.1. If f is real, we get (7.2.6) by observing that |∂f |2 = 1

2 |df |2, and

equality holds if and only if X = gradgf is (real) holomorphic, if and onlyif f is a Killing potential, cf. Section 2.6.

Remark 7.2.1. In the above argument, the complex function F canbe convenientlt regarded as a section of the trivial complex line bundleL = M × C equipped with its natural holomorphic structure and with thehermitian inner product defined by (7.2.10), for which |1|2pg =

∫M e−pg vg.

In particular, the curvature form of the corresponding Chern connection ∇is k ω − ρ. With respect to this hermitian inner product, the (hermitian)adjoint of ∂ is ∂∗pg := epg ∂∗ e−pg and D+

g is then the usual Dolbeault laplacian

LF = (∂∗)L ∂F , as defined in Section 1.16.

prop-killing-fano Proposition 7.2.2. For any Kahler metric g in MΩ, of Ricci potentialpg, denote by Dg the operator on C∞(M,R) defined by

DgDg (7.2.14) Dgf = epgδ(e−pgf) = ∆gf + (df, dpg),

for any real function f . Denote by DR0 = C∞(M,R)∩D0 the space of smooth

real functions f such that∫M f e−pg vg = 0. Then, Dg − 2k Id preserves DR

0 ,

its restriction to DR0 is semi-positive and, for any f in DR

0 ,

futaki-realfutaki-real (7.2.15) Dgf − 2k f = 0,

if and only if f is a Killing potential with respect to g.

Proof. The operator Dg defined by (7.2.14) is clearly self-adjoint withrespect to the volume form e−pg vg. Moreover, since

∫M (Dgf−2k f)f e−pg vg =∫

M |df |2 e−pg vg−2k

∫M f2 e−pg vg, it follows from the second part of Propo-

sition 7.2.1) that the restriction of Dg − 2k Id to DR0 and that its ker-

nel is the space of Killing potentials f with respect to g normalized by∫M f e−pg vg.

Remark 7.2.2. By the proof of Proposition 7.2.1, we know that a com-plex function F such that

∫M F e−pg vg = 0 is a holomorphic potential with

respect to g if and only if D+F − k F = 0, i.e.

futaki-complexfutaki-complex (7.2.16) ∆gF − 2k F + (dF, dpg)− i(dcF, dpg) = 0.


A real function f is then a Killing potential, i.e. a real holomorphic potential,with respect to g if and only if (7.2.15) is satisfied as well as (dcf, dpg) = 0.Notice that the latter also reads LJgradfpg = 0 and is then automaticallysatisfied if f is a Killing potential since Jgradf is then a Killing vector field.Conversely, any real function f satisfying

∫M f e−pg vg = 0 and (7.2.15)

satisfies∫M |df |

2 e−pg vg − 2k∫M f2 e−pg vg = 0, hence is a Killing potential

with respect to g according to Proposition 7.2.1. In particular, the conditionDgf − 2k f = 0 automatically implies that LJgradgfpg = 0.

Proposition 7.2.3. Let X be any (real) holomorphic vector field on M .For any Kahler metric g inMΩ, of Ricci potential pg, denote by fXg the real

potential of X with respect to X, normalized by∫M fXg vg = 0. Denote by

FΩ : h→ R the Futaki character determined by Ω. Then

futaki-fanofutaki-fano (7.2.17)1

VFΩ(X) = −2k

∫M fXg e−pg vg∫M e−pg vg

.

Proof. Denote by F = fXg + ihXg the complex potential of X, normal-

ized by∫M F vg = 0. From (7.2.16) we infer

(7.2.18) ∆gF − 2k F + (dF, dpg)− i(dcF, dpg) = C,

for some (complex) constant C. By integrating over M along vg, and bytaking into account that δcd = 0, we get

C V =

∫M

(dF, dpg) vg =

∫MF∆pg vg =

∫MF (sg − s) vg

= FΩ(X)− iFΩ(JX).

(7.2.19)

By integrating along e−pg vg, we get instead:

(7.2.20) C

∫Mepg vg = −2k

∫MF e−pg vg.

By comparing these two expressions of C, we get (7.2.17).

Remark 7.2.3. If, instead, the real potential of X, call it then fXg , is

normalized by∫M fXg e−pg vg, a similar argument shows that

(7.2.21) FΩ(X) = 2k

∫MfXg vg.

rem-futaki-ding Remark 7.2.4. In view of (7.5.5) below, (7.2.17) shows that, up to thefactor −2k V , FΩ(X) is equal to the derivative of the Ding functional, intro-

duced in Section 7.5 below, along the lift X of X on MΩ, i.e. that FΩ = 0if and only if the Ding functional is constant on each orbit of H(M,J), cf.Remark 4.12.4 in Section 4.12.

prop-soliton-compact Proposition 7.2.4. Let g be a Kahler metric in MΩ and suppose thatg is a Kahler-Ricci soliton, as in Definition 8. As usual, denote by K(M, g)the identity component of the group of isometries of (M, g) and by H(M,J)the identity component of the group of automorphisms of (M,J). Then,K(M, g) is a maximal compact subgroup of H(M,J).

7.3. THE MODIFIED MONGE-AMPERE EQUATION 171

Proof. The statement is similar to Theorem 3.5.1 and the proof is alsoquite similar, even simpler since h = hred, as M is Fano, hence simply-connected. Consider the conjugate Futaki twisted laplacian D−g , defined by

D−g F = D+F = epg∂∗(e−pg ∂F ). Like D+g , D−g is a second order elliptic

operator on C∞(M,C), self-adjoint and semi-positive with respect to thevolume form e−pg vg, with the same spectrum 0 = λ0 < λ1 < . . . < λr < . . .as D+

g . On the space D0 of complex functions F satisfying∫M F e−pg vg = 0,

cf. (7.2.11), the two operators D+g −k Id and D−g −k Id are then semi-positive,

of the form

D+g F − k F =

1

2epgδ(e−pg dF )− k F +

i

2LJgradgpgF,

D−g F =1

2epgδ(e−pg dF )− k F − i

2LJgradgpgF,

D+D-fanoD+D-fano (7.2.22)

cf. (2.5.8)–(2.5.9). Moreover, since Jgradgpg is a Killing vector field, D+g −

k Id and D−g −k Id commute, cf. Proposition 3.3.2. It follows that D−g −k Id

acts on the kernel ker(D+g − k Id

)of D+

g − k Id as a semi-positive hermitian

operator, and coincides there with (−iLJgradgpg)|kerD+g −k Id = (epgδ(e−pgd−

2k Id)|kerD+g −k Id. In particular, the kernel of this action in ker

(D+−k Id

) ∼= h

is k⊕ Jk, and h then splits as

(7.2.23) h = k⊕ Jk⊕⊕λ>0h(λ),

whith h(λ) = X ∈ h | [Jgradgpg, X]λJX. Notice that this decompositionis formally identical to (3.4.5), with Jgradgpg instead of K = Jgradgsg. Therest of the argument is the same as for Theorem 3.5.1.

7.3. The modified Monge-Ampere equationssMAfano

In view of Lemma 7.1.1, the Einstein condition ρgφ = k ωφ, which isequivalent to the condition pgφ ≡ 0, can then be rewritten as the followingMonge-Ampere equation:

MA1MA1 (7.3.1) (ω0 + ddcφ)m = e−pg0−2k φ+ 1VEω0 (ωφ) ωm0 ,

where, we recall, φ is normalized by Iω0(φ) = 0. In particular, for anysolution to (7.3.1) in I−1

ω0(0), the Mabuchi K-energy of ωφ is given by

(7.3.2)1

VEω0(ωφ) = − log

1

V

∫Me−pg0−2k φvg0 = log

1

V

∫Mepg0+2k φvgφ .

The continuity method introduced by T. Aubin — cf. Section 1.20 —consists in replacing the Einstein equation ρgφ = k ωφ by the following familyof equations:

ρgφ = k(ω0 + tddcφ

)= k

(ωφ − (1− t)ddcφ

),

= tk ωφ + (1− t)k ω0,

KEtKEt (7.3.3)


parametrized by a real parameter t in the closed interval [0, 1]; equivalently,for each t in [0, 1], the Ricci potential has the following expression:

(7.3.4) pgφ = 2k(1− t)φ− 2k(1− t)V

∫Mφ vgφ ,

for ωφ = ω0 + ddcφ in MΩ. In view of Lemma 7.1.1, the latter equation isequivalent to the following modified Monge–Ampere equation:

MAtMAt (7.3.5) (ω0 + ddcφ)m = e−pg0−2tk φ+c(t,φ) ωm0 ,

where c(t, φ) is defined by

ctphi-Ectphi-E (7.3.6) c(t, φ) =1

VEω0(ωφ)− 2k(1− t)

V

∫Mφ vgφ ,

and φ is normalized by Iω0(φ) = 0. From (7.3.5), we also infer

ctphictphi (7.3.7) c(t, φ) = − log1

V

∫Me−pg0−2tk φvg0 = log

1

VΩ

∫Mepg0+2tk φvgφ .

For t = 1, (7.3.3) is the initial Kahler-Einstein equation. For t = 0,(7.3.3) reduces to the equation

rho-MA0rho-MA0 (7.3.8) ρgφ = k ω0,

and (7.3.5) specializes to

MA0MA0 (7.3.9) (ω0 + ddcφ)m = e−pg0+c(0,φ) ωm0 ,

with Iω0(φ) = 0, where, by (7.3.6)-(7.3.7), c(0, φ) is given by

c0c0 (7.3.10) c(0, φ) =1

VEω0(ωφ)− 2k

V

∫Mφ vgφ = − log

1

V

∫Me−pg0 vg0 .

prop-MA0 Proposition 7.3.1. For any chosen element ω0 in MΩ, the equation(7.3.8) has a unique solution, φ0, in I−1

ω0(0) and we then have

EnegEneg (7.3.11) Eω0(ωφ0) ≤ 0,

for ωφ0 = ω0 + ddcφ0.

Proof. The existence and uniqueness of a solution φ0 of (7.3.8), equiv-alently of the Monge-Ampere equation (7.3.9), is a direct consequence of theCalabi-Yau theorem 1.20.1. From (7.3.10), we infer

(7.3.12)1

VEω0(ωφ0) =

2k

V

∫Mφ0 vgφ0

− log1

V

∫Me−pg0 vg0 ,

where the rhs is the sum of two terms which are both non-positive: thefirst one by Proposition 4.2.2, as Iω0(φ0) = 0, the second one by the Jenseninequality1, since

∫M pg0 vg0 = 0.

1 In general, any real function f defined on a compact oriented riemannian (M, g), offnjensenvolume form vg and of total volume V =

∫Mvg, satisfies the Jensen inequality:

jensenjensen (7.3.13)1

V

∫M

fvg ≤ log1

V

∫M

efvg,

with equality if and only if f is constant. This is easily deduced fron the concavity of thegraph of the log function, cf. also the footnote 3 of the page 50.


rembounded Remark 7.3.1. Denote by M+Ω the subspace of elements of MΩ whose

Ricci form is positive definite, hence of the form k ω, where ω is the Kahlerform of an element ofMΩ. Then,M+

Ω coincides with the space of solutions(gφ0 , ωφ0 = ω0 + ddcφ0) of (7.3.8)-(7.3.9) when ω0 runs over all elements ofMΩ. By changing the notation, Proposition 7.3.1 has then the followingconsequence: for any ω in MΩ, there exists ω+ in M+

Ω such that

(7.3.14) Eω0(ω+) ≤ Eω0(ω),

for any base-point ω0 in MΩ. Indeed, by (7.3.11), we have: Eω0(ω+) −Eω0(ω) = Eω(ω+) − Eω(ω) = Eω(ω+) ≤ 0, if ω is chosen as the base-pointin (7.3.8)-(7.3.9) and ω+ = ωφ0 is then chosen to be the unique solution to(7.3.8)-(7.3.9) of base-point ω. In particular, if the K-energy Eω0 happensto be bounded from below on M+

Ω , it is bounded from below on the wholespace MΩ.

propcurve Proposition 7.3.2 ([20], [23] Proposition 4.4.1). There exists T in (0, 1]and a unique smooth curve φtt∈[0,T ), in I−1

ω0(0) such that, for each t in

[0, T ), φt is a solution to (7.3.5) and φt extends to [0, T ′) for no T ′ > Tin (0, 1].

Proof. (Sketch) Denote by H0 the subspace of C∞(M,R) defined byH0 = f | 1

V

∫M ef vg0 = 1 and let V the map from [0, 1] × I−1

ω0(0) to H0

defined by

(7.3.15) V(t, φ) = logvgφvg0

+ pg0 + 2tk φ− log1

VΩ

∫Mepg0+2tk φvgφ .

Then, φ in I−1ω0

(0) is a solution to (7.3.5) if and only if V(t, φ) = 0. Moreover,by (3.2.4), for any fixed t the derivative along the second variable of V at(t, φ) is the linear map

(7.3.16) f 7→ −(∆gφf − 2tk f) +

∫M (∆gφf − 2tk f) epg0+2k φvgφ∫

M epg0+2k φvgφ

from TφI−1ω0

(0) = f |∫M fvgφ = 0 to TV(t,φ)H0 = ψ |

∫M ψ eV(t,φ)vg0 = 0.

If (t, φ) is a solution to (7.3.5), so that V(t, φ) = 0, this then reduces to

linearlinear (7.3.17) f 7→ −(∆gφf − 2tk f) +1

V

∫M

(∆gφf − 2tk f) vg0

from TφI−1ω0

(0) = f |∫M fvgφ = 0 to T0H0 = ψ |

∫M ψvg0 = 0. For each

t in [0, 1], this map is an isomorphism of Frechet spaces. This is because, by(7.3.3),

riccitriccit (7.3.18) rgφ > tk gφ,

for any solution (t, φ) of (7.3.5), with t in [0, 1), so that, by Proposition3.6.1, 2tk is smaller than the smallest positive eigenvalue of ∆gφ . This im-plies that the map f 7→ ∆gφf − 2tk f is then a Frechet isomorphism from

f |∫M fvgφ = 0 to itself, hence that the map (7.3.17) is a Frechet isomor-

phism from f |∫M fvgφ = 0 to ψ |

∫M ψvg0. By introducing appropriate

Holder spaces and by using the ellipticity of ∆gφ , cf. Section 1.20 andChapter 5, we infer: For any solution (t, φ) of (7.3.5), t ∈ [0, 1), the map


V : (t, φ) 7→ (t,V) induces an isomorphism from a neighborhood of (t, φ) in(0, 1)× I−1

ω0(0) to some neighbourhood of (t, 0) in (0, 1)×H0.

In particular, by specializing to t = 0, we get ε in (0, 1] and a unique

smooth curve φt, t ∈ [0, ε), in I−1ω0

(0), whose image by V is the curve (t, 0) in[0, 1]×H0. If the curve φt cannot be extended to t = ε, we get Proposition7.3.2, with T = ε. If the curve φt smoothly extends to ε, then φε is a solutionto (7.3.5). By the above, we again get an isomorphism of a neighbourhoodof (ε, φε) in [0, 1] × Iω0 with a neighbourhood of (ε, 0) in [0, 1] ×H0, hencea unique smooth extension of the previous curve to [0, ε′), fro some ε′ > εin (0, 1]. This completes the proof of Proposition 7.3.2, by choosing T equalto the upper bound of the ε’s.

defi-maximal Definition 9. The smooth curve φtt∈[0,T ) in I−1ω0

(0) determined byProposition 7.3.2 is called the maximal solution of the modified Monge-Ampere equation (7.3.5).

remT Remark 7.3.2. It follows from the proof of Proposition 7.3.2 that themaximal solution φtt∈[0,T ) does not extend to T unless, possibly, if T = 1.If T = 1 and if φtt∈[0,T ) does extend to T = 1, then φT = φ1 is the relativeKahler potential with respect of ω0 of a Kahler-Einstein metric. It was shownby S.–T. Yau in [198], cf. also Lemma 3.2 and Lemma 3.3 in [185], thatthis happens if and only if |φt|C0 = max (supM φt,− infM φt) is uniformlybounded (Notice that, by Proposition 4.2.2, the condition Iω0(φt) = 0 impliessupM φt ≥ 0 and infM φt ≤ 0.)

The maximal solution φtt∈[0,T ) of (7.3.5) is determined by

maximalmaximal (7.3.19) (ω0 + ddcφt)m = e−pg0−2tk φt+c(t,φt)ωm0 ,

where φt is normalized by the Donaldson condition

donaldson-conddonaldson-cond (7.3.20) Iω0(φt) = 0,

for each t in [0, T ), and where, by Lemma 7.1.1, the constant c(t, φt) hastherefore the following expression:

ctphit-Ectphit-E (7.3.21) c(t, φt) =1

VEω0(ωφt)−

2k(1− t)V

∫Mφt vgφt .

Notice that (7.3.19) also implies the following tautological expression ofc(t, φt):

ctphitctphit (7.3.22) c(t, φt) = − log1

V

∫Me−pg0−2tk φtvg0 = log

1

VΩ

∫Mepg0+2tk φtvgφt .

propct Proposition 7.3.3. Along the maximal solution φtt∈[0,T ) of (7.3.19),c(t, φt) is a non increasing function of t. In particular

ctboundedctbounded (7.3.23) c(t, φt) ≤ c(0, φ0) = − log1

V

∫Me−pg0vg0 ≤ 0.

Proof. By differentiating both sides of (7.3.19), as functions of t, andby using (3.2.4), we get

dotmaximaldotmaximal (7.3.24) ∆gφtφt = 2k φt + 2tk φt − c(t, φt).


Since Iω0(φt) = 0 for any t in [0, T ), we have∫M φtvgφt = 0, whereas, by

Proposition 4.2.2,∫M φtvgφt ≤ 0. We thus infer from (7.3.24):

dotcdotc (7.3.25) c(t, φt) =2k

V

∫Mφtvgφt ≤ 0.

We thus get (7.3.23), where the last inequality is deduced from the Jensensinequality and

∫M pg0 vg0 = 0.

propEt Proposition 7.3.4 ([23] Theorem (5.7)). Along the maximal solutionφtt∈[0,T ) of (7.3.5), 1

V Eω0(ωφt) is a non-increasing function of t. In par-ticular,

EphitboundedEphitbounded (7.3.26)1

VEω0(ωφt) ≤

1

VEω0(ωφ0) ≤ 0,

for any t in [0, T ). If, moreover, the K-energy Eω0 is bounded from belowon MΩ, then T = 1.

Proof. By (7.3.21), we have:

EphitEphit (7.3.27)1

VEω0(ωφt) =

2k(1− t)V

∫Mφt vgφt + c(t, φt),

along φt. By Proposition 7.3.3, c(t, φt) is non-increasing and we nowcheck that

∫M φt vgφt is also non-increasing. Since Iω0(φt) = 0, we have∫

M φt vgφt = 0; by (3.2.4), the derivative of∫M φt vgφt is then equal to

−∫M φt∆gφt

φt vgφt . By using (7.3.24), this can be rewritten as

− 12k

∫M ∆gφt

φt (∆gφt− 2kt) φt vgφt = − 1

2k

∫M

((∆gφt

− 2kt)φt)2vgφt

− t∫M φ (∆gφt

− 2kt) φ vgφt . By (7.3.18) and Proposition 3.6.1, the self-adjoint operator ∆gφt

− 2kt, acting on the space of real functions of mean

value zero with respect to vgφt , is positive definite for any t in [0, T ), so that

−t∫M φt

(∆gφt

− 2kt)φt vgφt ≤ 0, whereas we evidently have:

− 12k

∫M

((∆gφt

− 2kt) φt)2vgφt ≤ 0. This completes the proof of the first

assertion, hence of the first inequality in (7.3.26) (the last one is is (7.3.11)).Assume that Eω0 ≥ C on MΩ for some positive real number C and

assume, for a contradiction, that T < 1. From (7.3.27) and Proposition7.3.3, we infer

(7.3.28) − 1

V

∫Mφt vgφt ≤

c(0, φ0)− C2k(1− t)

≤ c(0, φ0)− C2k(1− T )

,

It then follows from Proposition 7.3.5 below that φt would extend to asolution φT of (7.3.5) for t = T , which contradicts the definition of T , cf.Remark 7.3.2 above.

The following statement is essentially Proposition (3.6) in [23]:

propbounded Proposition 7.3.5. If either − 1V

∫M φt vgφt or 1

V

∫M φt vg0 is unifor-

maly bounded along the maximal solution φtt∈[0,T ) of the Monge-Ampereequation, then T = 1 and φt extends to a solution φ1 of (7.3.5) for t = 1,which is then the relative Kahler potential of a Kahler-Einstein metric inMΩ.


Proof. By Remark 7.3.2, it is sufficient to show that a uniform upperbound of − 1

V

∫M φt vgφt — equivalently, of 1

V

∫M φt vg0 by Proposition 4.2.2,

since Iω0(φt) = 0 — provides a uniform upper bound of maxM φt and of− infM φt, hence a uniform upper bound of |φt|C0 = inf (maxφt,−minM φt),the C0-norm of φt. This requires general arguments involving the Greenfonctions of metrics in MΩ. In general, for any (connected) compact rie-mannian manifold (M, g) of dimension n > 1, the Green function Gg of gis the kernel of the Green operator Gg defined in Section 1.17 and is thendetermined by:

green-def-fanogreen-def-fano (7.3.29) f(x) =1

V

∫Mf vg +

∫MGg(x, ·) ∆gf vg,

and

green-normgreen-norm (7.3.30)

∫MGg(x, ·)vg = 0,

for any (smooth) function f and any point x on M , cf. e.g. [19, Chapter 4].Let (gφ, ωφ = ω0+ddcφ) be any element ofMΩ. Then, by using (1.15.6),

we get

minDeltaphitminDeltaphit (7.3.31) ∆g0φ = −Λω0ddcφ = m− Λω0ωφ ≤ m,

and

maxDeltaphitmaxDeltaphit (7.3.32) −∆gφφ = Λωφddcφ = m− Λωφω0 ≤ m,

cf. Remark 1.20.1). Let xm, resp. xM , be any point of M such thatminM φ = φ(xm) and maxM φ = φ(xM ). Denote by −Ag0 , resp. −Agφ , theminimum of Gg0 , resp. Ggφ , on M × M , cf. [17, Theorem 4.13]. From(7.3.29)-(7.3.30) and (7.3.31)-(7.3.32), we get:

maxM

φ =φ(xM ) =1

V

∫Mφ vg0 +

∫MGg0(xM , ·) ∆g0φ vg0

=1

V

∫Mφ vg0 +

∫M

(Gg0(xM , ·) +Ag0) ∆g0φ vg0

≤ 1

V

∫Mφ vg0 +m

∫M

(Gg0(xM , ·) +Ag0) vg0

=1

V

∫Mφ vg0 +mVAg0 ,

supphitsupphit (7.3.33)

and

−minM

φ =− φ(xm) = − 1

V

∫Mφ vgφ −

∫MGgφ(xm, ·) ∆gφφ vgφ

= − 1

V

∫Mφ vgφ −

∫M

(Ggφ(xm, ·) +Agφ) ∆gφφ vgφ

≤ − 1

V

∫Mφt vgφ +m

∫M

(Ggφ(xm, ·) +Agφ) vgφ

= − 1

V

∫Mφ vgφ +mVAgφ .

infphitinfphit (7.3.34)

7.4. THE TIAN CONSTANT 177

Suppose now that φtt∈[0,T ) is the maximal solution to (7.3.5) and that

leqCleqC (7.3.35) − 1

V

∫Mφt vgφt ≤ C,

for some positive constant C independent of t. From (7.3.33) and Proposi-tion 4.2.2, we then infer:

maxM

φt ≤1

V

∫Mφt vg0 +mVAg0

≤ −mV

∫Mφt vgφt +mVAg0

≤ m(C + VΩAg0

).

maxphitmaxphit (7.3.36)

From (7.3.34), we get

minphitminphit (7.3.37) −minM

φt ≤ −1

V

∫Mφt vgφt +mVAgφt .

For any δ in (0, T ), (7.3.18) implies rgφt > kδ gφt for any t ∈ [δ, T ). By Myers’

theorem — cf. e.g. [28, Theorem 6.51] — this implies diamgφt<√

2m−1π√kδ

,

where diamgφtdenotes the diameter of (M, gφt). This, in turn, implies

(7.3.38) V Agφt <γ(m)

kδ,

for any t in [δ, T ), where γ only depends on m, cf. [23, Theorem (3.2)]. Wethus get

(7.3.39) −minM

φt ≤ C +mγ(m)

kδ,

for any t in [δ, T ), hence

(7.3.40) −minM

φt ≤ maxs∈[0,δ]

(−min

Mφs

)+ C +

mγ(m)

kδ,

for any t in [0, T ).

7.4. The Tian constantsstianconstant

proptianpositive Proposition 7.4.1 (G. Tian [181]). Let (M,J) be a Fano manifold andfix the Kahler class Ω = 2π

k c1(M,J), k > 0. Fix any base-point (g0, ω0) inMΩ. There then exists a positive real number α and a positive real constantCα, only depending on α and on (M,J, ω0), such that

alphaalpha (7.4.1)1

V

∫Me2kα(maxM φ−φ) vg0 ≤ Cα,

for any φ in Mω0.

Proof. Proposition 7.4.1 relies on the following general statement. Ingeneral a (smooth) real function f defined on a Kahler manifold (M, g, J, ω)is called plurisubharmonic if the J-invariant 2-form ddcf is everywhere posi-tive, meaning that the associated symmetric bilinear formX,Y 7→ ddcf(X, JY ),which is the J-invariant part D+df of the hessian of f , up to a factor 2 — seeSection 1.23 for the notation — is everywhere positive definite. By (1.15.6),we then have ∆gf ≤ 0 everywhere.


lemmahormander Lemma 7.4.1 ([104] Theorem 4.4.5). Let a, b any two real number witha < b. For any positive real number R, denote by BR(0) the open ball of Cmcentered at the origin 0 and of radius R. Let Pa,b,R denote the the set of thosepluriharmonic functions ψ defined on BR(0) which satisfy the following twoconditions:

(7.4.2) ψ(0) ≥ a, ψ(ζ) ≤ b,

for any ζ in BR(0). Then, there exists a positive real number C = C(a, b, R)such that

hormanderhormander (7.4.3)

∫BR/2(0)

e− ψ

(b−a) vg0 ≤ C,

for any ψ in Pa,b,R.

Proof. This is an easy generalization of Theorem 4.4.5 in L. Hormander’sbook [104].

In view of Lemma 7.4.1, the proof of Proposition 7.4.1 then goes asfollows (we here closely follow unpublished notes kindly communicated by

Y. Rubinstein). Let φ be any element of Mω0 , so that ω = ω0 + ddcφ isthe Kahler form of a Kahler metric g0. Without loss of generality, we canassume maxM φ = 0. From (7.3.33) — which holds for any φ in Mω0 — wethen infer

(7.4.4)

∫Mφ vg0 ≥ −mAg0V

2Ω .

Since φ ≤ 0, for any open subset U ⊂M we infer supU φ ≥ 1Vol(U)

∫U φ vg0 ≥

1Vol(U)

∫M φ vg0 ≥ −mAg0V

2Ω , hence

supphiUsupphiU (7.4.5) supUφ ≥ −

mAg0V2

Ω

Vol(U),

where Vol(U) denotes the volume of U with respect to g0. Since M is com-pact, we may assume M = ∪Nj=1Uj , for some positive integer N , where each

Uj is an open subset isomorphic to the open ball B0(R) in Cm. We then setUj = Bxj (R), where xj denotes the center of Uj in the chosen identificationτj : Uj 7→ B0(R). We may moreover assume that the Uj are chosen in sucha way that M = ∪Nj=1Bxj (R/6), with Bxj (R/6) = τ−1(B0(R/6)). In view

of (7.4.5), in each Bxj (R/6), there exists yj such that

phiyjphiyj (7.4.6) φ(yj) ≥ −mAg0V

2

Vol(Bxj (R/6)).

On the other hand, on each Uj = Bxj (R), ωo admits a Kahler potential,say ψj , so that ω0|Uj = ddcψj . By adding a constant to ψj , we may assume

ψj(yj) = 0. Moreover, each ψj admits a (non-negative) upper bound onBxj (5R/6) and we denote by C1 the greatest of them, so that ψj − C1 ≤ 0for any j = 1, . . . , N . As a Kahler potential of ω on Uj , φ + ψj is plusi-subharmonic, as well as φ+ ψj − C1, and we have

H1C1H1C1 (7.4.7) φ+ ψj − C1 ≤ 0,

7.4. THE TIAN CONSTANT 179

on Bxj (5R/6), hence also on Byj (2R/3) ⊂ Bxj (5R/6), whereas, because of(7.4.6),

H2H2 (7.4.8) (φ+ ψj − C1)(yj) ≥ −C1 −mAg0V

2

Vol(Bxj (R/6)).

By applying Lemma 7.4.1 to φ + ψj − C2 on the ball Byj (2R/3) ⊂ Uj ∼=B0(R), for any j = 1, . . . , N , we get

∫Byj (R/3) e

−aj φe−aj (C1−ψj) v0vg0

vg0 ≤ Cj ,

by setting aj =(C1 +

mAg0V2

Vol(Bxj (R)

)−1, where v0 denotes the volume form

determined by the holomorphic chart τj : Uj → B0(R) and the constant

Cj is determined by aj and R; we infer∫M e−aj φ vg0 ≤ Cj , where Cj is

the product of Cj and the upper bound of v0vg0

on Byj (R/3). Finally, since

Bxj (R/6) ⊂ Byj (2R/3), the open balls Byj (2R/3) cover M and we end upwith

(7.4.9)

∫Me−aφ vg0 ≤ C,

with a = minj aj > 0 and C =∑N

j=1 Cj . By substituting φ −maxM φ, we

obtain (7.4.1) for any φ in Mω0 .

defitianconstant Definition 10. The Tian constant, αM , is defined as the upper boundof the set of those positive numbers α such that there exists a positive realnumber Cα so that (7.4.1) holds for any φ in MΩ.

remtianconstant Remark 7.4.1. The Tian constant as defined above is independent ofthe chosen base-point (g0, ω0) in MΩ. Indeed, if we substitute any otherbase-point (g0, ω0 = ω0 + ddcψ), and if (7.4.1) holds for some pair α,C

and for any φ in Mω0 , then (7.4.1) holds for the pair α, C and for any

φ = φ − ψ in Mω0 , with, e.g. Cα = supMvg0vg0

eα(supM ψ−infM ψ). The Tian

constant is also independent of the scaling k of Ω: if Ω = 2πk c1(M,J) is

replaced by Ω′ = 2πk′ c1(M,J) and ω′0 = k

k′ω0 is chosen as a base-point of

MΩ′ , then, any φ′ in Mω′0is of the form φ′ = k

k′ φ, with φ in Mω0 , so that∫M ek

′α(supM φ′−φ′) vg′0 =(kk′

)m ∫M ekα(supM φ−φ) vg0 . The Tian constant αM

is then an invariant of the Fano manifold (M,J).

thmtianconstant Theorem 7.4.1 (G. Tian [181]). Let (M,J) be a Fano manifold of Tianconstant αM , equipped with the Kahler class Ω = 2π

k c1(M,J). Assume that

tiantian (7.4.10) αM >m

m+ 1.

Then, MΩ contains a Kahler-Einstein metric.

Proof. We first observe that the integral in (7.4.1) is invariant under

the natural R-action a ·φ = φ+a on MΩ: There is then no loss in restrictingour attention to those φ in MΩ which belong to I−1

ω0(0). Let α be any positive

number α satisfying (7.4.1) and let φtt∈[0,T ) be the maximal solution to themodified Monge-Ampere equation (7.3.5), as defined in Definition 9. From


(7.4.1) and (7.3.5), we infer:

logCα ≥ log1

VΩ

∫Me2kα(supM φt−φt) vg0

= log1

VΩ

∫Me2kα supM φt+2k(t−α)φt+pg0−c(t,φt) vgφt

= 2kα supM

φt + log1

V

∫Me2k(t−α)φt+pg0 vgφt − c(t, φt).

(7.4.11)

By using the Jensen inequality (7.3.13), we infer:

logCα ≥ 2kα supM

φt +1

V

∫M

(2k(t− α)φt + pg0

)vgφt − c(t, φt)

≥ 2kα1

V

∫Mφt vg0 +

2k(t− α)

V

∫Mφt vgφt + inf

Mpg0 − c(t, φt).

(7.4.12)

By Proposition 4.2.2,∫M φt vg0 ≥ − 1

m

∫M φt vgφt ≥ 0, whereas, by Proposi-

tion 7.3.3, c(t, φt) ≤ c(0, φ0). The above inequality can then be rewrittenas:

tianinequalitytianinequality (7.4.13) logCα ≥ 2k((m+ 1)

mα− t

)(− 1

VΩ

∫Mφt vgφt

)+ inf

Mpg0 − c(0, φ0).

We observe that the coefficient in front of − 1VΩ

∫M φt vgφt in (7.4.13) is pos-

itive for each t in [0, 1] if and only if α is greater than mm+1 and that α can

be chosen that way if and only if the Tian constant αM satisfies (7.4.10). Ifso, we get

(7.4.14) − 1

VΩ

∫Mφt vgφt ≤

logCα − infM pg0 + c(0, φ0)

2k( (m+1)

m α− 1)

and we conclude by using Proposition 7.3.5.

Remark 7.4.2. In the case when Aut(M,J) is not reduced to the iden-tity, Theorem 7.4.1 can be significantly improved by considering any com-pact subgroup — possibly non connected — of Aut(M,J), and by substi-tuting the Tian constant relative to G, αGM , defined a the upper bound of

the set of positive real numbers α such that (7.4.1) holds for any φ in MGω0

,the space of G-invariant relative Kahler potentials with respect to somearbitrary G-invariant ω0 in MΩ.

7.5. The Ding functionalssding

The Ding functional, Fω0 , was first introduced on Fano manifolds byWei-Yue Ding in [68], as a functional on MΩ — Ω = 2π

k c1(M,J), k > 0 —defined by

ding1ding1 (7.5.1) Fω0(ωφ) = 2k

(Jω0(ωφ)− 1

V

∫Mφ vg0

)− log

∫M e−pg0−2kφ vg0∫M e−pg0 vg0

,

for any ωφ = ω0 +ddcφ inMΩ (notice that this expression is independent ofthe chosen normalization of the Ricci potential pg0). By using (4.2.7), this

7.5. THE DING FUNCTIONAL 181

can be re-written as

ding2ding2 (7.5.2) Fω0(ωφ) = −2k Iω0(φ)− log

∫M e−pg0−2kφ vg0∫M e−pg0 vg0

.

It follows that Fω0 can be regarded as the restriction of −2k Iω0 to the space

of relative Kahler potentials φ in Mω0 normalized by the condition

(7.5.3)

∫Me−pg0−2kφ vg0 =

∫Me−pg0 vg0 .

In the setting of this Chapter, where the relative Kahler potential φ is nor-malized by Iω0(φ) = 0, Fω0 has then the following alternative expression:

ding2bisding2bis (7.5.4) Fω0(ωφ) = − log1

V

∫Me−pg0−2kφ vg0 + log

1

V

∫Me−pg0 vg0 .

Proposition 7.5.1. The derivative of the Ding functional Fω0 at (g, ω)in MΩ is given by

FderivativeFderivative (7.5.5) (dFω0)g(f) = 2k

∫M e−pg f vg∫M e−pg vg

for any (g, ω = ω0 + ddcφ) in MΩ — identified with I−1ω0

(0) in Mω0 — andany f in TωMΩ, with

∫M f vg = 0. In particular, ω is critical for Fω0 if and

only if pg ≡ 0, hence if and only if (g, ω) is Kahler-Einstein.

Proof. We readily deduce from (7.5.4) that

(7.5.6) (dFω0)g(f) = 2k

∫M e−pg0−2kφ f vg0∫M e−pg0−2kφ vg0

.

We then deduce (7.5.5) by using (7.1.4). From (7.5.5), we readily inferthat (g, ω) is critical for Fω0 if and only if pg is constant, hence identicallyzero. By the very definition (1.19.10)-(1.19.11) of the Ricci potential pg, thismeans that (g, ω) is Kahler-Einstein.

It follows from (7.5.5) that the Ding functional Fω0 is the primitivevanishing at ω0 of the 1-form $ on MΩ defined by

(7.5.7) $(g) = 2ke−pg vg∫M e−pg vg

,

riemannian dual of the vector field p defined by

(7.5.8) p(g) = 2ke−pg∫

M e−pg vg− 2k,

in the same way as the Mabuchi K-energy was defined as the promitivevaniching at ω0 of the 1-form σ(g) = sg vg, cf. Section 4.10, in particularLemma 4.10.1, for the notation.

lemmaDding Lemma 7.5.1. The covariant derivatives of p and $ relative to Mabuchiconnection D are given by

DhatpDhatp (7.5.9) Df p = 2k(Dg − 2k) f e−pg∫

M e−pg vg,


and

DvarpiDvarpi (7.5.10) Df$ = 2k(Dg − 2k) f e−pg vg∫

M e−pg vg,

for any g in MΩ and any f in TgMΩ = f ∈ C∞(M,R) |∫M f vg = 0,

where f is defined by

tildef-fanotildef-fano (7.5.11) f = f −∫M f e−pg vg∫M e−pg vg

.

Proof. As in Lemma 4.10.1 we only make the computation for Df p, as(7.5.10) readily follows from (7.5.9). By Proposition 4.5.1, Df p is given atg by

(7.5.12) Df p = f · p− (dp(g), df)g,

where f · p(g) stands for the first variation of p along f . By (7.1.11) and(3.2.4), the latter is given by

f · p = − 2k e−pg∫M e−pg vg

[−∆gf + 2k f +

1

V

∫Mpg∆gf vg

]+

2k e−pg( ∫M e−pg vg

)2 ∫M

(2k f +

1

V

∫Mpg ∆gf vg

)e−pg vg

=2k e−pg∫M e−pg vg

[∆gf − 2k f

],

calcul1calcul1 (7.5.13)

whereas

calcul2calcul2 (7.5.14) −(dp(g), df)g = 2k(dpg, df) e−pg∫

M e−pg vg.

By putting (7.5.13)–(7.5.14) together we get (7.5.9).

prop-ding-convex Proposition 7.5.2. The Ding functional Fω0 is convex with respect tothe Mabuchi connection D, i.e. its hessian with respect to D is semi-positiveon MΩ. More precisely,

ding-convexding-convex (7.5.15) (DfdFω0)(f) ≥ 0,

for any g in MΩ and any f in TgMΩ, with equality at g if and only if f isa Killing potential with respect to g.

Proof. Since $ = dFω0 , from (7.5.10) we infer that (DfdFω0)(f) =

2k∫M 〈(Dg−2k)f ,f〉 e−pgvg∫

M e−pg vg. We then conclude by using Proposition 7.2.2.

rem-bo Remark 7.5.1. Proposition 7.5.2 implies that the Ding functional isconvex along each geodesic of MΩ, as is the Mabuchi K-energy, cf. Sec-tions 4.10 and 4.15. The convexity of the Ding functional along (weak)geodesics of MΩ was first established by B. Berndtsson in [27].

propFE Proposition 7.5.3 ([70] Formula (3.8), [170] Lemma 2.3). The Ding func-tional is related to the Mabuchi K-energy by

ding3ding3 (7.5.16) Fω0(ωφ) =1

VEω0(ωφ)− log

1

V

∫Me−pgφvgφ + log

1

V

∫Me−pg0vg0 ,

7.5. THE DING FUNCTIONAL 183

for any (gφ, ωφ = ω0 + ddcφ) in MΩ. In particular, Fω0 is bounded fromabove by Eω0 in the following sense:

FEFE (7.5.17) Fω0(ω) ≤ 1

VEω0(ω) + log

1

V

∫Me−pg0vg0 .

Proof. By integrating epgφ along vgφ , when the Ricci potential pgφ isgiven by (7.1.4), we easily get the following expression of the K-energy onMΩ = I−1

ω0(0):

(7.5.18)1

VEω0(ωφ) = − log

1

V

∫Me−pg0−2kφ vg0 + log

1

V

∫Me−pgφ vgφ .

By comparing with (7.5.4), we readily get (7.5.16). We then deduce (7.5.17)

by using the Jensen inequality (7.3.13) log 1V

∫M e−pgφvgφ ≥ − 1

V

∫M pgφ vgφ ≥

0.

The following definition is due to G. Tian, cf. Definition 4.5 in [185](beware however that the functional denoted Fg in [185] is the opposite ofthe Ding functional as defined above):

Definition 11. The Ding functional is said to be proper with respectto the Aubin functional Iω0 , or simply proper, if there exists an increasingfunction h = h(s) defined on R, with lims→+∞ h(s) = +∞, such that

FproperFproper (7.5.19) Fω0(ωφ) ≥ h(Iω0(ωφ)),

for any ωφ = ω0 + ddcφ in MΩ.

Remark 7.5.2. It is easy to check that the properness of Fω0 as definedin (7.5.19) is independent of the chosen base-point ω0 in MΩ.

Proposition 7.5.4 (G. Tian). If the Ding functional Fω0 is proper,the maximal solution φtt∈[0,T ) smoothly converges to the relative Kahlerpotential of a Kahler-Einstein metric.

Proof. Assume that Fω0 is proper and consider Fω0(ωgφt ),

where φtt∈[0,T ) is the maximal solution to the deformed Monge-Ampereequation (7.3.5) relative to any base-point ω0 inMΩ, as defined in Definition9 (in particular, Iω0(φt) = 0 for any t in [0, T )). By using (7.5.17), (7.3.26)and (7.5.19), we thus get

(7.5.20) h(Iω0(ωφt)) ≤ Fω0(ωφt) ≤ −c(0) + log1

V

∫Me−pg0 vg0 .

We infer that Iω0(ωφt) is uniformly bounded from above and we concludethat ωφt smoothly converges to a Kahler-Einstein metric by using Proposi-tion 7.3.5, together with Proposition 4.2.2 and (4.2.5).

CHAPTER 8

Polarized Kahler manifolds

spolar

8.1. Polarized Kahler manifoldssspolarized

In general, a symplectic manifold (M,ω) is said to be (pre)quantized bya hermitian complex line bundle (L, h) if ω can be realised as the curvatureform of some hermitian connection ∇ of L, so that

polarpolar (8.1.1) R∇ = iω,

cf. Section 1.19.

propquantize Proposition 8.1.1. A symplectic manifold (M,ω) can be (pre)quantizedas above if and only if the de Rham class [ ω2π ] is integral, i.e. belongs to the

image of H2(M,Z) in H2dR(M,R).

Proof. A detailed proof of this basic and well-known fact would gobeyond the scope of these notes and we only give a sketchy argument. Recall

that for any hermitian connection ∇ on L, the real 2-form R∇

2πi is closed and

represents the (real) Chern class cR(L) of L in H2(M,R) (cR(L) is actually

represented by R∇

2πi for any C-linear connection ∇ on L, but we here onlyconsider hermitian connections). It turns out that cR(L) is the image inH2dR(M,R) of an element of H2(M,Z), called the (integral) Chern class

of L, denoted by c(L), and that the map L 7→ c(L) determines a groupisomorphism of H2(M,Z) with the set of classes of isomorphisms of complexline bundles over M , equipped with the abelian group structure induced bythe (complex) tensor product (we then have: c(L1 ⊗ L2) = c(L) + c(L2)),cf. e.g. [100] Chapter 1. It follows that a necessary condition that (M,ω)be (pre)quantized by some complex line bundle if that [ ω2π ] be integral, asdefined above (equivalently, the periods

∫C ω of ω on all closed 2-cycles C of

M are integers). Conversely, if [ ω2π ] is integral, it is the real Chern class of

a complex line bundle L, well-defined up to a torsion element in H2(M,Z).For any hermitian inner product h and any h-hermitian connection ∇ on L,−iR∇ and ω then belong to the same de Rham class, so that R∇ = i(ω+dα),for some real 1-form α. By replacing ∇ by ∇+iα, which is again a hermitianconnection on L, we get R∇+iα = R∇ − idα = iω.

We now consider the case when the symplectic form ω is the Kahler formof a Kahler structure (g, J, ω) on M . If ω is (pre)quantized as above, then∇ is the Chern connection of a uniquely defined holomorphic structure onL, as its curvature R∇ = i ω is J-invariant, cf. Proposition 1.6.1.

We accordingly change our point of view and say that a Kahler man-ifold (M, g, J, ω) is polarized by a holomorphic line bundle L if L admitsa hermitian inner product h such that the Kahler form ω is equal to thecurvature form of the corresponding Chern connection, cf. Section 1.7. We

185

186 8. POLARIZED KAHLER MANIFOLDS

then say that the Kahler manifold is polarized by the hermitian holomor-phic line bundle (L, h) and the Kahler form will be occasionally written ωh.By Proposition 8.1.1, a Kahler manifold (M, g, J, ω) can be polarized by aholomorphic line bundle L if and only if [ ω2π ] is integral in H2

dR(M,R).

defipositive Definition 12. Let (M,J) be a complex manifold and L a holomorphicline bundle over M . Then, L is said to be ample if there exists a Kahlermetric (g, J, ω) on M which is polarized by L. Equivalently, L is ample ifthere exists a hermitian inner product h on L such that the curvature formof the corresponding Chern connection is positive, i.e. is the Kahler form ofa Kahler metric on (M,J). The holomorphic hermitian line bundle (L, h) isthen called positive.

Let (M, g, J, ω) be a Kahler manifold polarized by a holomorphic linebundle L and denote by h a hermitian inner product on L, unique up torescaling, such that the Kahler form ω be the curvature form of the in-duced Chern connection. By Proposition 1.7.1, any (local) non-vanishingholomorphic section s of L provides a (local) Kahler potential, by

(8.1.2) ω =loc −1

2ddc log |s|2h,

where |s|2h denotes the square norm of s with respect to h. If L denotes thecorresponding C∗-principal bundle, regarded as as the bundle of non-zeroelements of L, and π the natural (holomorphic) projection from L to M , wethen have that

fakepotentialfakepotential (8.1.3) π∗ω = −ddc log rh,

where rh stands for the restriction to L of the norm function on L determinedby h. For any polarized Kahler manifold (M,L, h), the function − log rh on

L then appears as a substitute of a (in general missing) globally definedKahler potential.

rem-Fubini-Study Remark 8.1.1. According to (6.2.3), the standard complex projectivespace, equipped with the Fubini-Study Kahler structure of holomorphic sec-tional curvature c = 2 defined in Section 6.2, satisfies

rhofsrhofs (8.1.4) ρΛ∗ = ωFS ,

and is then polarized by the dual tautological bundle Λ∗ = O(1). From nowon, unless otherwise specified, the term Fubini-Study metric will implicitymean Fubini-Study metric of holomorphic sectional curvature +2, i.e. theKahler metric polarized by the dual tautological bundle for some hermitianinner product.

Any (compact) complex submanifold, M , of Pm = P(Cm+1) inheritsa Kahler structure, whose metric and Kahler form are the restrictions ofgFS and ωFS to M (note that ωM := ωFS |M is closed again, so that theinduced almost hermitian structure is still Kahler by Proposition 1.1.1).This metric is then polarized by the restriction to M of the dual tautologicalline bundle Λ∗ to M , equipped with the hermitian inner product inducedby the standard hermitian inner product of Cm+1.

A projective manifold is a compact complex manifold that can be realizedas a complex submanifold of some complex projective. Any holomorphic

8.1. POLARIZED KAHLER MANIFOLDS 187

embedding of a projective manifold in a complex projective space then turnsit into a polarized Kahler manifold. Conversely, any compact polarizedKahler manifold is a projective manifold. This is the main content of thecelebrated Kodaira embedding theorem, of which a precise statement is givenin Section 8.3.

We close this section by giving an alternative description of the reducedautomorphism group Hred(M,J) defined in Section 2.4 for a polarized com-pact Kahler manifold (M,L). Let L be a holomorphic line bundle overa compact complex manifold (M,L) and denote by π the natural (holo-morphic) projection from L to M . By an automorphism of L we mean a

biholomorphic map, Φ, from L to itself such that there exists an element Φof Aut(M,J) and a commutative diagram

diagram-Ldiagram-L (8.1.5) L

π

Φ // L

π

MΦ // M

with the following property: for any x in M , the restriction of Φ to π−1(x)is a C-linear isomorphism from π−1(x) to π−1(Φ(x)). The automorphismsof L form a group denoted by Aut(L); the identity component of Aut(L)

is denoted by H(L). Two elements Φ, Φ′ of Aut(L) induce the same Φ inAut(M,J) if and only if there exists an invertible holomorphic section, sayζ, of the holomorphic vector bundle End(L) = L∗ ⊗L such that Φ′ = ζ Φ.Since L is of rank 1, we have EndL = M × C, so that ζ can be viewed asa non-vanishing holomorphic complex function, hence a (non-zero) constantnumber, as M is compact. We then have the following exact sequences ofgroup homomorphisms:

diagram-exactdiagram-exact (8.1.6) 1 // C∗ // Aut(L) // AutL(M,J) // 1

1 // C∗ // H(L)?

OO

// HL(M,J)?

OO

// 1

where AutL(M,J), resp. HL(M,J), denotes the group of thoes elementsof Aut(M,J), resp. H(M,J), which lift to Aut(L), resp. to H(L). Bydefinition, HL(M,J) is a subgroup of H(M,J) and its Lie algebra, call ithL(M,J) = hL, is then a Lie subalgebra of h(M,J) = h. It turns out thatHL(M,J), hL(M,J), are actually independent of L. More precisely, we have(cf. [30] lemme 1.1):

propautML Proposition 8.1.2. Let (M, g, J, ω) be a compact Kahler manifold andsuppose that it is polarized by some holomorphic line bundle L. Then,

(8.1.7) HL(M,J) = Hred(M,J);

equivalently,

(8.1.8) hred = hL.

Proof. From the above, we infer that HL(M,J) can be viewed as thequotient of H(L) by the natural action of C∗. In view of Theorem 3.1 in


[120], the Lie algebra of H(L) is the space of those complete C∗-invariant(real) holomorphic vector fields defined on the total space of L, which projecton a (real) holomorphic vector field of M , and hL is then the quotient of thisLie algebra by the space of vector fields generated by the natural action ofC∗ on L, namely the space of vector fields of the form aT + b iT , for any tworeal numbers a, b, where T denotes the (real) vector field on L generated bythe natural action of S1, so that T (u) = iu for any u in L.

Let X be any element of hL ⊂ h and denote by X any lift of X in theLie algebra of the group of automorphisms of L (X is then well-defined upto vector fields of the form aT + b iT as above). Denote by ∇ the Chernconnection on L such that R∇ = iω and by H∇ be the associated horizontaldistribution on the total space of L, cf. Section 1.6. Since X projects to X,it can be written as

(8.1.9) X = X − hT + f iT,

where X denotes the horizontal lift of X with respect to H∇ and f, h arereal functions, defined up to additive constants. Since X is C∗-invariant,and X is already C∗-invariant — as ∇ is a C-linear connection — f, hare actually constant on each fiber: they can therefore be considered asreal functions defined on M and we then normalize them by

∫M fωm =∫

M hωm = 0. It remains to express the condition that LXJ = 0, whereJ stands for the complex structure operator defined on the total space ofL. In order to computeLXJ , it is convenient to use auxiliary vector fields,

namely horizontal lifts, Y , of vector fields Y on M and sections, s, of L,viewed as vertical vector fields on L, constant on each fibre, cf. Section 1.6.As already mentioned, cf. (1.6.11), the brackets of these vector fields on thetotal space of L are given by

(8.1.10) [Y , s] = ∇Y s,

whereas their brackets with T, iT are given by

(8.1.11) [T, Y ] = [iT, Y ] = 0,

as ∇ is a C-linear connection, and by

(8.1.12) [T, s] = −is, [iT, s] = s,

as T (u) = iu and (iT )(u) = −u, whereas, we recall, s, as a vertical vectorfield, is constant on each fibre. We also recall — cf. (1.6.12) in Section 1.6

— that for any two vector fields X,Y on M , the bracket [X, Y ] is the sum

of a horizontal vector field, namely the horizontal lift [X,Y ] of the bracket[X,Y ], and of a vertical vector field, namely the linear vertical vector fieldR∇X,Y = ω(X,Y )T . On the other hand, for any u in L, with π(u) = x,

we have that: TuL = H∇u ⊕ T Vu L ∼= TxM ⊕ Lx and J is then J on TxMand i on Lx (cf. Section 1.6). We are now ready to compute LXJ . Since

8.1. POLARIZED KAHLER MANIFOLDS 189

df(T ) = df(iT ) = dh(T ) = dh(iT ) = 0, we first get

(LXJ )(s) = [X, is]− J [X, s]

= [X, is]− i[X, s]− h[T, is] + hi[T, s] + f [iT, is]− fi[iT, s]= ∇X(is)− i(∇Xs) = 0,

(8.1.13)

for any section s of L. Then,

(LXJ )(Y ) = [X, JY ]− J [X, Y ]

= [X, JY ]− J [X, Y ]

+ dh(JY )T − df(JY )iT − dh(Y )iT − df(Y )T

= ˜(LXJ)(Y ) + ω(X, JY )T − ω(X,Y )iT

+ dh(JY )T − df(JY )iT − dh(Y )iT − df(Y )T

= (ω(X, JY ) + dh(JY )− df(Y ))T

− (ω(X,Y ) + df(JY ) + dh(Y )) iT

(8.1.14)

for any vector field Y on M . This part of LXJ vanishes if and only ifX = gradf + Jgradh, if and only if X belongs to hred. It follows that hL iscontained in hred.

Conversely, for any X = gradf + Jgradh in hred, the vector field X :=X − hT + f iT is a C∗-invariant, holomorphic vector field on L, whichprojects on X: it then belongs to hL, provided that its flow is complete. Inorder to check that X is complete for any X in hred, we use the followingcompleteness criterion taken from [1]:

Proposition 8.1.3 ([1] Proposition 2.1.20). Let Z be a (smooth) vectorfield, k ≥ 1, defined on some manifold N , and let φ a (smooth) real properfunction defined on N . Suppose that there exist two non-negative constantsA,B such that

AMAM (8.1.15) |dφ(Z)| ≤ A |φ|+B.

Then, the flow of Z is complete.

For a proof of this criterion, the reader is referred to [1], where theargument is given for the more general sitation when Z is Ck, k ≥ 1, andφ is C1. In the current situation, we can choose φ = r2 on L — the squarenorm function r2 determined by h — which is proper as M is compact. Wethen have dφ(X) = −2fr2 = −2fφ and (8.1.15) is then evidently satisfiedwith A = 2 SupM |f | and B = 0. This completes the proof of Proposition8.1.2.

remalternative Remark 8.1.2. For any compact Kahler manifold (M,J, g0, ω0) polar-

ized by an ample hermitian holomorphic line bundle (L, h0), the space Mω0

defined in Section 4.1 can be identified with the space, H>0(L), of hermit-ian inner products on L such that the curvature form of the correspondingChern connection is positive, i.e. the Kahler form of a Kahler structure inMΩ, with Ω = 2π c1(L). The identification goes as follows, cf. [74]. Anyhermitian inner product h on L is conformal to h0, as L is of (complex) rank


1, hence of the form h = e−2φ h0, for some real function φ. By Proposition1.7.2, the Chern curvature form of h, say ω, is then given by

(8.1.16) ω = ω0 + ddcφ.

It follows that h belongs to H>0(L) if and only if φ belongs to Mω0 . The ob-

vious advantage of considering H>0(L), instead of Mω0 , is that its definitionis independent of the choice of ω0 in MΩ, as well as the natural projection,p say, from H>0(L) to MΩ, which is simply the map h 7→ ρ∇, where, werecall, ρ∇ denotes the corresponding Chern curvature form. Again, p makesH>0(L) into a R-principal bundle over MΩ, via the R-action on H>0(L)defined by a · h = e−2a h, for any real number a. The whole structure ofMω0 , as described in Section 4.1, can then be transported on H>0(L), inparticular the connection 1-form τ and its primitive Iω0 , cf. Section 8.7below.

8.2. The Bergman density of a polarized Kahler manifoldssbergman

In this section (M, g, J, ω) denotes a (connected) compact Kahler mani-fold polarized by a hermitian holomorphic line bundle (L, h) in the sense of(8.1.1). The (pointwise) hermitian inner product h on L together with thevolume form vg of g then determine a hermitian inner product, 〈·, ·〉h, onthe space Γ(L) of (smooth) sections of L defined by

globalinnerglobalinner (8.2.1) 〈s, s〉h =

∫M

(s, s)h vg,

for any two s, s in Γ(L). By restriction, this induces a hermitian innerproduct on the (finite dimensional) space H0(M,L) of holomorphic sectionsof L, still denoted by 〈·, ·〉h.

defibergman Definition 13. For any compact Kahler manifold (M, g, J, ω) polarizedby a hermitian holomorphic line bundle (L, h), the Bergman density, BL,h,is the function defined on M by

bergmanbergman (8.2.2) BL,h(x) :=N∑j=0

|sj(x)|2h,

for any orthonormal basis s = s0, s1, . . . , sN of H0(M,L) with respect tothe (global) hermitian inner product 〈·, ·〉h.

rem-bergman-op Remark 8.2.1. The Bergman operator, ΠL, is defined as the orthogonalprojector from the Hilbert space L2(M,L) of L2-sections of (L, 〈·, ·〉) tothe (finite dimensional) subspace H0(M,L); for any s in L2(M,L) we then

have ΠL(s) =∑N

j=0〈s, sj〉h sj for any orthonormal basis s = s0, . . . , sN of

H0(M,L). The Bergman kernel is then defined by BL(x, y) =∑N

j=0 sj(y)⊗sj(x) in Ly ⊗ L∗x, for any pair (x, y) in M ×M and BL,h can then regardedas the restriction of the Bergman kernel to the diagonal M of M ×M (formore information on the Bergman kernel we refer the reader to [140] andreferences therein).

The following theorem concerns the behaviour of the Bergman densityBLk,h(k) when the positive integer k tends to infinity. Here, BLk,h(k) is

8.2. THE BERGMAN DENSITY 191

the Bergman density of the polarized Kahler manifold (M,Lk) when Lk

is equipped with the hermitian inner product h(k) induced by h, so that:

(8.2.3) BLk,h(k)(x) =

Nk∑i=0

|s(k)i (x)|2

h(k) ,

for any orthonormal basis s(k) of H0(M,Lk) with respect to the hermitian

inner product 〈·, ·〉h(k) of H0(M,Lk) determined by h(k) and the volume formvgk = km vg. We then have:

thmzeldichandco Theorem 8.2.1 (S. Zelditch [200], D. Catlin [52], Z. Lu [138]). For anycompact Kahler manifold (M, g, J, ω) of complex dimension m, polarized bya hermitian holomorphic line bundle (L, h), the Bergman density BLk,h(k) of

(M,Lk) admits an asymptotic expansion

zeldichzeldich (8.2.4) BLk,h(k) '1

(2π)m

+∞∑r=0

φrkr,

when k tends to infinity, where, for each non-negative integer r, φr is asmooth function on M which is a universal polynomial expression of thecurvature R of M and the covariant derivatives of R up to the order 2r− 1.

The asymptotic expansion is understood in the following sense: for anynon-negative integers q, `, there exists a positive constant Cq,` such that

mamamama (8.2.5) |BLk,h(k) −1

(2π)m

q∑r=0

φrkr|C` ≤

Cq,`kq+1

,

where | |C` stands for the usual C`-norm.Moreover, each constant Cq,` is uniform with respect to the Cs-norm of

ω, with s = ` + 2(m + q + 2), and (8.2.5) then holds when the C`-norm in(8.2.5) includes the derivatives of ω up to the order `.

Finally, the first two coefficients φ0, φ1 in (8.2.4) are given by

lulu (8.2.6) φ0 ≡ 1, φ1 =sg4,

where s denotes the scalar curvature of (M, g, ω).

Proof. The fact that the family of Bergman densities BLk,h(k) admitsan asymptotic expansion as above can be regarded as a special case of amore general theorem due to L. Boutet de Monvel and J. Sjostrand [40].In the current setting, a weak form of Theorem 8.2.1, namely the estimate(8.2.5) for q = 0 and ` = 2, is due to G. Tian [182] — cf. also [37], [38],[65]; the above full asymptotic asumption has been first given by S. Zeldich[200] and D. Catlin [52], cf. also [169]. The explicit determination of thecoefficients φ0 and φ1 is due to Z. Lu [138], which also includes the explicitdetermination of φ2 and φ3. The formulation of Theorem 8.2.1 given hereis extracted from the more general Theorems 4.5 and 4.6 in [140]). Theproof of Theorem 8.2.1 goes far beyond the scope of these notes: we referthe reader to the above references and to [30].

As a direct consequence of Theorem 8.2.1 we get:


propdimasymptotic Proposition 8.2.1. Let M be any compact Kahler manifold (M, g, J, ω)of complex dimension m, polarized by a hermitian holomorphic line bundle(L, h). Then, the dimension d(Lk) = Nk +1 of H0(M,Lk) has the followingasymptotic expansion

dimasymptoticdimasymptotic (8.2.7) d(Lk) ' 1

(2π)m

+∞∑r=0

ar km−r,

meaning that, for any non-negative integer q, we have

(8.2.8) d(Lk) =1

(2π)m

q∑r=0

ar km−r +O(km−r−1),

with ar =∫M φr vg. In particular,

aa (8.2.9) a0 = V =

∫Mvg, a1 =

1

4

∫Msg vg.

Proof. By the very definition of the Bergman density, we have thatd(Lk) =

∫M BLk,h(k) vgk = km

∫M BLk,h(k) vg, where gk denotes the metric

determined by ωk = k ω. We then conclude as for Proposition 8.9.1, byusing Theorem 8.2.1.

remchern Remark 8.2.2. For k large enough, Lk is very ample and all cohomologyspace Hj(M,Lk) are reduced to 0 except for H0(M,Lk). Then, d(Lk) isequal to the holomorphic Euler characteristic χ(L), defined by

(8.2.10) χ(L) =m∑j=0

(−1)j hj(L),

where hj(L) denotes the (complex) dimension of Hj(M,L). In general, χ(L)is expressed by the Riemann-Roch formula which can be written as

(8.2.11) χ(L) = (ch(L) ∪ Td(M,J))[M ],

where ch(L) denotes the Chern character of L and Td(M,J) the Todd classof the tangent bundle (TM, J), viewed as a complex vector bundle of rankm over M . For k large enough, we then have

RRkRRk (8.2.12) d(Lk) = (ch(Lk) ∪ Td(M,J))[M ],

where the rhs is actually a polynomial in k of degree m. To make thispoint precise, recall that the Chern character is the additive characteristic

class associated with the power series ex =∑∞

j=0xj

j! , whereas the Todd

class is the multiplicative characteristic class associated to the power seriesx

1−e−x = 1 + x2 +

∑∞j=1(−1)j−1 B2j

(2j)! x2j , where the B2j are positive rational

numbers, namely the so-called Bernoulli numbers. For any complex vectorbundle E of rank r over M , we then have: ch(E) =

∑rj=1 e

γj and Td(E) =∏rj=1

γj

1−e−γj, where the γj ’s denote the Chern roots, formally defined by

c(E) =∏rj=1(1 +γj), where c(E) = 1 +

∑rj=1 cj(E) denotes the total Chern

class of E; the Chern roots γj defined that way don’t live in H2(M,Z), onlyin some extension of H2(M,Z), but each symmetric polynomial functionof the γj ’s is a well-defined polynomial function of the Chern classes cj(E),hence a well-defined element inH∗(M) (for all the material mentioned in this

8.3. THE KODAIRA EMBEDDING THEOREM 193

remark, we refer the reader to [100, Chapter 1]). In the current situation,where L is of rank 1, the (unique) Chern root γ1 is equal to the first Chernclass c1(L), which is represented by ω

2π , as the Kahler structure is polarized

by L; similarly, the first Todd class Td1(M,J) is equal to 12c1(M,J), which is

represented by ρ4π , where ρ denotes the Ricci form of the Kahler structure (cf.

Section 1.19). The rhs of (8.2.12) is then equal to∫M (1+

∑mi=1 k

i(ω2π

)i)(1+

ρ4π + . . .), where only forms of degree m contribute non trivially. The righthand side is then a polynomial of degree m in k, whose leading coefficientis equal to 1

(2π)m

∫M

ωm

m! and the second coefficient is equal to 1(2π)m

∫M

12ρ ∧

ωm−1

(m−1)! = 1(2π)m

∫M

sg4ωm

m! . We thus retrieve (8.2.9).

Theorem 8.2.1 plays a prominent role in this chapter, as will be clearin the next sections. In particular, Theorem 8.2.1 appears as a crucial toolin the proof of Donaldson’s Theorem 8.7.1. A weighted version of Theorem8.2.1 plays a similar role in Mabuchi’s improvement of Theorem 8.7.1 quotedin Remark 8.7.3.

8.3. The Kodaira embedding theoremsskodaira

We first introduce the canonical map, ϕL, associated to any pair (M,L)formed of a (connected) compact complex manifold (M,J) and a holomor-phic line bundle L over M (no Kahler metric on M is needed at this level).

Denote by E = H0(M,L) the (complex) vector space of holomorphicsections of L; we assume that its dimension is positive, equal to N + 1.Denote par E∗ the (complex) dual of E and by P(E∗) the corresponding(complex) projective space. We denote by DL the set of points x of M suchthat s(x) = 0 for all s in E, and by M0 the open subset M \DL.

A point [α] of P(E∗) can be viewed either as a complex line in E∗ oras a complex hyperplane in E, namely the kernel, H[α] = (E∗/[α])∗, of anynon-zero element α of E∗ in the complex line [α]. Both viewpoints are usefuland will be used indifferently. In particular, the tautological line bundle Λover P(E∗) is naturally described by:

tauttaut (8.3.1) Λ[α] = [α] ⊂ E∗,

whereas the dual tautological line bundle Λ∗ = O(1) is best described by

tautdualtautdual (8.3.2) Λ∗[α] = E/H[α],

for any [α] in P(E∗).The canonical map ϕL : M0 → P(E∗) is the holomorphic map defined

as follows:

kodaira1kodaira1 (8.3.3) ϕL(x) = s ∈ E | s(x) = 0,

for any x in M0 (since x sits in M0 = M \ DL, the rhs of (8.3.3) is acomplex hyperplane in E, hence determines an element of P(E∗) in thesecond acceptation).

For any x in M0 denote par evx : E → Lx, the evaluation map at x,defined by: evx(s) = s(x). The map ϕL is then alternatively defined by

kodaira2kodaira2 (8.3.4) ϕL(x) = ev∗x(L∗x),


where ev∗x : L∗x → E∗ denotes the (algebraic) dual map of evx and ev∗x(L∗x)the image of L∗x in E∗ (notice that evx is surjective for any x in M0, so thatev∗x is injective).

Whichever formulation we adopt to define ϕL, we get the following nat-ural identification

LLambdastarLLambdastar (8.3.5) ϕ∗L(Λ∗) = L|M0,

of the restriction of L to M0 with the pull-back of the dual tautological linebundle Λ∗ by ϕL. This means that

idLidL (8.3.6) Lx ∼= Λ∗ϕL(x) = E/HϕL(x),

for any x in M0: to any u in Lx we associate the (well-defined) element [s]in E/HϕL(x) such that s(x) = u. We similarly get a natural identification

(8.3.7) ϕ∗L(Λ) = L∗|M0,

i.e. a natural isomorphism

idL*idL* (8.3.8) L∗x∼= ΛϕL(x) = ϕL(x) ⊂ E∗,

for each x in M0, defined as follows: to any ζ in L∗x we associate ev∗x(ζ).The link between these two viewpoints is given by the duality identity:

dualityduality (8.3.9) 〈ev∗x(ζ), s〉 = 〈ζ, s(x)〉

for any s in E and any ζ in L∗x.For any x in M0, the tangent space TϕL(x)P(E∗) is naturally identi-

fied with the space Hom(HϕL(x), E/HϕL(x)) of C-linear homomorphismsfrom the corresponding hyperplane HϕL(x) to the quotient E/HϕL(x), cf.Section 6.1. By (8.3.6), Hom(HϕL(x), E/HϕL(x)) is the same as the spaceHom(HϕL(x), Lx). The tangent map to ϕL at x is then tautolgically givenby

devkodairadevkodaira (8.3.10) (dϕL)x(X) = s ∈ HϕL(x) 7→ (dXs)(x),

where (dXs)(x) stands for the derivative of s at x along X (this makes sense,as s(x) = 0; e. g. (dXs)(x) = (∇Xs)(x) for any C-linear connection on L).

For any basis s = s0, s1, . . . , sN of E, denote by σ = σ0, σ1, . . . , σNthe (algebraic) dual basis of E∗, so that 〈σi, sj〉 = δij , for any i, j in0, 1, . . . , N . For any such choice, we get the following expression of ϕL:

kod3kod3 (8.3.11) ϕL(x) = [N∑i=0

sj(x)⊗ σj ] ⊂ Lx ⊗ E∗,

via the natural identification P(Lx ⊗E∗) ∼= P(E∗) for any x in M0. Indeed,for any s in E and any x in M0, we evidently have that

(8.3.12) 〈N∑i=0

sj(x)⊗ σj , s〉 = s(x),

so that the kernel of∑N

i=0 sj(x)⊗ σj in E coincides with HϕL(x). Note that

the expression∑N

i=0 sj(x)⊗σj appearing in the rhs of (8.3.11) is independentof the choice of the basis s of E.

8.3. THE KODAIRA EMBEDDING THEOREM 195

On the other hand, s identifies P(E∗) with the standard projective spacePN via the map s : P(E∗)→ PN defined by

hatshats (8.3.13) s([σ]) = (〈σ, s0〉 : 〈σ, s1〉 : . . . : 〈σ, sN 〉),for any non-zero σ in E∗, where [σ] denotes the corresponding element of

P(E∗). By substituting [σ] =∑N

j=0[sj(x)⊗ σj ] in P(Lx ⊗ E∗) = P(E∗), wethus get

kod-corkod-cor (8.3.14) js(x) := s ϕL(x) = (s0(x) : s1(x) : . . . : sN (x)),

where the s0(x), s1(x), . . . , sN (x), which are elements of the fibre Lx, are hereregarded as the homogeneous coordinates of js(x) in PN via any auxiliary(holomorphic) local trivialization, σ, of L in a neighbourhood of x; if si =λi σ, i = 0, 1, . . . , N , where each λi is a holomorphic function defined on thisneighbourhood, we then have

kod-cor-lambdakod-cor-lambda (8.3.15) js(x) = (λ0(x) : λ1(x) : . . . : λN (x)).

defiample Definition 14. Let (M,J) be a compact complex manifold and L aholomorphic line bundle over M . Then, L is said to be very ample if themap ϕL is defined eveywhere — i.e. if M0 = M — and if ϕL is an embeddingfrom M to P((H0(M,L))∗).

If V ⊂ E = H0(M,L) is a (complex) non-zero vector subspace ofH0(M,L), we can similarly define a holomorphic map from M \ DV toP(V ∗), where DV denotes the set of points x of M such that s(x) = 0 forall s in V . We then have

propvample Proposition 8.3.1. A holomorphic line bundle L over a compact com-plex manifold is very ample if and only if there exists a complex vectorsubspace V of E = H0(M,L) such that ϕV is defined everywhere and is anembedding from M to P(V ∗).

Proof. Let V be any complex vector subspace of E = H0(M,L) suchthat ϕV is defined everywhere — i.e. DV = ∅ — and is an embedding.Since DV = ∅, DL = ∅ a fortiori, so that ϕL is defined eveywhere as well.Moreover, the image of ϕL misses the projective subspace P((E/V )∗), where(E/V )∗ is viewed as the subspace of elements α of E∗ such that α|V ≡ 0.Now, we have a natural submersion, pV , from P(E∗)\P((E/V )∗) onto P(V ∗)and we then have ϕV = pV ϕL, i.e. the following commutative diagram:

diagramdiagram (8.3.16) MϕL

//

ϕV

''OOOOOOOOOOOOO P(E∗) \ P((E/V )∗)

pV

P(V ∗)

It is then clear that ϕL is an embedding from M to P(E∗) \ P((E/V )∗),hence to P(E∗), whenever ϕV is an embedding from M to P(V ∗).

Remark 8.3.1. If L is a very ample holomorphic line bundle over acompact complex manifold of (complex) dimension m, it can be shown thatthere always exists a subspace V ofH0(M,L) of (complex) dimension smallerthan or equal to 2(m+ 1) such that ϕV is an embedding from M to P(V ∗),cf. [66] Proposition 11.9 for a stronger statement.


exbasic Example 1. (i) By Proposition 6.1.1, for any complex projective spaceP(V ), H0(P(V ),O(1)) = V ∗, so that (H0(P(E),O(1)))∗ = V and ϕO(1) isthen the identity map from P(V ) to itself. More generally, for any k > 0,

H0(P(V ),O(k)) =⊙k(V ∗), (H0(P(V ),O(k)))∗ =

⊙k V and ϕO(k) is then

the Veronese embedding from P(V ) to P(⊙k V ) defined by

veroneseveronese (8.3.17) x→ x⊗k,

for any complex line x in E. In particular, the Veronese embeddding as-sociated to O(k) over the complex projective line P1 is the holomorphicembedding from P1 to Pk defined, in term of homogeneous coordinates, by

(8.3.18) (z : t)→ (zk : zk−1t : . . . : ztk−1 : tk).

(ii) Let M be any (compact) projective manifold M and let ϕ : M →P(V ) be a holomorphic full embedding from M to some complex projectivespace P(V ) (here, full means that ϕ(M) is not contained in a proper pro-jective subspace of P(V )). Denote by L = ϕ∗O(1) the restriction of O(1)to M . Then, L is very ample. This is a consequence of Proposition 8.3.1,by choosing V = H0(P(V ),O(1)) ⊂ E = H0(M,L); then, ϕV = ϕ whichis an embedding by assumption (notice that H0(P(V ),O(1)) ⊂ H0(M,L) isa proper inclusion in general, cf. [93] for a more precise discussion on thispoint).

The Kodaira embedding theorem can be formulated as follows:

thmkodaira Theorem 8.3.1 (K. Kodaira [122]). Let (M,J) be a compact complexmanifold and let L a holomorphic line bundle over M . Then L is ample ifand only if there exists a positive integer k0 such that Lk is very ample forany integer k ≥ k0.

Proof. In one direction, the theorem is clear: if Lk is very ample,for some positive integer k, then (M,J) is holomorphically embedded inthe complex projective space P((H0(M,Lk)∗) via the map ϕLk and Lk isthen identified with the restriction of the dual tautological line bundle ofP((H0(M,Lk)∗); since the latter is ample, Lk is then ample, as well as Litself (more details on the induced Fubini-Study Kahler metrics below). Thedifficult part of the theorem is to show that if L is ample, for k large enoughthe map ϕLk is defined everywhere and is an embedding, i.e. an injectiveimmersion as M is compact. By the very definition (8.3.3)-(8.3.4) of theKodaira map and the expression (8.3.10) of its derivative, this amounts tochecking the following two conditions:

(1) For any two (distinct) points x, y of M , there exits s in Ek suchthat s(x) = 0 and s(y) 6= 0.

(2) For any x in M and any X in TxM , there exists s in Ek such thats(x) = 0 and (dXs)(x) 6= 0.

The proof of these two facts makes use of the vanishing theorem 1.22.5 andof the sheaf theory machinery, which goes outside the scope of these notes.For a complete proof, we refer the reader to the original paper [122] or tomore recent references like [53], [93], [193], [66], [140].

8.4. BERGMAN METRICS ON POLARIZED MANIFOLDS 197

rembergmanbalanced Remark 8.3.2. As observed in [200] the fact that the canonical mapϕLk is defined everywhere when L is ample and k is large enough directlyfollows from Theorem 8.2.1. Indeed, when k is large, it follows from thistheorem that the Bergman density BLk,h(k) is everywhere positive, meaning

that the set DL is empty, cf. also [140].

Let L be a very ample holomorphic line bundle over a compact (projec-tive) complex manifold M and, as before, denote by ϕL the canonical mapfrom M to the complex projective space P(E∗) — whith E = H0(M,L) —which, by Theorem 8.3.1, is a holomorphic embedding. We then have:

propautMLkodaira Proposition 8.3.2. The group AutL(M,J) coincides with the group ofthose elements of Aut(P(E∗)) wich preserve the image of M by ϕL.

Proof. Let Φ be any element of AutL(M,J). Then, up to the overall

action of C∗, Φ determines a linear automorphism, Φ, of E, defined bys 7→ (Φ(s))(x) := Φ(s(Φ−1 · x)), which, in turn, determines a well-defined

automorphism, Φ say, of P(E∗). For any x in M , we then have: Φ(ϕL(x)) =

Φ(ev∗x(L∗x)) = ev∗Φ·x(Φ(L∗x)) = ev∗Φ·x(L∗Φ·x) = ϕL(Φ · x). This means that Φ

preserves M and that the restriction of Φ to M coincides with Φ. Conversely,any automorphism Ψ of P(E∗) is determined by a linear automorphism of E∗

— cf. the proof of Proposition 6.1.2 — hence determines an automorphismΨ of the dual tautological line bundle Λ∗ of P(E∗), defined up to the actionof C∗; if Ψ preserves M and induces Φ in Aut(M,J), then Ψ restricts toan automorphism of L over Φ, defined up the action of C∗, so that Φ isan element of AutL(M,L), cf. (8.3.5). The two constructions are clearlyinverse to each other.

rem-bergmandensity Remark 8.3.3. In the context of the Kodaira map ϕL, the Bergmandensity defined by (8.2.2) can be alternatively defined as follows, cf. [81]:

bergman-bisbergman-bis (8.3.19) BL,h(x) = maxs∈H0(M,L)∫M |s|

2h vg=1

|s(x)|2h,

for any x in M . Indeed, for any orthonormal basis s = s0, s1, . . . , sNof H0(M,L), any s in H0(M,L) such that

∫M |s|

2h vg = 1 is of the form

s =∑N

i=0 ai si, with∑N

i=0 |ai|2 = 1. Then, |s(x)|2 = |∑N

i=0 ai si(x)|2 ≤(∑N

i=0 a2i )∑N

i=0 |si(x)|2 = BL,h(x) (by using the hermitian Cauchy–Schwarzinequality). Conversely, let s0, s1, . . . , sN be any orthonormal basis ofH0(M,L) such that s1(x) = . . . = sN (x) = 0 (if x ∈ DL this holds for anybasis of H0(M,L); if x /∈ DL, choose any orthonormal basis s0, . . . , sNof the hyperplane ϕL(x) of H0(M,L) and complete it into an orthonormalbasis of H0(M,L) by adding s0); then, BL,h(x) = |s0(x)|2.

8.4. Bergman metrics on polarized manifoldsssbergmanmetrics

The canonical embedding ϕL attached to any very ample holomorphicline bundle L over M is defined without using any metric on M or anyhermitian inner product on L or any basis of E = H0(M,L). On the otherhand, any choice of a basis s = s0, . . . , sN of E determines a (positivedefinite) hermitian inner product, 〈·, ·〉s, on E, namely the unique one forwhich s is orthonormal. This, in turn, induces a hermitian inner product


on E∗, also denoted by 〈·, ·〉s for simplicity, hence a Fubini-Study metric onP(E∗) — of holomorphic sectional curvature equal to 2, cf. Remark 8.1.1 —polarized by the dual tautological line bundle Λ∗ of P(E∗), equipped by the(fiberwise) hermitian inner product induced by the hermitian inner product〈·, ·〉s of E∗. The pull-back of this metric by the Kodaira embedding ϕLis then a Kahler metric, denoted by gM,s, whose Kahler form, ωM,s, is thecurvature form of the (pointwise) hermitian inner product, hs, of L inducedby the above hermitian inner product of Λ∗ via the isomorphism L = ϕ∗L(Λ∗).We then have:

lemmainduced Lemma 8.4.1. For any basis s = s0, s1, . . . , sN of E, the induced her-mitian inner product hs on L at any point x of M is given by

FSnormFSnorm (8.4.1) |u|2hs =|u|2h∑N

i=0 |si(x)|2h,

for any u in Lx and any auxiliary hermitian inner product h on L. Equiv-alently, hs is the unique hermitian inner product on L such that

unitunit (8.4.2)

N∑i=0

|si(x)|2hs = 1,

for any x in M .

Proof. We first consider the induced hermitian inner product on L∗x,for each x in M , via the identification of L∗ with ϕ∗LΛ given by (8.3.8). Anyζ in L∗x is then identified with ev∗x(ζ) in E∗, whose explicit expression isgiven by (8.3.9). From (8.3.9) we readily infer

(8.4.3) |ζ|2hs := |ev∗x(ζ)|2E∗ =N∑i=0

|〈ev∗x(ζ), si〉|2 =N∑i=0

|〈ζ, si(x)〉|2,

for any x in M and any ζ in L∗x. The induced hermitian inner product onL is then given by

(8.4.4) |u|2hs =|〈ζ, u〉|2

|ζ|2hs=

|〈ζ, u〉|2∑Ni=0 |〈ζ, si(x)〉|2

,

for any non-zero ζ in L∗x. In particular, for any j = 0, 1, . . . , N , we have that

(8.4.5) |sj(x)|2hs =|〈ζ, sj(x)〉|2∑Ni=0 |〈ζ, si(x)〉|2

.

We then get (8.4.2), hence (8.4.1).

Lemma 8.4.1 can be reformulated as follows:

propinduced Proposition 8.4.1. Let (M,L) be a compact Kahler manifold polarizedby a (ample) holomophic line bundle L. Suppose that L is very ample anddenote by ϕL the Kodaira embedding of M in P(E∗), with E = H0(M,L).For any basis s = s0, s1, . . . , sN of E, consider the Fubini-Study metricon P(E∗) determined by the hermitian inner product 〈·, ·〉s of E∗, as definedabove, and denote by ωM,s the Kahler form of the induced Kahler metricgM,s on M via ϕL. For any hermitian inner product h on L, denote by ωh


the curvature form of the Chern connection of (L, h). Then, ωM,s and ωhare related by

FSinducedFSinduced (8.4.6) ωM,s = ωh +1

2ddc log (

N∑i=0

|si(x)|2h).

Proof. If π denotes the natural projection of L to M as in (8.1.3), wehave π∗(ωh) = −1

2ddc log r2

h, where r2h denotes the square norm function on

L determined by h, whereas π∗ωM,s = −12dd

c log r2hs

, where r2hs

denotes thesquare norm function on L determined by h(s). We then readily infer (8.4.6)from (8.4.1).

Remark 8.4.1. In (8.4.6), the Kahler form ωM,s only depends on thechoice of the basis s of E and it is easy to check directly that the right handside of (8.4.6) is actually independent of h. Also observe that ωh need notbe positive — although it can be chosen so by an appropriate choice of h, asL is ample — and that the very ampleness of L implies that

∑Ni=0 |si(x)|2h

has no zero on M . Observe finally that∑N

i=0 |si(x)|2h is not, in general, theBergman density of ωM,s, nor of ωh, when the latter is positive.

thm-tianconvergence Theorem 8.4.1 (G. Tian [182]). Let (M, g, J, ω) be a compact Kahlermanifold of complex dimension m, polarized by a hermitian holomorphicline bundle (L, h), and suppose that Lk is very ample for all k ≥ k0. Forany k ≥ k0, consider: the hermitian inner product on Ek = H0(M,Lk)

determined by the induced pointwise hermitian product h(k) and the volumeform vgk = km vg corresponding to the polarization of M by (Lk, h(k)), as in(8.2.1); the induced Fubini-Study metric of P(E∗k); the Kahler form ωM,s(k)

of the Kahler metric induced by the canonical embedding ϕLk , where s(k)

stands for any orthonormal basis of Ek with respect to the above hermitianinner product; the renormalized induced Kahler form ωFS,k = 1

k ωM,s(k) .Then, when k tends to +∞ the sequence ωFS,k C

∞ converges to ω.

Proof. From Proposition 8.4.1 applied to the pair (M,Lk), we get

FS-tianFS-tian (8.4.7) ωFS,k = ω +1

2kddc logBLk,h(k) ,

for any k ≥ k0, whereBLk,h(k) denotes the Bergman density of (M, (Lk, h(k))),as defined in Section 8.2. By Theorem 8.2.1, BLk,h(k) is C∞ bounded from

above when k tends to +∞: 1kdd

c logBLk,h(k) then C∞ converges to zero.

rem-tian Remark 8.4.2. As already mentioned in the proof of Theorem 8.2.1,Theorem 8.4.1 has been first obtained by G. Tian as a consequence of aweak form of Theorem 8.2.1 established in [182], also cf. [37], [38].

rem-bergmanmetric Remark 8.4.3. The metrics ωFS,k = 1k ωM,s appearing in Theorem 8.4.1

all belong to the Kahler class Ω = 2π c1(L) for any k ≥ k0. These metricsare called Bergman metrics or algebraic Kahler metrics, cf. [80], relative tothe ample line bundle L. Tian’s theorem can then be roughky reformulatedas follows: The space of Bergman metrics relative to L is dense inM2π c1(L).


For any basis s of E = H0(M,L), the induced (fiberwise) hermitianinner product hs on L determines in turn a hermitian inner product on Edefined by

globalinnersglobalinners (8.4.8) 〈s, s〉hs =

∫M

(s(x), s(x))hs vM,s,

for any two elements s, s of E, where vM,s = 1m!ω

mM,s denotes the volume

form of ωM,s, cf. (8.2.1),

In the case when (M,L) = (P(V ),Λ∗) we have E = V ∗ — cf. Proposition6.1.1 — and ϕL is then the identity map fromM = P(V ) to P(E∗). Then, theglobal scalar product 〈·, ·〉hs defined by (8.4.8) coincides, up to normalization,with the previously defined hermitian inner product 〈·, ·〉s of H0(P(V ),Λ∗),with respect to which s is orthonormal, cf. below in this section. This,however, is no longer true in general and we now make this point moreprecise.

Denote by B(E) the space of bases of E = H0(M,L), by B0(E) =B(E)/C∗ the set of bases of E modulo homothety and by π the naturalprojection from B(E) to B0(E). Both spaces B(E) and B0(E) have a naturalcomplex structure and π makes B(E) into a holomorphic C∗-principal bundleover B0(E).

The complex manifold B(E) admits a simply transitive right action ofthe complex linear group GL(N + 1,C), defined by

sgammasgamma (8.4.9) (s · γ)i =

N∑j=0

γji sj , i = 0, . . . , N,

for any s in B(E) and any γ in GL(N + 1,C). This action clearly descendsto a (right) action of GL(N + 1,C), in fact of PGL(M + 1,C) = GL(N +1,C)/C∗, on B0(E), whereas B0(E) admits in addition a left action of thegroup HL(M,J) = Hred(M,J), cf. Section 8.1, determined by

Phi-actionPhi-action (8.4.10) (Φ · s)i(x) = Φ(si(Φ

−1(x)), i = 0, . . . , N,

for any s in B(E), any x in M and any Φ in HL(M,J), where Φ denotes any

lift of Φ in Aut0(L). Since Φ is well-defined up to a constant in C∗, (8.4.10)determines a well-defined (left) action on B0(E), which clearly commuteswith the (right) action of PGL(N + 1,C).

For any s in B(E) we denote by js the embedding of M into the standardprojective space PN = P(CN+1), defined by

js1js1 (8.4.11) js = s ϕL,where s : P(E∗)→ PN is defined by (8.3.13); equivalently:

js2js2 (8.4.12) js(x) = (s0(x) : . . . : sN (x)),

for any x in M , where (s0(x) : . . . : sN (x)) has to be thought of as the

element ( s0(x)σ(x) : . . . : sN (x)

σ(x) ) of P(CN+1) for any non-vanishing holomorphic

local section of L in some neighbourhood of x, cf. (8.3.14)–(8.3.15). Noticethat js only depends on the projection of s in B0(E). We then have

jsgammajsgamma (8.4.13) js γ = tγ js,


for any γ in GL(N + 1,C) — where tγ denotes the transpose of γ, definedby: tγij = γji — which intertwines the right action of GL(N+1,C) on B(E)

and the standard left action of GL(N + 1,C) on PN , and

PhijsPhijs (8.4.14) jΦ·s = js Φ−1,

for any Φ in HL(M,J). In terms of the embedding js : M → PN , the Kahlerform ωM,s appearing in Proposition 8.4.1 and the corresponding volume formvM,s are given by

omegas-vsomegas-vs (8.4.15) ωM,s = j∗s ωFS , vM,s = j∗s vFS ,

for any s in B(E).

From (8.3.11), (6.3.16) and Lemma 8.4.1, we infer

(8.4.16)(µFS(js(x))

)i,j

= i((si(x), sj(x))hs −δi,j

N + 1),

for any x in M , where µFS denotes the momentum map µFS : PN →su(N + 1) determined by the standard action of SU(N + 1) on PN and theFubini-Study metric (of holomorphic sectional curvature c = 2) of PN , cf.Section 6.3. By integrating over M with respect to the induced volume formvM,s, we get:

IMIM (8.4.17)

(∫M

(µFS js) vM,s

)i,j

= i

(〈si, sj〉hs −

V

N + 1δij

),

where V =∫M vM,s denotes the total volume of M for the induced metric

— the same for all bases s of H0(M,L), as ωM,s sits in the same de Rhamclass as ω — and 〈si, sj〉hs is defined by (8.4.8).

In the case when (M,L) = (P(V ),Λ∗), so that ϕL is the identity, cf.above, we have that

∫PN µFS vFS = 0 — cf. (6.3.12) — and, from (8.4.17),

we then infer that 〈si, sj〉hs = VN+1 δij , i.e. that 〈·, ·〉FS coincides with the

initial hermitian inner product of E up to normalization. As already an-nounced, this is no longer true in general however: by (8.4.17), the discrep-ancy between the hermitian inner product on H0(M,L) determined by s,by decreeing that s is orthonormal, and the hermitian inner product definedby (8.4.8) is measured by the quantity

IMsIMs (8.4.18) µL(s) :=1

V

∫M

(µFS js) vM,s,

which can be viewed as the center of mass of js(M) ⊂ PN , when the latteris viewed as an adjoint orbit in su(N + 1).

Notice that µL(s) remains unchanged when s is replaced by a homotheticbasis, so that µL descends to a map, still denoted by µL, from B0(E) tosu(N + 1). We then have:

lemmamoment Lemma 8.4.2. The map µL : B0(E) → su(N + 1) is equivariant underthe (right) action of SU(N + 1), and invariant under the (left) action ofHL(M,J).

Proof. For any s in B(E) amd any γ in SU(N + 1), s and sγ deter-mine the same hermitian inner product on E, hence Fubini-Study metricon P(E∗), so that vM,sγ = vM,s, whereas µFS sγ = Adγ−1(µFS s): we


then have µL(sγ) = Adγ−1 µL(s). This proves the first assertion. Thesecond assertion follows from (8.4.14) and (8.4.15), from which we infer:µL(Φ · s) =

∫M (Φ−1)∗

(µFS js vM,s

)=∫M µFS js vM,s = µL(s).

8.5. The Ricci form of a complex hypersurfacesshypersurface

Let (M, g, J, ω) be a compact Kahler manifold of (complex) dimensionm ≥ 2 and let L be any holomorphic line bundle over M . Assume that Ladmits a holomorphic section s, such that the zero divisor is a smooth hy-persurface Σ of M . We assume that Σ is endowed with the Kahler structureinduced by the Kahler structure of M . In what follows, the (real) tan-gent bundles TM and TΣ will be considered as holomorphic complex vectorbundles, of (complex) rank m and m − 1 respectively, equipped with thehermitian inner product induced by g, cf. Section 1.8. We then have thefollowing exact sequence of holomorphic vector bundles over Σ:

exactexact (8.5.1) 0→ TΣ→ TM|Σ → L|Σ → 0,

where, for any x in Σ, the map from TxM to Lx is (∇s)(x), the covariantderivative at x of s with respect to any C-linear connection ∇ on L (sinces(x) = 0, (∇s)(x) is actually independent of the chosen connection). Forany ξ in Lx, we then denote by Xξ the unique element of TxM such that(∇Xξs

)(x) = ξ and which is g-orthogonal to TxΣ ⊂ TxM . We thus get an

isomorhism of complex vector bundles over Σ:

Theta-isoTheta-iso (8.5.2) TM|Σ = TΣ⊕ L|Σ,where, at each point x of M , Lx is identified with the 1-dimensional complexsubspace of TM generated by Xξ. This direct sum is (hermitian) orthogonal,by the very definition of Xξ, and is (hermitian) isometric whenever L|Σ isequipped with the hermitian inner product defined by

xi-hxi-h (8.5.3) |ξ|2h = |Xξ|2g,

for any x in Σ and any σ in Lx. We denote by ∇h the corresponding Chernconnection and by ρh the curvature form of ∇h. Denote by K−1

M = ΛmC TM ,

resp. K−1Σ = Λm−1

C TΣ, the anti-canonical line bundle ofM , resp. of Σ. Fromthe exact sequence (8.5.1), we get the following isomorphism of complex linebundles over Σ:

(8.5.4)(K−1M

)|Σ = K−1

Σ ⊗ L|Σ,

which is actually an isomorphism of holomorphic line bundles (in contrastwith (8.5.2). It is also an isomorphism of hermitian line bundles, providedthat LΣ is equipped with the hermitian inner product h defined by (8.5.3).Since the Ricci form of any Kahler manifold coincides with the curvatureform of the anti-canonical line bundle equipped with its natural holomorphicstructure and the hermitian inner product induced by the metric, we inferthat the Ricci form, ρΣ, of Σ and the Ricci form, ρM , of M are related by

(8.5.5) ρΣ = (ρM )|Σ − ρh.

In order to obtain a more workable expression, we fix an auxiliary hermitianinner product, say h0, on L, and, for any point x of Σ, we choose any localholomorphic trivialization, say σ, of L on some open neighbourhood U of

8.6. THE DONALDSON GAUGE FUNCTIONAL AS A KAHLER POTENTIAL 203

x, so that s = λσ on U . If ∇ denotes any C-linear connection on L, wethus get: ∇Xs = dλ(X)σ(x) for any X in TxM (since λ(x) = 0). We thushave: dλ(Xξ)σ(x) = ξ, hence |ξ|2h0

= |dλ|2g|Xξ|2g|σ(x)|2h0. Since ξ|2h := |Xξ|2g,

it follows that h = h0

|σ|2h0|dλ|2g

= h0

|∇s|2h0,gon Σ. We eventually infer

rhoSigmarhoSigma (8.5.6) ρΣ = (ρM )Σ − (ρh0)|Σ +1

2ddc log

(|∇s|2h0,g

)|Σ

for any auxiliary hermitian inner product h0 on L, of curvature form ρh0 ,and for any C-linear connection ∇ on L, where |∇s|2h0,g

denotes the squarenorm of ∇s, as a section of T ∗M|Σ ⊗ L|Σ, with respect to g and h0.

ex-tian Example 2. As a main example, we consider the case when M is thestandard complex projective space CPm, equipped with the standard Fubini-Study Kahler structure (gFS , ωFS) — of holomorphic sectional curvature 2,cf. Remark 8.1.1 — whose Ricci form is ρFS = (m + 1)ωFS , cf. (6.2.6), Lis the holomorphic line bundle O(k), for some positive integer k, equippedwith the standard hermitian inner product h0, whose curvature form is ρh0 =k ωFS , and the chosen holomorphic section s of L is therefore identified witha homogeneous polynomial of degree k defined on Cm+1, denoted by F ,cf. Proposition 6.1.1. We choose s = F as in Example ??, so that Σ isdefined in CPm+1 by the equation F (z0, z1, . . . , zm+1) = 0 in homogeneouscoordinates. The covariant derivative of F , as a section of O(k), restrictedto Σ, is represented on the pull-back of Σ in Cm+2 \ 0 by dFΘ. In view of(??) and of (8.5.6), we get the Tian formula:

(8.5.7) ρΣ = (m+ 1− k) (ωFS)|Σ +1

2ddc log

∑mi=0

∣∣∣ ∂F∂zi ∣∣∣2(∑m

i=0 |zi|2)k−1.

8.6. The Donaldson gauge functional as a Kahler potentialssgaugefunctional

Lemma 8.4.2 suggests that the map µL : B0(E)→ su(N + 1) might be amomentum map for the natural action of SU(N +1) on B0(E) — rather, on

the quotient, B0(E) = HL(M)\B0(E) — for some symplectic structure de-

fined on B0(E) and we now show that this is indeed the case. More precisely,

we construct a Kahler structure on B0(E), compatible with the natural com-plex structure, JB(E), of B(E), in such a way that µL be a momentum mapfor the action of SU(N + 1) with respect to the corresponding Kahler form.Our treatment closely follows O. Biquard’s one in [30, Lemma 2.3].

Fix h0 in H>0(L), the space of hermitian inner products on L whoseChern curvature form is positive, and denote by ω0 the corresponding cur-vature form. By definition of H>0(L), ω0 is the Kahler form of a Kahlermetric in MΩ, with Ω = 2π c1(L). Recall — cf. Remark 8.1.2 — that

the 1-form τ on Mω0 defined by (4.1.2) in Section 4.1 and its primitive,the Donaldson gauge functional Iω0 , can be viewed as defined on H>0(L).

Denote by h the map from B(E) to H>0(L) given by

(8.6.1) h : s 7→ hs =h0∑N

j=0 |sj |2h0

,


cf. Lemma 8.4.1. We thus get the following commutative diagram

diagram-basesdiagram-bases (8.6.2) B(E)h //

π

H>0(L)

∼= // Mω0

B0(E) // H>0(L)/R∼= //MΩ

.

In this diagram: the first vertical arrow is the C∗-principal bundle map πfrom B(E) to B0(E); the second vertical arrow is the R∗-principal bundlemap from H>0(L) to its quotient by the R-action described in Remark 8.1.2;

the third vertical arrow is the R∗-principal bundle map p from Mω0 to MΩ

described in Section 4.1; the last up and down horizontal arrows are theidentifications described in Remark 8.1.2 and h can be viewed as a principalbundle map along the group homomorphism λ 7→ log |λ| from C∗ to R.

We now consider the real function FL defined on B(E) by

FLFL (8.6.3) FL = Iω0 h,

and we define θL — a real 1-form on B(E) — by

thetatauthetatau (8.6.4) θL = −(h∗τ) JB(E) = dc(Iω0 h) = dcFL,

where dc is relative to the natural complex structure, JB(E), of B(E). Denoteby ωL the real 2-form on B(E) defined by

omegaLomegaL (8.6.5) ωL = dθL = ddc(Iω0 h) = ddcFL.

thmolivier Theorem 8.6.1. (i) The 2-form ωL = ddcFL defined by (8.6.5) descends

to a real 2-form, ωL, on B0(E) = HL(M)\B0(E), which is there the Kahlerform of a SU(N + 1)-invariant Kahler structure compatible with the natural

complex structure, JB0(E), of B0(E).

(ii) With respect to this symplectic structure, µL is the momentum of theinduced action of SU(N + 1).

Proof. At each point s of B(E), the tangent space TsB(E) is naturallyidentified with the linear Lie algebra gl(N + 1,C): each A in gl(N + 1,C)is identified with the vector in TsB(E) tangent to the curve st = s exp(tA)in B(E) at t = 0. In the sequel, we denote by A+, resp. A−, the hermitian,resp. anti-hermitian, part of A. For any h in H>0(L), the tangent spaceThH>0(L) is naturally identified with the space of real functions on M : eachreal function f is then determined by the curve ht = e−2tfh, cf. Remark8.1.2. Theorem 8.6.1 is then a consequence of the following series of lemmas.

lemmaq* Lemma 8.6.1. The differential of h at s is given by

q*q* (8.6.6) (h∗)s(A) =

∑Ni,j=0A

+ij(si, sj)h0∑N

k=0 |sk|2h0

= µiA+

s

for any A in gl(N + 1,C), by setting µiA+

s := µiA+

FS js, where, we recall,

µiA+

FS denotes the momentum of the induced vector field iA+ on the standardprojective space PN .

8.6. THE DONALDSON GAUGE FUNCTIONAL AS A KAHLER POTENTIAL 205

Proof. With the above convention we have that

(h∗)s(A) =d

dt |t=0

1

2log

N∑j=0

|(s · exp(tA))j |2h0

=Re(

∑Ni,j=0Aij(si, sj)h0)∑Nk=0 |sk|2h0

=

∑Ni,j=0A

+ij(si, sj)h0∑N

k=0 |sk|2h0

.

computeq*computeq* (8.6.7)

The second equality in (8.6.6) readily follows from the expression (8.3.14)of ϕL and the expression (6.3.11) of the momentum of a, for a = iA+ (wehere admit the case when A = λ I, for some complex number λ; then, a = 0and the “momentum” µas is the constant function Re(λ)).

lemmatheta Lemma 8.6.2. For any s in B(E) and any A in gl(N + 1,C) = TsB(E),we have

thetatheta (8.6.8) θL(A) =1

V

∫MµA−

s vM,s,

with µA−

s = µA−

FS js.

Proof. From (8.6.4), we infer θL(A) = dc(I h)(A) = −d(I h)(iA) =

−τ((h∗)s(iA)), where, we recall, τ is defined by (4.1.2). By Lemma 8.6.1,

we thus get: θL(A) = − 1V

∫M µ

i(iA)+

s vM,s = 1V

∫M µA

−s vM,s.

lemmavartheta Lemma 8.6.3. For any a in u(N + 1) and any B in gl(N + 1,C) =TsB(E), we have that

varmuhatvarmuhat (8.6.9)d

dt |t=0µas·exp(tB) = ωFS(B, a) js.

Proof. Since µas = µa s ϕL, we have ddt |t=0

µas·exp(tB) = dµa(B) js =

ωFS(B, a) js.

In the following lemma, for any A in gl(N + 1,C) we set A⊥ = A −A|M , where A|M denotes the orthogonal projection on TM with respect to

the Fubini-Study metric of PN , and we write ωFS(A⊥, B⊥))|M instead of

ωFS(A⊥, B⊥)) js.

Lemma 8.6.4. For any s in B(E) and any A,B in gl(N + 1,C) =TsB(E), we have that

olivierolivier (8.6.10) d θL(A,B) =1

V

∫M

(ωFS(A⊥, B⊥))|M vM,s.

Proof. We use the general formula:

dthetadtheta (8.6.11) dθL(A,B) = A · (θL(B))−B · (θL(A))− θL([A,B])

where A,B are viewed as vector fields on B(E) via the overall identificationgl(N + 1,C) = TsB(E) for any s in B(E) (these vector fields must not be


confused with the induced vector field on Pm, denoted A, B...). By (8.6.9)we have

(8.6.12) A · (µB−|M ) = ωFS(A, B−)|M ,

whereas, by (3.2.4) in Lemma 3.2.1, with φt = 12 log

∑Nj=0 |(s · exp(tA))j |2h0

,

and by using (8.6.7), we get

(8.6.13) A · vM,s = −∆M,s(µiA+

|M ) vM,s.

We thus obtain

A · (θL(B)) =1

V

∫M

(ωFS(A, B−)|M − µB−|M ∆M,s(µiA

+

|M )) vM,s

=1

V

∫M

(ωFS(A, B−)|M − (dµiA+

|M , dµB−|M )M,s) vM,s

=1

V

∫M

((ωFS(A, B−))|M − ωM,s(A+|M , B−|M )) vM,s

=1

V

∫M

((ωFS(A, B−))|M − ωM,s(A|M , B−|M )) vM,s.

computeABcomputeAB (8.6.14)

Observe that

omegaA-B-omegaA-B- (8.6.15)

∫MωM,s(A−|M , B−|M ) vM,s = 0,

for any A,B in gl(N+1,C), as A−|M and B−|M are both hamiltonian vectorfields on M with respect to ωM,s. In general, the orthogonal projection ofthe gradient of a function on the tangent bundle of a submanifold is thegradient of the restriction of the fuction with respect to he induced metric,so that

(8.6.16) iA+|M = gradωM,s(µiA+

|M ),

and similarly for B−|M . Finally, since [A,B]− = [A−, B−] + [A+, B+] =

[A−, B−]− [iA+, iB+], we get:

computebracketcomputebracket (8.6.17) θL([A,B]) =1

V

∫M

(ωFS(A−, B−)|M − ωFS(A+, B+))|M vM,s.

By plugging (8.6.14) and (8.6.17) into (8.6.11) and by using (8.6.15), weeasily get (8.6.10).

Lemma 8.6.5. The 2-form dθL descends to a U(N +1)-invariant Kahler

form on B(E).

Proof. Since any two bases s, sγ induce the same hermitian innerproduct on E, as soon as γ belongs to U(N + 1), the map h from B(E) toH>0(L) is U(N + 1)-invariant, hence also FL, θL and dθL. On the otherhand, from (8.6.10), we readily infer that dθL(A,B) = 0, whenever A is a

multiple of the identity, so that A = 0, or A belong to the Lie algebra ofHL(M), cf. Proposition 8.3.2; since dθL is (evidently) closed, it follows thatdθL is invariant under the (left) action of HL(M,J) and the (right) action of

C∗, and actually descends to a 2-form ωL on B0(E) = HL(M)\B(E)/C∗. OnB0(E), dθL is clearly JB0(E)-invariant and the corresponding symmetric form

8.7. BALANCED METRICS 207

is gL(A,B) = 1V

∫M ((gFS(A⊥, B⊥))|M ) vM,s, which descends to a positive

definite metric on B0(E).

This proves the first assertion of Theorem 8.6.1. To prove the secondassertion, we use the fact that θL is invariant under the action of SU(N +1)on B0(E): it follows that for any a in su(N + 1), the “momentum” of theinduced vector field a with respect to dθL is simply θL(a), cf. Remark 9.6.2.By (8.6.8), θL(a) is equal to 〈µL(s), a〉. Since µL is HL(M)-invariant, cf.

Lemma 8.4.2, we infer that it is the momentum of a on B0(E) with respectto the induced symplectic form ωL.

8.7. Balanced metricsssbalanced

defi-balanced Definition 15. A pair (M,L) formed of a (compact) projective mani-fold M and a very ample holomorphic line bundle L over M is balanced ifthere exists a basis s of E = H0(M,L) such that µL(s) = 0. Then, (M,L)is said to be balanced with respect to s.

Equivalently, (M,L) is balanced with respect to s if s is orthonormal,up to normalization, with respect to the hermitian inner product 〈·, ·〉h(s)

on H0(M,L) determined by the induced pointwise hermitian product hs, asdefined in (8.4.8).

rembalancedbergman Remark 8.7.1. It readily follows from Lemma 8.4.1 that (M,L) is bal-anced with respect to s if and only if the corresponding Bergman densityBL,hs — cf. Section 8.2 — is constant. This constant is then equal to N+1

V ,

where, we recall, N+1 stands for the (complex) dimension of H0(M,L) andV denotes the total volume determine by the Kahler class 2π c1(L).

propunique Proposition 8.7.1. If (M,L) is balanced with respect to a basis s ofE = H0(M,L), then s is uniquely determined up to the actions of C∗,SU(N + 1) and HL(M).

Proof. By Lemma 8.4.2, if (M,L) is balanced with respect to s it isalso balanced with respect to any other basis deduced from s by the actionof C∗, SU(N + 1) and HL(M). It remains to check that (M,L) is notbalanced with respect to s · γ, whenever γ is an element of SL(N + 1,C)not in SU(N + 1) (it is in fact sufficient to assume that γ = exp(ia), whena is in su(N + 1) but a is not in hL). This is a direct consequence of µLbeing the momentum map of the SU(N + 1)-action on B0(E)/HL(M) withrespect to the symplectic form dθL = ωL given by (8.6.10). Indeed, for anya as above, we have that

d

dt|µaL(s · exp(i ta))|2 =

2

V

∫MgFS(a⊥, a⊥) vM,s·exp(i ta)(8.7.1)

which is positive for any real value of t, cf. [30] and Remark 9.6.2 below. Itfollows that µL(s · exp(i ta)) is different from 0 except for t = 0.

Remark 8.7.2. In [76], S. K. Donaldson considers the space, Herm+(E)of (positive definite) hermitian inner product on E — denoted by M in [76]


— which is naturally identified with the quotient B(E)/U(N + 1), and themaps Hilb from H>0(L) to H(E) and FS from H(E) to H>0(L), defined by

(8.7.2) Hilb(h) =N + 1

V〈·, ·〉h, FS(H) = hs,

for any h in H>0(L) and any H = [s] in Herm+(E) = B(E)/U(N +

1). We thus get the following diagram, where the map H from B(E) to

Herm+(E) is defined by H(s) = [s] (here, and above, [s] denotes the classof s mod U(N + 1), identified with the unique element of Herm+(E) withrespect to which s is orthonormal):

(8.7.3) H>0(L)Hilb

--Herm+(E)

FS

mm

B(E)h

ddIIIIIIIII H

99ssssssssss

As noticed by Donaldson, (M,L) is balanced with respect to s if and onlyif FS Hilb(hs) = hs or, equivalently, Hilb FS([s]) = [s].

It has been observed by S. Zhang [201] and H. Luo [139] that the conceptof balanced pair (M,L) is closely connected to the concept of Mumford-Chow stability of M as a submanifold of P(H0(M,L)∗). This in turn isconjectured to be closely connected to the existence of Kahler metrics withconstant scalar curvature or, more generally, with extremal Kahler metrics.A strong evidence is then given by the following theorem of S. Donaldson:

thmdonaldson-uniqueness Theorem 8.7.1 (S. Donaldson [75] Theorems 1, 2, 3). Let M be a com-pact complex manifold and let L be an ample holomorphic line bundle overM . Suppose that Lk is very ample for k ≥ k0 and, for any k ≥ k0, denoteby Ek = H0(M,Lk) the space of holomorphic sections of Lk and by Nk + 1the dimension of Ek. Assume moreover that H(M,L) = H0(M,J) = 1.Then:(i) If the pair (M,Lk) is balanced, for some k ≥ k0, then it is balanced withrespect to a uniquely defined element of FS(Ek).(ii) Suppose that, for each sufficiently large positive integer k, (M,Lk) is bal-

anced with respect to some basis s(k) of H0(M,Lk) and that the correspond-ing normalized induced Fubini-Study metrics ωFS,k := 1

kωM,s(k) converge inC∞ to some limit ω∞ as k tends to +∞. Then, ω∞ is of constant scalarcurvature.(iii) Suppose that (M,J) admits a Kahler metric ω of constant scalar cur-vature polarized by L. Then, there exists k0 > 0 such that for any k ≥ k0

the pair (M,Lk) is balanced with respect to some basis s(k) of H0(M,Lk) —essentially unique by (i) — and the corresponding sequence of renormalizedmetrics ωFS,k := 1

kωM,s(k) then C∞ converges to ω.

Proof. Part (i) follows from Proposition 8.7.1. Part (ii) can be ob-tained by applying Theorem 8.2.1 to the family of Kahler forms ωFS,k =1k ωM,s(k) , which we simply denote ωk, k ≥ k0; the corresponding metricis similarly denoted by gk. Then, by Lemma 8.4.1, ωk is the Kahler formof M when M is polarized by L, equipped with the pointwise hermitian

8.8. S1-ACTIONS ON POLARIZED KAHLER MANIFOLD 209

product of L, say hk, induced by h(s(k)): for each u in L, we then have

|u|2hk := |u⊗k|2/k. For each k ≥ k0, the estimate (8.2.5) for q = 1 now reads

mamapmamap (8.7.4) |BLp,h

(p)k

− 1

(2π)m− 1

(2π)msgk4p|2C` ≤

C1,`

p2,

where h(p)k denotes the induced hermitian inner product on Lp. The main

point here is that the constants C1,` are independent of k, as the ωk areC∞ bounded by assumption. Now, by assumption B

Lk,h(k)k

= BLk,h(k) is

constant, hence equal to d(Lk)Vk

, with Vk = km V ; for p = k, (8.7.4) then

reads

mamapkmamapk (8.7.5) |k (d(Lk)

Vk− 1

(2π)m)− 1

(2π)msgk4|2C` ≤

C1,`

k,

for each k ≥ k0. On the other hand, from Proposition 8.2.1, we infer that

(8.7.6)d(Lk)

Vk=

1

kmd(Lk)

V=

1

(2π)m+

k−1

(2π)m

∫M

sg4vg +O(k−2).

We infer that for each ` and each k ≥ k0, we have that

(8.7.7) |sgk −∫M sg vg

V|2C` ≤

c`k,

for some positive constant c` independent of k. It follows that sgk C∞

converges to the constant∫M sg vgV , which is then equal to sg∞ (note that the

mean value of the scalar curvature of all metrics of Kahler forms ωk are thesame,as well as the scalar curvature of g∞, as they all have the same Kahlerclass 2π c1(L)). This completes the proof of Part (ii) of Theorem 8.7.1 (moredetails in [75]). Part (iii) is the hard core of Theorem 8.7.1 and its proofgoes beyond the scope of these notes: the reader is referred to Donaldson’soriginal paper [75]; cf. also the Bourbaki Seminar [30] by O. Biquard.

remmabuchi Remark 8.7.3. Donaldson’s theorem has been extended by T. Mabuchito include general extremal polarized Kahler metrics and any group HL(M).Again, the proof of Mabuchi’s theorem lies beyond the scope of these notesand the reader is referred to original papers [143], [144], [145], [146].

8.8. S1-actions on polarized Kahler manifoldssdonaldson

Let M be a connected smooth manifold and L a hermitian complex linebundle over M , equipped with a C-linear connection ∇ which preserves thehermitian inner product. The curvature R∇ of ∇ is of the form R∇ = iρ∇,where ρ∇ — the curvature form — is a closed real 2-form, cf. Section 1.19.Suppose that M admits a smooth (left) action of S1 which preserves ρ∇;equivalently, LXρ∇ = d(ιXρ

∇) = 0, where X denotes the generator of theS1-action, i.e. the (real) vector field on M defined by

generatorgenerator (8.8.1) X(x) =d

dt |t=0eit · x,

for any x in M . We say that this action is ρ∇-hamiltonian if there exists asmooth real function hX on M such that ιXρ

∇ = −dhX ; the ρ∇-momentum


hX is then well defined up to an additive constant (notice however that wedon’t assume that ρ∇ be a symplectic form).

For any choice of hX , we consider the vector field X on (the total spaceof) L defined by

hatXhatX (8.8.2) X = X − hX T,

where X denotes the horizontal lift of X with respect to ∇ — cf. Section1.6 — and T the generator of the natural action of S1 on L. We then have

propliftaction Proposition 8.8.1. Any ρ∇-hamiltonian action of S1 on M can belifted into an action of S1 on L which preserves the hermitian inner productof L and the horizontal distribution H∇ of ∇. For any such lifted action, thegenerator is given by (8.8.2) for an appropriate choice of the ρ∇-momentumhX , uniquely defined up to an additive integer.

Proof. For any S1-action on M , the generator, X, of this action canbe lifted into a real vector field on L and any lift of X has the form X −f1 T − f2 iT , where f1, f2 are any real functions defined on M , regarded asdefined on L by pullback. Since ∇ is a hermitian connection, X certainlypreserves the hermitian inner product of L, and so does T , whereas iT doesnot; it follows that a lift of X on L preserves the hermitian inner product ofL if and only if it has the form (8.8.2) for any real function hX defined on

M . Note that for any such function, X commutes with T : this is becauseT preserves ∇ — in fact any C-linear connection — and dhX(T ) is clearly

zero. Moreover, X preserves ∇, i.e. the horizontal distribution H∇, if andonly if hX is a ρ∇-momentum: indeed, for any vector field Z on M , we havethat

[X, Z] = [X, Z]− [dhX T, Z]

= [X,Z] + (ρ∇(X,Z) + dhX(Z))T,(8.8.3)

where [X,Z] is the horizontal lift of the bracket [X,Z]; the latter expression

is horizontal, then equal to [X,Z], for any Z — meaning that X preservesH∇ — if and only if the coefficient of T is identically zero, meaning that hX

is a ρ∇-momentum.We now show that X can be made the generator of a lifted action of S1

on L by adding an appropriate real constant to hX , well defined up to an

integer. Denote by ΦXt the flow of X on M and by ΦX

t the flow of X on L;

then, the two flows are related by π ΦXt = ΦX

t π for any value of t, whereπ denotes the projection from L to M . Morever, since the two summands,X and −hX T of X commute — this is because dhX(X) = 0 — the flow

ΦXt is the product of the flows of X and −hX T which both preserve the

hermitian inner product of L; since ΦX2π(x) = x for any x in M , it follows

that ΦX2π(ξ) = ζ(x) ξ for any ξ in Lx, for some ζ(x) in S1. We now claim that

ζ is constant. Indeed, for any vector field Z on M , the derivative of ζ along

Z is determined by ΦX2π · Z = Z − idζ(Z)

ζ T , where Z denotes the horizontal

lift of Z on L. Since ΦXt preserves the horizontal distribution for all values

of t, in particular for t = 2π, ΦX2π · Z must be horizontal and dζ(Z) is then


equal to zero. It follows that ζ = ei2πc, for some real constant c which we canchoose in the interval [0, 2π) and is then uniquely defined. If we substitute

hX + c to hX and if we still denote the corresponding lift of X by X, we

then have ΦX2π(ξ) = ξ for any ξ in L. This means that eit · ξ := ΦX

t (ξ) is aS1-action on L, which clearly lifts the action of S1 on M .

remrhoham Remark 8.8.1. To any action of S1 which preserves ρ∇ as above but apriori is not assumed to be ρ∇-hamiltonian, we associate the map ζ : M →S1 defined by ΦX

2π(ξ) = ζ(x) ξ for any ξ in Lx: ζ is then the holonomy of∇ along the loop ΦX

t (x). The above argument can then be interpreted asfollows: The action of S1 is ρ∇-hamiltonian if and only if there exists a

smooth real function, hX , which integrates ζ, i.e. such that ζ(x) = e2πihX ,and hX is then a ρ∇- momentum of this action.

This action of S1 on L induces a linear (left) action of S1 on the spaceΓ(L) of smooth sections of L, defined as follows:

(8.8.4) (eit · s)(x) = ΦXt (s(ΦX

−t(x)).

In analogy with the usual Lie derivative — cf. Section 1.4 — the oppositeof the derivative of this action can be regarded as a Lie derivative along Xoperating on the space Γ(L), denoted by LX , with

(8.8.5) LXs := − d

dt |t=0(eit · s).

We then have:

proplieXs Proposition 8.8.2. For any s in Γ(L), the Lie derivative LXs has thefollowing expression

lieXslieXs (8.8.6) LXs = ∇Xs+ ihX s.

Proof. This is a simple computation: for any x in M ,

(LXs)(x) = − d

dt |t=0ΦXt (s(ΦX

−t(x))

= −X(s(x)) + s∗(X(x))

= −X(s(x)) + hX(x)T (s(x)) + X(s(x)) + (∇Xs)(x)

= (∇Xs)(x) + ihX(x)s(x),

where we used (1.6.10) of Section 1.6.

As an illustration of Proposition 8.8.1, we consider the case when L =K∗(M) is the anti-canonical line bundle of a Kahler manifold (M, g, J, ω).Recall that the Kahler structure of M induces a hermitian inner producton K∗(M), whose curvature form is the Ricci form ρ, cf. Section 1.19. Wethen have

propcanonical Proposition 8.8.3. Consider any action of S1 on M , of generator X,which preserves the whole Kahler structure (g, J, ω) and which is hamiltonianwith respect to ω, with momentum hX . Then, this action also preserves theRicci form ρ and is ρ-hamiltonian, of ρ-momentum 1

2∆hX .


Moreover, this action lifts canonically as a holomorphic action on theanti-canonical line bundle K−1

M , whith generator X = X− 12∆hX T , and the

induced linear action on Γ(K−1M ) coincides with the usual Lie derivative.

Proof. Since S1 preserves the metric and the complex structure, italso preserves the Ricci form ρ ; morever, it readily follows from Lemma1.23.4 that X is also ρ-hamiltonian, of ρ-momentum equal to 1

2∆hX , up

to an additive constant. If ΦXt denotes the flow of X, so that the S1-

action on M is given by eit · x = ΦXt (x), the canonical lifted action on

K∗(M) is simply given by eit · Ψ := (ΦXt )∗(Ψ), for any Ψ in K∗(M). This

makes sense, as ΦXt is a biholomorphic transformation of M for each t.

The induced linear action on Γ(K∗(M)) is then the usual Lie derivative,as defined in Section 1.4. Now, the Lie derivative LX and the Levi-Civitacovariant derivative D acting on sections of T 1,0M ∼= TM are related byLX = DX−DX, cf. Section 1.23; the operator LX−DX , acting on sectionsof K∗(M) then coincides with the induced action of the endomorphism DXon elements of K∗(M) = Λm(T 1,0M). This action can be easily madeexplicit as follows. Consider any point x of M and any element Ψ in K∗x(M),which we can choose of the form Ψ = ε1∧. . .∧εm, where the εj , j = 1, . . . ,m,

form a basis of T 1,0x M ; we can even assume that this basis is hermitian-

orthonormal, meaning that εj = 1√2(ej − iJej), where e1, Je1, . . . , em, Jem

is a g-orthonormal basis of TxM ; we then have

(8.8.7) (LX −DX)(ε1 ∧ . . . ∧ εm) = −m∑j=1

ε1 ∧ . . . ∧DεjX ∧ . . . ∧ εm,

where the rhs is equal to −∑m

j=1(DεjX, εj) ε1 ∧ . . .∧ εm, where the bracket

(·, ·) here just mean the C-bilinear extension of g.Now, −

∑mj=1(DεjX, εj) is equal to −1

2

∑mj=1(D(ej−iJej)X, ej + iJej),

hence to 12δX

[ − i2δ(JX

[) (this is because DJejX = JDejX, as X is (real)

holomorphic); finally, δX[ = 0, as X is Killing, whereas δ(JX[) = −∆hX ,

since X[ = dchX . We then get

(8.8.8) LXΨ = DXΨ +i

2∆hX Ψ,

for any section Ψ of K∗(M). By comparing with (8.8.6), we conclude thatthe choice of the ρ-momentum map corresponding the canonical lift of theS1-action on K∗(M) — meaning that its generator is X = X − 1

2∆hX T —

is exactly 12∆hX , i.e. the ρ-momentum map of X which integrates to 0.

Remark 8.8.2. From Lemma 1.23.4 we immediately infer that ιXρ =−d(1

2∆f) — hence that 12 ∆f is a ρ-momentum of X — for any hamiltonian

Killing vector field X = Jgradgf defined on any compact Kahler manifold.The new information given by Proposition 8.8.3 is that if X is the gen-erator of a S1-action on M , then 1

2 ∆f is exactly the K∗(M)-momentumcorreseponding to the canonical lift of this action on K∗(M).

We now consider the case when the pair (M,L) is a polarized Kahlermanifold as defined in Section 8.1, so that the curvature form ρ∇ is theKahler form ω. Propositions 8.8.1 and 8.8.2 can then be specialized asfollows


propliftpolar Proposition 8.8.4. Let (M,L) be a polarized Kahler manifold and con-sider an action of S1 on M which preserves the whole Kahler structure(g, J, ω) and is hamiltonian with respect to ω. Then, for an appropriatechoice of the momentum hX , uniquely determined up to an additive integer,the lift X = X − hX T is the generator of a holomorphic action of S1 on L.

Moreover, the induced linear action on Γ(L) given by (8.8.6) preservesthe subspace H0(M,L) of holomorphic section.

Proof. For any choice of hX , we already showed in the proof of Propo-sition 8.1.2 that X is a (real) holomorphic vector field on L. When hX is

so chosen that X is the generator of a S1-action — cf. Proposition 8.8.1 —this action is then holomorphic. This proves the first assertion.

A section s of L is holomorphic if and only if ∂s = 0, where the Cauchy-Riemann operator of the holomorphic structure is related to the Chern con-nection ∇ by ∂ = ∇0,1, cf. Section 1.6; equivalently, a section s of L isholomorphic if and only if

holsectionholsection (8.8.9) ∇JZs = i∇Zs,

for any (real) vector field Z on M . For any vector field Z on M and for anysmooth section s of L, we have:

∇Z(LXs) = ∇Z(∇Xs) + idhX(Z) s+ ihX∇Zs= ∇X(∇Zs) +R∇X,Zs−∇[X,Z]s+ idhX(Z) s+ ihX∇Zs= i(ω(X,Z) + dhX(Z)) s

+∇X(∇Zs)−∇[X,Z]s+ ihX∇Zs= ∇X(∇Zs)−∇[X,Z]s+ ihX∇Zs.

Now, assume that s is holomorphic and substitute JZ to Z in the last termof the rhs: we then get ∇JZs = i∇Zs, whereas ∇[X,JZ]s is equal to ∇J [X,Z]s,as X is (real) holomorphic, hence to i∇[X,Z]s as s is holomorphic; we infer∇JZ(LXs) = i∇Z(LXs); by (8.8.9) this means that LXs is a holomorphicsection.

remLmoment Remark 8.8.3. The momentum hX appearing in Propostion 8.8.4 willbe called a L-momentum of X: L-momenta of X are then well defined upto an additive integer and

∫M hX vg is then well defined as an element of

R/ZV , where V =∫M vg. In general,

∫M hX vg mod ZV is different from 0.

As a matter of fact,∫M hX vg appears as the leading term in the asymptotic

expansion of the trace of the induced linear action of S1 in the spaces ofholomorphic sections H0(M,Lk), cf. Proposition 8.8.5 below.

The trace, wX(L), of the restriction of LX — or rather of the corre-sponding hermitian operator −iLX — to E = H0(M,L) is defined by

wLXwLX (8.8.10) wX(L) = −iN∑j=0

∫M

(LXsj , sj) vg,

for any orthonormal basis s = s0, . . . , sN of E. Equivalently, wX(L) isthe weight of the induced action of S1 on the complex line ΛN+1(E∗).


proptrace Proposition 8.8.5. The trace, wX(L), of the restriction of −iLX toH0(M,L) is given by

tracetrace (8.8.11) wX(L) =

∫M

(hX +1

2∆hX)

N∑j=0

|sj |2 vg,

for any orthonormal basis s = s0, s1, . . . , sN of H0(M,L).

Proof. From (8.8.10) and (8.8.6) we infer

wX(L) = −i∫M

N∑j=0

(∇Xsj , sj) vg +

∫MhX

N∑j=0

|sj |2 vg.

For any holomorphic section s of L, the hermitian inner product (∇Xs, s)can be written as

nablaXssnablaXss (8.8.12) (∇Xs, s) =1

2(d|s|2)(X)− i

2(d|s|2)(JX).

Now, X is Killing, hence divergence-free, whereas δ(JX) = −∆hX ; we thenreadily infer (8.8.11).

We now consider the case of two hamiltonian actions of S1 on the po-larized Kahler manifold (M,L). For each generator, X and Y , we choose aL-momentum, hX and hY . The induced action on H0(M,L) is then givenby (8.8.6). We then have :

propbitrace Proposition 8.8.6. The trace, wX,Y (L), of the restriction of the prod-uct −LX LY = (−iLX) (−iLY ) to H0(M,L) is given by

wX,Y (L) =

∫MhXhY

N∑j=0

|sj |2 vg

+1

2

∫M

(−∆(hXhY ) + g(X,Y ))

N∑j=0

|sj |2 vg

+1

4

∫Mδ((∆hX)dhY )

N∑j=0

|sj |2 vg

bitracebitrace (8.8.13)

for any orthonormal basis s = s0, . . . ,N of H0(M,L).

Proof. From (8.8.6) we readily infer

−(LX LY )(s) = −∇X(∇Y s)− idhY (X) s

− ihX∇Y s− ihY∇Xs+ hXhY s,

(8.8.14)

for any s in H0(M,L). In this expression, dhY (X) = fX , hY — cf. Section1.2 — and−〈∇X(∇Y s), s〉 = −〈X·(∇Y s, s), s〉+〈∇Y s,∇Xs〉 = 〈∇Y s,∇Xs〉,as X is divergence-free. Since the trace of −(LX LY ) is symmetric in X,Y ,


we infer

wX,Y (L) =1

2

N∑j=0

〈∇Xsj ,∇Y sj〉

− iN∑j=0

〈(hX∇Y sj + hY∇Xsj , sj〉

+

∫MhXhY

N∑j=0

|sj |2 vg.

bitraceprovbitraceprov (8.8.15)

From (8.8.12), we infer that the contribution of the second line in the rhsof (8.8.15) is equal to −1

2∆(hXhY ); this is because, δ(hXY + hYX) =

−(dhX , dchY )− (dhY , dchX) = 0, whereas δ(hXJY +hY JX) = −hX∆hY +(dhX , dhY )− hY ∆hX + (dhY , dhX) = −∆(hXhY ). In order to calculate thecontribution of the first line, we first establish the following identity

(∇Xs,∇Y s) + (∇Y s,∇Xs) =1

2(X · Y · |s|2 + JX · JY · |s|2)

− g(X,Y ) |s|2,biidentitybiidentity (8.8.16)

for any s in H0(M,L) and any X,Y in h (note that the rhs is symmetric inX,Y , as [JX, JY ] = −[X,Y ]); indeed, we have that

X · Y · |s|2 = (∇X(∇Y s), s) + (s,∇X(∇Y s))+ (∇Y s,∇Xs) + (∇Xs,∇Y s),

XYsXYs (8.8.17)

whereas

JX · JY · |s|2 = (∇JX(∇JY s), s) + (s,∇JX(∇JY s))+ (∇JY s,∇JXs) + (∇JXs,∇JY s)= i(∇JX(∇Y s), s)− i(s,∇JX(∇Y s))+ (∇Y s,∇Xs) + (∇Xs,∇Y s),

as ∇JY s = i∇Y s; moreover,

∇JX(∇Y s) = ∇Y (∇JXs) +R∇Y,JXs+∇[JX,Y ]s

= i∇Y (∇Xs) + ig(X,Y ) s+ i∇[X,Y ]

= i∇X(∇Y s)− ω(X,Y ) s+ ig(X,Y ) s;

it follows that

JX · JY · |s|2 = −(∇X(∇Y s), s)− (s,∇X(∇Y s))+ (∇Y s,∇Xs) + (∇Xs,∇Y s)− 2g(X,Y ) |s|2;

JXJYsJXJYs (8.8.18)

(8.8.16) readily from (8.8.17)-(8.8.18). Since X is divergence-free, the lhs of(8.8.17) integrates to zero, whereas∫

M(JX · JY · |s|2) vg =

∫Mδ((∆hX)dhY )|s|2 vg;

this completes the proof of (8.8.13).


remsymmetryw Remark 8.8.4. The symmetry in X,Y of wX,Y (L) is not directly ap-parent in the last term of the rhs of (8.8.13). This, in fact, can be rewrittenin the following symmetric form:

(8.8.19)1

4

∫M

(∆hX∆hY − (ddchX , ddchY )−∆(g(X,Y )))

N∑j=0

|sj |2 vg,

as follows from the following simple computation:

(ddchX , ddchY ) = (dchX , δddchY ) + δ((ιXddchY )

= (dhX , d∆hY ) + δ(LX(dchY ))−∆(ιXdchY ))

= (dhX , d∆hY )−∆(g(X,Y ))

(8.8.20)

(note that LX(dchY ) = dc(LXhY ), as X is (real) holomorphic, so thatδ(LX(dchY )) = δdc(LXhY ) = 0).

8.9. The Futaki character and the Futaki-Mabuchi bilinear formsas asymptotic invariants

ssfutakiasymptoticIn this section, we still consider a compact Kahler manifold (M, g, ω)

polarized by (L, h), which admits a hamitonian S1-action of generator X.If hX is a L-momentum of X with respect to ω — cf. Proposition 8.8.4and Remark 8.8.3 — then khX is a Lk-momentum with respect to k ω.This is because the lifted action of S1 on L decends to an action on Lk

via the map u 7→ ⊗k u from L to Lk which realizes L as a k-fold branchedcovering of Lk and the Lk-momentum of X corresponding to this S1-actionis clearly k hX . For any positive integer k, we denote by wX(Lk) the trace— as defined above — of the induced linear action on Ek = H0(M,Lk).Note that wX(Lk) depends on the chosen L-momentum hX ; by Proposition8.8.4, if hX is replaced by hX + ` for some integer `, then wX(k) is replacedby wX(k) + ` (Nk + 1). When k tends to infinity, we have the followingasymptotic expansion:

proptraceasymptotic Proposition 8.9.1. Let M be any compact Kahler manifold (M, g, J, ω)of complex dimension m, polarized by a hermitian holomorphic line bundle(L, h). Consider any hamiltonian isometric action of S1 on M and hX an L-momentum for the generator X. Then the trace wX(Lk) of the correspondinglinear action on H0(M,Lk) admits an asymptotic expansion

traceasymptotictraceasymptotic (8.9.1) wX(Lk) ' 1

(2π)m

+∞∑r=0

bXr km+1−r,


(8.9.2) wX(Lk) =1

(2π)m

q∑r=0

bXr km+1−r +O(km−r).

Moreover, the first two coefficients bX0 , bX1 are given by

bb (8.9.3) bX0 =

∫MhX vg, bX1 =

1

4

∫Msg h

X vg.

8.9. ASYMPTOTIC INVARIANTS 217

Proof. From (8.8.11), we infer that

wkaswkas (8.9.4) wX(Lk) =

∫M

(khX +1

2∆hX)

Nk∑j=0

|s(k)j |

2h(k) k

mvg,

for any orthonormal basis s(k) = s(k)0 , s

(k)1 , . . . , s

(k)Nk with respect to the

hermitian inner product 〈·, ·〉h(k) on H0(M,Lk) determined by ωk = k ω and

the induced hermitian inner product h(k) on Lk (notice that if ω is replacedby ωk = k ω, then hX is replaced by k hX , and vg by km vg, whereas ∆hX

remains unchanged). Proposition 8.9.1 is then a direct consequence of (8.9.4)and of Theorem 8.2.1.

Considering two hamiltonian S1actions, we similarly get:

propbitraceasymptotic Proposition 8.9.2. Let M be any compact Kahler manifold (M, g, J, ω)of complex dimension m, polarized by a hermitian holomorphic line bundle(L, h). Consider any two hamiltonian isometric actions of S1 on M and lethX , hY be L-momenta for the generators X,Y of these actions. Then thetrace wX,Y (k) of the corresponding linear action on H0(M,Lk) admits anasymptotic expansion

bitraceasymptoticbitraceasymptotic (8.9.5) wX,Y (Lk) ' 1

(2π)m

+∞∑r=0

bX,Yr km+2−r,


(8.9.6) wX,Y (Lk) =1

(2π)m

q∑r=0

bX,Yr km+2−r +O(km−r).

Moreover, the first two coefficients bX,Y0 , bX,Y1 are given by

bibbib (8.9.7) bX,Y0 =

∫MhXhY vg, bX1 =

1

4

∫Msg h

XhY vg +1

2

∫Mg(X,Y ) vg.

Proof. From (8.8.13), we infer that

wX,Y (Lk) = km+2

∫MhXhY

N∑j=1

|s(k)j |

2h(k) vg

+km+1

2

∫M

(−∆(hXhY ) + g(X,Y ))N∑j=1

|s(k)j |

2h(k) vg

+km

4

∫Mδ((∆hX)dfY )

N∑j=1

|s(h)j |

2h(k) vg

bitraceasbitraceas (8.9.8)

Proposition 8.9.2 is then a direct consequence of (8.9.8) and of Theorem8.2.1.

remchernequiv Remark 8.9.1. Proposition 8.9.1 can also be obtained by using the S1-equivariant Riemann-Roch formula relative to the given S1-action on L. Inthe current situation, the S1-equivariant Riemann-Roch reads

RRequivRRequiv (8.9.9)m∑r=0

(−1)rtr(e−itLX |Hr(M,Lk)

)=

∫MchS1(Lk)TdS1(M,J),


for any positive integer k, see [16], [26], [31]. In the rhs, chS1 and TdS1

denote the S1-equivariant Chern character of L and the S1-equivariantTodd class of (TM, J) respectively or, better their representatives in the deRham picture of the S1-equivariant cohomology : we then have chS1(Lk) =

ek( ω2π

+hX t), where ω2π + hX t represents the S1-equivariant Chern class of

L, relative to the chosen lifted action determined by the choice of the L-momentum fX , and t must be regarded as a formal variable; similarly,TdS1(M,J) is represented by 1 + 1

2( ρ2π + 1

2∆hX t) + . . .), where ρ2π + 1

2∆hX t

represents the first S1-equivariant Chern class of the (complex) tangent bun-dle (TM, J) or, better, of the anti-canonical line bundle K−1

M , relative to thecanonical lifted action (cf. Proposition 8.8.3). In this formula, both sideshave to be regarded as elements of the ring R[[t]] of formal power series in twith real coefficients and the formula then asserts the equality of the coef-ficients of t` for all `. In particular, if k ≥ k0, the constant term in the lhsis equal to d(Lk), whereas the coefficient of t is equal to the weight wX(Lk)as defined above (for k large enough).

If we are only interested in the first two dominant terms in k — namelythe terms in km+1 and km — and the first two terms in t, i.e. the constantterm — already computed in Remark 8.2.2 —- and the term in t, and if kis large enough, Formula (8.9.9) can be written as

d(Lk) + wX(Lk) t+ . . . =

∫M

(( ω2π + hX t)m

m!km +

( ω2π + hX t)m+1

(m+ 1)!km+1 + . . .)

(1 +ρ

4π+

1

4∆hX t+ . . .),

RRequivshortRRequivshort (8.9.10)

where, in the rhs, we only retained the terms which contribute non trivially.We then easily recover (8.2.9) and (8.9.3), cf. [77].

By bringing together Propositions 8.9.1 and 8.2.1 we get

propdonaldsonstability Proposition 8.9.3. Let M be any compact Kahler manifold (M, g, J, ω)of complex dimension m, polarized by a hermitian holomorphic line bundle(L, h). Consider any hamiltonian isometric action of S1 on M and hX

an L-momentum for X. Then, the quotient wX(Lk)k d(Lk)

admits the following

expansion

futaki-donaldsonfutaki-donaldson (8.9.11)wX(Lk)

k d(Lk)= hX +

1

4F[ω](−JX) k−1 +O(k−2)

when k tends to +∞, where hX =∫M hXvg∫M vg

denotes the mean value of the

chosen L-momentum hX and F[ω] the normalized Futaki character — seeSection 4.12 — of the Kahler class [ω] = 2π c1(L).

Proof. From Propositions 8.9.1 and 8.2.1 we readily infer

(8.9.12)wX(Lk)

k d(Lk)=bX0a0

+a0b

X1 − a1b

X0

a20

k−1 +O(k−1),

8.9. ASYMPTOTIC INVARIANTS 219

when k tends to +∞. From (8.2.9) we getbX0a0

=∫M hXvg∫M vg

= hX , whereas

a0bX1 −a1b0a2

0= 1

41∫M vg

∫M sg(h

X − hX)vg); since X = gradωhX , (fX − hX) is

the real potential of the (real) vector field −JX; from (4.12.1), we then get∫M sg(h

X − hX)vg = F[ω](−JX).

Similarly, by bringing together Propositions 8.9.2 and 8.2.1 we get

propbidonaldsonstability Proposition 8.9.4. Let M be any compact Kahler manifold (M, g, J, ω)of complex dimension m, polarized by a hermitian holomorphic line bun-dle (L, h). Consider any two hamiltonian isometric actions of S1 on Mand L-momenta hX , hY for the generators X,Y of these actions. Then,

1k2d(Lk)

(wX,Y (Lk)− wX(Lk)wY (Lk)

d(Lk)

)admits the following expansion

1

k2d(Lk)

(wX,Y (Lk)− wX(Lk)wY (Lk)

d(Lk)

)= BΩ(−JX,−JY )

+k−1

4VΩ

( ∫Msg(h

X − hX)(hY − hY ) vg

+ 2

∫Mg(X,Y ) vg − SΩBΩ(−JX,−JY )

)+O(k−2),

bifutaki-donaldsonbifutaki-donaldson (8.9.13)

where BΩ(−JX,−JY ) =∫M (hX − hX)(hY − hY ) vg denotes the Futaki-

Mabuchi inner product of −JX = gradghX and −JY = gradgh

Y , cf. Section4.11.

Proof. From Propositions 8.9.2, 8.9.1 and 8.2.1 we infer

wX,Y (Lk)

k2 d(Lk)− wX(Lk)wY (Lk)

k2(d(Lk))2=a0b

X,Y0 − bX0 bY0

a20

+ k−1

(bX,Y1

a0− bX,Y0 a1 + bX0 b

Y1 + bY0 b

X1

a20

+2bX0 b

Y0 a1

a30

)+O(k−2).

(8.9.14)

By replacing the coefficients by their values given by (8.2.9)-(8.9.3)-(8.9.7),we get (8.9.13).

Remark 8.9.2. The fact that for any S1-action on a compact polarizedKahler manifold (M, g, J, ω), generated by a hamitonian Killing vector field

X, the (normalized) Futaki number F[ω](−JX) appears as a coefficient inthe asymptotic expansion of wX(k)/kdk(L) as shown in (8.9.11) has firstbeen observed by S. Donaldson in [77] and proved to be a fact of paramountimportance in the definition of the K-stability introduced by G. Tian. Thisis because the expression wX(k)/kdk(L) and its asymptotic expansion stillmake sense not only for smooth projective manifolds but also for a largeclass of singular varieties, in particular for singular fibres appearing in de-generations of M used in the definition of K-stability.

CHAPTER 9

The scalar curvature as a momentum map

smomentum9.1. The symplectic viewpoint: the space Jω0

sssymplecticviewIn this chapter, M will denote a (connected, oriented) compact manifold

of even dimension n = 2m and we choose a Kahler structure (g0, J0, ω0) onM . The de Rham class of ω0 is denoted by Ω.

In previous chapters, we fixed the complex structure J0 and a Kahlerclass Ω on M and we considered the space MΩ of Kahler metrics (g, J0, ω)on the complex manifold (M,J0) such that the de Rham class of the Kahlerclass ω is Ω. This space is naturally acted on on the right by H(M,J0), cf.Section 3.2.

In this chapter, we instead consider the space of Kahler structures (g, J, ω0)on M whose Kahler form is fixed — equal to ω0 — and whose complex struc-ture, J , belongs to the same Teichmuller class as J0, meaning that thereexists a diffeomorphism Φ in Diff0M , the identity component of the groupof diffeomorphisms of M , such that J = Φ · J0 (such a Φ is then uniquelydefined up to right composition with an element of H(M,J0)).

We then define Jω0 as the connected component of (g0, J0, ω0) in thisspace.

In the above, Φ · J0 denotes the natural (left) action of Φ on J0, so thatJ(X) = (Φ ·J0)(X) = Φ∗(J0(Φ−1

∗ (X))), for any point x of M and any vectorX in TxM . More generally, Φ· will denote the left action of Φ on any kindof tensor fields; in particular, the right action (ω, γ) 7→ γ∗ω of H(M,J0) onMJ0 can also be written as (ω, γ) 7→ γ−1 · ω.

The identity component, Sp0(M,ω0), of the group of symplectomor-phisms of (M,ω0) acts naturally — on the left — on Jω0 , by (ψ, J)→ ψ ·J .

In contrast with H(M,J0), which is a (finite-dimensional) complex Liegroup, Sp0(M,ω0) is an infinite dimensional “Lie group”, whose Lie algebra,denoted by sp, is the space of real vector fields X such that LXω0 = 0.

Equivalently, sp is the space of real vector fields X whose symplecticdual 1-form X[ω0 := −ιXω0 is closed (cf. Cartan formula (1.4.4)).

We denote by ham the Lie subalgebra of hamiltonian vector fields, de-fined as the space of elements Z of sp such that Z[ω0 = dfZ , for some realfunction fZ called the momentum of Z with respect to ω0; alternatively,Z = gradω0

fZ , where gradω0denotes the symplectic gradient with respect

to ω0, cf. Section 1.2. Unless otherwise specified, the momentum fZ will benormalized by

∫M fZωm0 = 0. The corresponding subgroup of Sp0(M,ω0)

will be denoted by Ham(M,ω0)1

1 Roughly speaking, Ham(M,ω0) is defined as the set of those elements of Sp0(M,ω0)Hamwhich are the time one value of the flow — in the sense of (4.6.4) — of time dependent

221

222 9. THE SCALAR CURVATURE AS A MOMENTUM MAP

The spaces MΩ and Jω0 are closely related according to the followingproposition:

propid Proposition 9.1.1. (i) There exists an Sp0(M,ω0)-invariant map, p,from Jω0 to the quotient MΩ/H(M,J0) and an H(M,J0)-invariant map, q,from MΩ to the quotient Jω0/Sp0(M,ω0), which are inverse to each other,hence induce a natural identification

identificationidentification (9.1.1) Jω0/Sp0(M,ω0)p

qMΩ/H(M,J0).

The maps p and q are defined as follows. For any J in Jω0 there exists Φ inDiff0(M), uniquely defined up to left composition by an element of H(M,J0),such that J = Φ · J0 and we then set

pp (9.1.2) p(J) = Φ∗ω0 = Φ−1 · ω0 mod H(M,J0).

Similarly, if ω belongs to MΩ, by Moser lemma — cf. Section 4.6 —thereexists Φ in Diff0(M), uniquely defined up to right composition by an elementof Sp0(M,ω0), such that ω = Φ∗ω0 and we then set

qq (9.1.3) q(ω) = Φ · J0 mod Sp0(M,ω0).

(ii) For any J in Jω0 the tangent space TJJω0 is described by

TJTJ (9.1.4) TJJω0 = A = −LZJ | Z ∈ sp + Jsp = sp⊕ J ham.

Via this identification, the differential of p at J = Φ · J0 is given by

dpdp (9.1.5) dp(−LZJ) = Φ∗(LZω0) = ddc(Φ∗f)

for each Z = X − JY in sp ⊕ J ham, where f denotes the normalized mo-mentum of the hamiltonian vector field Y , hence is determined by

ZZ (9.1.6) LZω0 = ddJf

and∫M fωm0 = 0 (here dc stands for the operator J0dJ

−10 , also denoted by

dJ0, whereas dJ := JdJ−1).(iii) Jω0 is a complex, hence Kahler, submanifold of Cω0.

Proof. (i) The symplectic form ω defined by (9.1.2) is J0-invariant,as the pair (J0, ω) is the image of the Kahler pair (J, ω0) by Φ−1·, andbelongs to Ω as Φ sits in Diff0M . Similarly, the complex structure J de-fined by (9.1.3) is compatible with ω0, hence belongs to Jω0 as Φ sits inDiff0M and the pair (J, ω0) is the image of the Kahler pair (J0, ω) byΦ·. The maps p and q are then well defined, from Jω0 to MΩ/H(M,J0)and from MΩ to Jω0/Sp0(M,ω0) respectively. Moreover, p and q are ev-idently Sp0(M,ω0)- and H(M,J0)-invariant respectively. They then deter-mine maps from Jω0/Sp0(M,ω0) toMΩ/H(M,J0) and fromMΩ/H(M,J0)to Jω0/Sp0(M,ω0), which are clearly inverse to each other (The Moserlemma used to define q has been described in Section 4.6). Note that forany J = Φ · J0 in Jω0 the riemannian metric g := ω0(·, J ·) determined bythe Kahler pair (J, ω0) and the riemannian metric g := ω(·, J0·) determined

vector fields of the form Xt = gradω0Ht, defined on M×[0, 1], see, e.g., [164] for a detailed

exposition.

9.1. THE SYMPLECTIC VIEWPOINT 223

by the Kahler pair (J0, ω), where ω is defined by (9.1.2), are linked togetherby

linklink (9.1.7) g = Φ∗g = Φ−1 · g.(ii) Let Jt = Φt · J0 be a curve in Jω0 , where Φt is a curve in Diff0M ,

with Φ0 = Φ. We then have

(9.1.8) J :=d

dt |t=0Φt · J0 = −LZJ,

whith Z := ddt |t=0

(Φt Φ−1). Moreover, the fact that Jt belongs to Jω0 for

all t implies that LZJ is symmetric with respect to the metric g determinedby (J, ω0). If ζ denotes the riemannian dual 1-form of Z with respect tothe metric g, this condition just means that dζ is J-invariant (cf. Lemma1.23.2). By Lemma 1.17.1, the latter condition is equivalent to the fact thatthe Hodge decomposition of ζ with respect to g is of the form

zetazeta (9.1.9) ζ = ζH + df + dJh,

where ζH is g-harmonic — equivalently, ζH is the real part of a J-holomorphic1-form — and dJh := Jdh. Notice that the symplectic dual 1-form of Z withrespect to ω0 is equal to Jζ = JζH+dJf−dh. It follows that ζ is of the form(9.1.9) if and only if Z is of the form Z = X−JY where X,Y both belong tosp: if so, X,Y are not uniquely determined by Z, because of the presence ofthe harmonic component ζH in (9.1.9); they can however be made uniquelydetermined by demanding that Y be hamiltonian; then, f is the normalizedmomentum of Y , also given by (9.1.6). Conversely, Z satisfies (9.1.6), forsome real function f , if and only if Jζ − dJf is closed, if and only if ζ isof the form (9.1.9). It follows that the tangent space TJJω0 is described by(9.1.4). For any such Z, we then have:

dp(−LZJ) =d

dt |t=0Φ∗(Φt Φ−1)∗ω0

= Φ∗(LZω0) = Φ∗(ddJf)

= ddJ0(Φ∗f) ∼= Φ∗f.

This gives (9.1.5). Note that the chosen normalization∫M fωm0 = 0 for

the momentum f implies the usual normalisation∫M Φ∗fωm = 0 for the

real function Φ∗f viewed as an element of the tangent space TωMΩ. Sincethe latter space is made of all real functions normalized that way we thendeduce from (9.1.5) that TJJω0 = A = −LZJ for all vector fields Z insp + Jsp.

As a subspace of TJJω0 , the tangent space of the orbit O of J under theaction of Sp0(M,ω0) is the space of LXJ for any X in sp. From Proposition9.1.1 we readily infer

orbitorbit (9.1.10) TJJω0 = TJO + J(TJO).

This sum is not direct in general. More precisely:

propdirectsum Proposition 9.1.2. For any J in Jω0, the intersection TJO ∩ J(TJO)is the space of elements of the form LZJ , for all elements Z of sp of theform Z = ZH + gradω0

fXg , where ZH is the dual vector field of a harmonic


1-form and fXg the real potential of a (real) holomorphic vector field withrespect to J .

In particular, the sum (9.1.10) is direct if and only if any J-harmonic1-form is g-parallel, where g denotes the associated riemannian metric g =ω0(·, J ·), and if h(M,J) splits as in (3.6.1).

Proof. Elements of TJO are of the form LZJ for ZH + gradω0hZ in sp,

where ZH is the dual vector field of a harmonic 1-form. Such an elementbelongs to J(TJO) if and only if there exists Z1 = Z1,H + gradω0

hZ1 insp such that LZJ = JLZ1J = LJZ1J . Equivalently, Z − JZ1 = (ZH −JZ1,H) + gradω0

(hZ) + gradghZ1) is a (real) holomorphic vector field. In

particular, hZ is the real potential of the (real) holomorphic vector field−(Z1 + JZ). Conversely, if Z = ZH + gradω0

hZ , where ZH is the dual

vector field of a harmonic 1-form and hZ is the real potential of a (real)vector field, X = XH + gradgh

Z + gradω0hX say, then LZJ = LJZ1J , with

Z1 = −(XH + JZH + gradω0hX). The last statement follows readily.

remdonaldson Remark 9.1.1. In view of (9.1.10) and of Proposition 9.1.1, Jω0 appearsas a complexification of the orbit of J0 under the action of the symplec-tomorphism group Sp0(M,ω0) or as a “complex orbit” of an (inexisting)complexification of Sp0(M,ω0) in the space of all almost complex structurescompatible with ω0 considered in the next section, cf. [73], [74]. The quo-tient Jω0/Sp0(M,ω0) — hence also the quotient MΩ/H(M,J0) by Propo-sition 9.1.1 — can then be regarded as dual symmetric space of the groupSp0(M,ω0), cf. [74].

remdavid Remark 9.1.2. For any fixed hamiltonian vector field Z = gradω0, de-

note by Z the induced vector field on Jω0 , defined by Z(J) = −LZJ . By

its very definition, Z is tangent to the orbit O and has a flow, ΦZt , defined

by ΦZt (J) = ΦZ

t · J , where ΦZt stands for the flow of Z on M . On the other

hand, the existence of a flow for the vector field JZ is not granted a priorion Jω0 . This is actually closely related to the existence issue of geodesicson MΩ, as shown by the following proposition, communicated to me by D.Calderbank:

propdavid Proposition 9.1.3. Let ωt = ω0 + ddcφt be a geodesic in MΩ startingfrom ω0, with tangent vector φ0. Let Φt be the corresponding Moser lift —cf. Section 4.6 — in Diff0(M), so that Φ∗tωt = ω0. Then, Jt := Φ−1

t · J0 is

an integral curve in Jω0 of JZ, with Z = gradω0φ0.

Proof. By definition, Φt is the “flow”, as defined in (4.6.4), of the 1-

parameter vector field Xt = −gradgt φt, with gt := ωt(·, J0·), see Section 4.6.

By (4.6.5), we have Jt = LΦ−1t ·Xt

Jt, whereas the 1-parameter vector field

Φ−1t ·Xt is given by:

Φ−1t ·Xt = −Φ−1

t · gradgt φt

= −gradΦ−1t ·gt

(Φ−1t · φt)

= −gradgt φ0

= Jt gradω0φ0.

(9.1.11)

9.2. THE SPACE OF SYMPLECTIC ALMOST COMPLEX STRUCTURES 225

Here, gt := Φ−1t · gt stands for the metric determined by the pair (ω0, Jt),

which is the image by Φ−1t of the pair (ωt, J0), and we used the fact that

Φ−1t · φt = Φ∗t φt = φ0, as Φt is the Moser lift of the geodesic ωt = ω0 +ddcφt,

cf. (4.6.8); in particular, gradgt φ0 = −Jt gradω0φ0. We thus get Jt =

LJtgradω0φ0Jt = JZ(Jt).

9.2. The space of symplectic almost complex structures as aKahler manifold

sssac

In this section, we fix a symplectic form ω0 on M . An almost com-plex structure J is said to be compatible with ω0, or simply symplectic, ifg := ω0(·, J ·) is a riemannian metric. The space of all symplectic almost-structures is denoted by ACω0 . This space is then the space of all (smooth)sections of the fiber bundle over M whose fiber at x is the space C(ω0(x))of all complex structures of TxM compatible with ω0(x), which is a hermit-ian symmetric space of non-compact type, cf. Appendix B.2. In particular,C(ω0(x)) is contractible, so that ACω0 is itself contractible. Via the Cayleytransformation J 7→ µ relative to any chosen J0 — cf. Appendix B.1 —ACω0 can also be viewed as an open set in the space of sections of the (com-plex) vector bundle of J0-anti-commuting endomorphisms of (TM, J0) Thiswill be used in the proof of Proposition 9.2.1 below).

The subspace of integrable almost complex structures in ACω0 will bedenoted by Cω0 ; Jω0 is then a subspace of Cω0 . By definition, a riemannianmetric g is attached to any J in ACω0 in such a way that the triple (g, J, ω0)is an almost Kahler structure. The volume form vg of g coincides with the

symplectic volume formωm0m! and is then independent of J in ACω0 .

The groups Sp0(M,ω0) and Ham(M,ω0) naturally act (on the left) onboth spaces ACω0 and Cω0 . As in the previous section, for any vector field

Z in sp, we denote by Z the induced vector field on ACω0 , defined by

hatZhatZ (9.2.1) ZJ = −LZJ.

An alternative useful expression is given by

Lemma 9.2.1. For any almost Kahler structure (g, J, ω), the Lie deriva-tive of J along any vector field Z preserving ω has the following expression

hatZbishatZbis (9.2.2) LZJ = −(DZ + (DZ)∗)J = J(DZ + (DZ)∗),

where D denotes the Levi-Civita connection of the metric g and (DZ)∗ theadjoint of DZ with respect to g as a field of endomorphisms of TM .

Proof. Since LZω = 0 and ω(X,Y ) = g(JX, Y ), we have

0 = (LZω)(X,Y )

= (LZg)(JX, Y ) + g((LZJ)X,Y )

= g(DJXZ, Y ) + g(JX,DY Z) + g((LZJ)X,Y )

= g((DZ + (DZ)∗)(JX), Y ) + g((LZJ)X,Y )

for any vector fields X,Y . The second expression in (9.2.2) follows from thefact that LZJ anticommute to J for any vector field Z.


We denote by Aut(TM,ω0) the group bundle of all automorphisms γof TM wich preserve ω0 in the sense that ω0(γX, γY ) = ω0(X,Y ) for anyvector fields X,Y , and by End(TM,ω0) the corresponding Lie algebra vectorbundle, whose elements are characterized by ω0(AX,Y ) + ω0(X,AY ) = 0.We denote by Gω0 , resp. Lω0 , the space of smooth sections of Aut(TM,ω0),resp. of End(TM,ω0): Gω0 can be viewed as an (infinite dimensional) Liegroup whose Lie algebra is Lω0 . For any a in Lω0 we denote by exp a theexponential of a in Gω0 .

The space ACω0 is naturally embedded in both Gω0 and Aω0 . In par-ticular, at each point J of ACω0 , the tangent space TJACω0 is naturallyidentified with the space of sections of End(TM,ω0) which anticommuteswith J , i.e. the space of field of endomorphisms of TM which anticommutewith J and are symmetric with respect to the metric g = ω0(·, J ·).

The group Gω0 acts transitively on the space ACω0 by

(9.2.3) γ · J = γJγ−1,

for any J in ACω0 and any A in Gω0 . More precisely, we have

lemmaa Lemma 9.2.2. For any pair J0, J of elements of ACω0, there exists aunique a in Lω0 which anticommutes with both J0 and J and joins J0 to Jin the following sense:

conjconj (9.2.4) J = exp(a) J0 exp(−a).

Proof. We denote by g = ω0(·, J ·) and g0 = ω0(·, J0·) the riemannianmetrics attached to J and J0 respectively. Define S by J = J0S; then, Sbelongs to Gω0 , as both J, J0 do, and is positive definite with respect to gand g0, as g0(SX, Y ) = g(X,Y ) and g(SX, Y ) = g0(X,Y ). It follows thatS = exp(2a) for a unique a in Lω0 which is again symmetric with respect to gand g0, hence anticommutes with J and J0. We thus have: J = exp(2a) J0 =exp(a) J0 exp(−a). Conversely, if (9.2.4) is satisfied where a anticommuteswith J0, then J0exp(−a) = exp(a)J0 and we get J = exp(2a) J0, so thatexp(a) is the positive square root of S = −J0J .

Any section, a, of End(TM,ω0) determines a vector field, a, on ACω0

defined by

hatahata (9.2.5) a(J) =d

dt |t=0exp(−ta)Jexp(ta) = [J, a],

for any J in ACω0 . Conversely, for any J in ACω0 , any J in TJACω0 is of

the form J = a(J), for some section a of End(TM,ω0). Such an a is welldefined up to the addition of any section of End(TM,ω0) which commutesto J ; in particular, a is uniquely defined by demanding that a anticommuteto J . We then have

valuevalue (9.2.6) a = −1

2JJ.

Since a is pointwise induced by a right action of Aut(TM,ω0), we have

liehatliehat (9.2.7) [a, b] = [a, b],for any two sections a, b of End(TM,ω0) (in the lhs, the bracket stands for

the bracket of the two vector fields a, b on ACω0 , whereas the bracket in the

9.2. THE SPACE OF SYMPLECTIC ALMOST COMPLEX STRUCTURES 227

rhs denotes the commutator [a, b] = ab − ba). This computation can alsoeasily be done directly in the vector space of sections of End(TM,ω0).

The actions of Gω0 and Sp(M,ω0) onACω0 do not commute but Sp(M,ω0)acts on Gω0 and on Lω0 in an obvious way. In particular, for any Z in spand any a in Lω0 , we have

(9.2.8) [Z, a] = LZa.

The space ACω0 comes naturally equipped with an almost hermitianstructure, whose almost complex structure J is defined by

boldJboldJ (9.2.9) JA = J A,

the metric g by

boldgboldg (9.2.10) g(A,B) =1

2

∫M

tr(AB) vg,

and the Kahler form, κ, by

kappakappa (9.2.11) κJ(A,B) =1

2

∫M

tr(JAB)vg.

The following identities are easily checked:

liehatJliehatJ (9.2.12) [a,Jb] = J[a, b], [Ja,Jb] = −[a, b],

for any two sections a, b of End(TM,ω0). Together with (9.2.7), they implythat the vector fields a preserve J.

propintJ Proposition 9.2.1. The almost complex structure J is integrable.

Proof. For any chosen J0 in ACω0 the Cayley transformation J 7→µ = (J + J0)−1(J0 − J) = (J − J0)(J + J0))−1 — cf. Appendix B.2 —realizes ACω0 as an open subset of the (complex) vector space of symmetricendomorphisms of TM which anti-commute to J0, namely the open set ofthose endomorphisms B which satisfy the additional (open) condition thatI + B and I − B are both invertible. Morever, by (B.1.7), the induced(integrable) complex structure coincides with J.

propkappaclosed Proposition 9.2.2. The Kahler form κ is closed.

Proof. The exterior differential dκ, is convenienly computed by usingvector fields of the form a, defined by (9.2.5), so that

kappahatkappahat (9.2.13) κJ(a, b) =

∫M

tr(J [a, b]) vg,

and the general formula

dκ(a, b, c) = a · κ(b, c) + b · κ(c, a) + c · κ(a, a)

− κ([a, b], c)− κ([b, c], a)− κ([c, a], b),dkappadkappa (9.2.14)

for any (fixed) three sections a, b, c of End(TM,ω0). By using (9.2.13) and(9.2.7), we easily check that the two lines in the rhs of (9.2.14) are separatelyequal to 0.


Since J is integrable and κ is closed, the natural almost hermitian struc-ture of ACω0 can be considered as a Kahler structure. In fact, we can easilydefine a “Levi-Civita connection”, say D, by first defining Da some distin-guished vector fields like a, Z or their image by J, by using the generalKoszul formula (1.1.1), then, for any vector field ξ, by metric duality, e.g.

(9.2.15) ((DAξ, a)) = A · ((ξ, a))− ((ξ,DAa)),

for any J in ACω0 and any vector A = J in TJACω0 , by using the fact thatthe a’s generate the tangent bundle of ACω0 . We already used a similarprocedure in Section 4.4 in order to define the Levi-Civita connection of theMabuchi metric on MΩ. We get that way, at J :

(9.2.16) DAa = [A, a+], DAZ = −(LZA)−,

by adopting the following convention : for any J in ACω0 , any a in Lω0 splitsas a = a+ + a−, where a+ = 1

2(a − JaJ) commutes and a− = 12(a + JaJ)

anticommutes with J (note that this splitting makes sense as JaJ belongsto Lω0 whenever a does). We also get DJa = JDa, so that

(9.2.17) DJ = 0.

This confirms our claim that ACω0 is Kahler. Of course, this Kahler struc-ture is a “globalization” of the (symmetric) Kahler structure of the algebraicmodel C(ω0) = Sp(ω0)/U(J0) described in Appendix B.

Proposition 9.2.3. The spaces Cω0 and Jω0 are Kahler subspaces ofACω0.

Proof. In both cases, it is sufficient to show that the tangent space isstable for J. For any J in Cω0 , TJCω0 is defined as the subspace of TJACω0

defined by the condition

NlinearNlinear (9.2.18) [JX, JY ]+[J, J ]−J [JX, Y ]−J [JX, Y ]−J [X, JY ]−J [X, JY ] = 0,

for any vector fields X,Y , obtained by linearizing the integrability conditionN(X,Y ) = 0. Equivalently, J can be viewed as (TM, J)-valued (0, 1)-formand, by considering the canonical Cauchy-Riemann operator ∂TM definedin the next section, the above condition reads

(9.2.19) d∂TMJ = 0.

It is a simple exercise to check — by using N = 0 — that (9.2.18) is satisfied

by J if and only if it is satisfied by JJ ; it follows that J(TJCω0) = TJCω0 .When J is integrable, we have that LJXJ = JLXJ for any vector field Z, cf.Lemma 1.1.1. This, together with (9.1.4), implies that J(TJJω0) = Jω0 .

The action of the group Gω0 on ACω0 is clearly hamiltonian and itsmomentum map is simply the natural inclusion of ACω0 in Lω0 , when thelatter is considered as a part of the dual L∗ω0

via the duality induced by themetric g.

The action of Sp(M,ω0) on ACω0 is symplectic, i.e. preserves κ, but ispresumably not hamiltonian. On the other hand, the action of Ham(M,ω0)is hamiltonian and its momentum can be identified with the hermitian scalarcurvature : this is the subject of the next sections.

9.3. THE CHERN CONNECTION OF AN ALMOST KAHLER STRUCTURE 229

remfujiki Remark 9.2.1. The natural Kahler structure of ACω0 described in thissection first appeared in Fujiki’s paper [84]. The fact — explained andgeneralized in Section 9.6 below — that the scalar curvature is a momentumfor restriction of the action of Ham(M,ω0) on Cω0 was also first observed in[84].

9.3. The Chern connection of an almost Kahler structuresschernsymplectic

propCRTM Proposition 9.3.1. The tangent space of any almost complex manifold(M,J) admits a canonical Cauchy-Riemann operator, given by

CRTMCRTM (9.3.1) ∂TMX Z = 2<([X0,1, Z1,0]1,0)

for any two vector fields X,Z. Equivalently

∂TMX Z =1

4([X,Z] + [JX, JZ] + J [JX,Z]− J [X, JZ])

= −1

4(JLZJ + LJZJ)(X)

= −1

2J(LZJ)(X) +N(X,Z).

CRbisCRbis (9.3.2)

Proof. Recall that the general notion of Cauchy-Riemann operator forany complex vector bundle E over any almost complex manifold (M,J) hasbeen introduced in Section 1.6. In the current case, E = (TM, J) is thetangent bundle itself, viewed as a complex vector bundle via J . Moreover,the complex tensor product (TM, J) ⊗C Λ0,1M is naturally identified withthe bundle End−(TM) of endomorphisms of TM which anticommute to J :a Cauchy-Riemann operator can be viewed as an operator acting on (real)vector fields and taking its values in End−(TM). The operator ∂TM definedby (9.3.1) is then a Cauchy-Riemann operator, called the canonical Cauchy-Riemann operator of (M,J). The equivalence of the various expressions in(9.3.2) with the rhs of (9.3.1) is easily checked.

remholovector Remark 9.3.1. When J is integrable, (9.3.1) is the same as (1.8.2) and(9.3.2) then reduces to (1.8.3). In this case, a (real) vector field Z belongs tothe kernel of ∂TM if and only if it preserves J and its (1, 0) part Z1,0 is thena holomorphic section of T 1,0M with respect to the natural holomorphicstructure of T 1,0M determined by any local holomorphic coordinate system,cf. Section 1.8. When J is a general almost-complex structure, the space of“holomorphic” (real) vector fields, defined as the kernel of ∂TM — does nolonger coincide in general with the space of (real) vector fields wich preserveJ . As a matter of fact, the latter space is no longer preserved in general bythe action of J .

The complex dual of (TM, J) ∼= T 1,0M is the bundle Λ1,0M of complex1-forms of type (1, 0). By (complex) duality, the canonical Cauchy-Riemannoperator ∂TM defined in Proposition 9.3.1 induces a Cauchy-Riemann oper-

ator on Λ1,0, say ∂Λ1,0M , defined by

CRT*MCRT*M (9.3.3) 〈∂Λ1,0MX ζ, Z〉 = ∂X〈ζ, Z〉 − 〈ζ, ∂TMX Z〉,

for any (1, 0)-form ζ, any (real) vector field X and any vector field Z (in(9.3.3), 〈·, ·〉 denotes the (complex) duality between (TM, J) ∼= T 1,0M and


Λ1,0M , whereas Z can be regarded indifferently as a real vector field or asa complex vecor field of type (1, 0)). We then infer:

CRdualCRdual (9.3.4) ∂Λ1,0MX ζ = (LX0,1α)1,0 = (ιX0,1dζ)1,0 = ιX0,1 ∂ζ,

for any (1, 0)-form ζ, where ∂ζ := (dζ)1,1 denotes the J-invariant part of dζ.As a real vector bundle, Λ1,0M is naturally identified with T ∗M , in the

same way as T 1,0M has been identified with TM , i.e. by identifying anyα in T ∗M with its (1, 0)-part α1,0 = 1

2(α + iJα) and any ζ in Λ1,0M with

2<(ζ) = ζ+ ζ. This identification is also an identification of complex vectorbundles of Λ1,0M , equipped with its natural structure of complex vectorbundle, with (T ∗M,−J); the latter can then be regarded as the complexdual of (TM, J), via the (complex) duality

cdualitycduality (9.3.5) 〈α,X〉C := α1,0(X) =1

2(α(X)− iα(JX)),

for any (real) 1-form α and any (real) vector field X.

In this setting, the canonical Cauchy-Riemann operator ∂Λ1,0is better

denoted by ∂T∗M and (9.3.4) then reads:

CRdual-bisCRdual-bis (9.3.6) (∂T∗M

X α)(Z) =1

2X · α(Z) +

1

2JX · α(JZ)− α(∂TMX Z),

for any (real) 1-form α and any two (real) vector fields X,Z, where ∂TMX Z,itself viewed as a real vector field, can be replaced by any expression in therhs of (9.3.2).

If J is a part of an almost hermitian structure (g, J, ω), (TM, J) comesequipped with a (fiberwise) hermitian inner product h defined by h(X,Y ) =g(X1,0, Y 0,1) = 1

2(g(X,Y )− iω(X,Y )). According to the general receipt ex-plained in Section 1.6 — cf. in particular Proposition 1.6.1 — this hermit-ian inner product, together with the canonical Cauchy-Riemann operator(9.3.1), determines a canonical Chern connection ∇2, defined by

(9.3.7) ∇XZ = ∂TMX Z + ∂TMX Z,

by setting ∂TMX Z := τ−1( ¯∂Λ1,0Mτ(Z)), where τ denotes the hermitian dualityfrom (TM, J) ∼= T 1,0M to (T ∗M,−J) ∼= Λ1,0M . Notice that τ and itsinverse, τ−1, are given by

hermitiandualhermitiandual (9.3.8) τ(X) = (X[)1,0 =1

2(X[ + iJX[), τ−1(ζ) = 2(<ζ)],

for any X in TM and any ζ in Λ1,0M , when τ is regarded as a (C-anti-linear) map from (TM, J) to Λ1,0M , whereas τ and τ−1, regarded as mapsfrom (TM, J) to (T ∗M,−J) and from (T ∗M,−J) to (TM, J), are simplythe riemannian isomorphims [ and ] defined in Section 1.2. We then have:

propcherntangent Proposition 9.3.2. For any almost-hermitian structure (g, J, ω), thecanonical Chern connection ∇ has the following expression

cherntangent1cherntangent1 (9.3.9) ∇XZ = 2<([X0,1, Z1,0]1,0) + (2<(ιX0,1(∂(Z[)1,0)]

2This connection is the second canonical hermitian connection as defined in A. Lich-nerowicz’s book [136], cf. also [89].

9.3. THE CHERN CONNECTION OF AN ALMOST KAHLER STRUCTURE 231

for any (real) vector fields X and Z. Alternatively,

(∇XZ, Y ) =1

2X · (Z, Y ) +

1

2JX · (Z, JY )

+1

4([X,Z] + [JX, JZ] + J [JX,Z]− J [X, JZ], Y )

− 1

4([X,Y ] + [JX, JY ] + J [JX, Y ]− J [X, JY ], Z),

cherntangent2cherntangent2 (9.3.10)

for any (real) vector fields X,Y, Z.

Proof. (9.3.9) is an immediate consequence of (9.3.1), (9.3.4) and (9.3.8).By using (9.3.6) and (9.3.2), we readily get (9.3.10).

By using (1.23.4), we now check that the canonical Cauchy-Riemannoperator ∂TM can be expressed in terms of the Levi-Civita connection D ofg as follows:

∂TMX Z =1

2(DXZ + JDJXZ)

− 1

4(DJZJ + JDZJ)X +

1

4(DJXJ − JDXJ)Z.

CRterCRter (9.3.11)

Similarly, from (9.3.4) we easily infer:

(∂TMX Z, Y ) =1

2(DXZ − JDJXZ, Y )

− 1

4(DJXJ + JDXJ)Z, Y ) +

1

4(Z, (DJY J + JDY J)X).

partialTMpartialTM (9.3.12)

For any almost-hermitian structure, the Chern connection ∇ and the Levi-Civita connection D are then related by

(∇XZ, Y ) = (DXZ, Y )− 1

2(J(DXJ)Z, Y )

+1

4((DJY J + JDY J)X,Z)− 1

4(DJZJ + JDZJ)X,Y ).

nablaDnablaD (9.3.13)

In the almost Kahler case, which is the case of main interest in this section,this expression much simplifies thanks to the following lemma:

lemmaAC Lemma 9.3.1. An almost hermitian structure (g, J, ω) is almost Kahlerif and only if DJ and the Nijenhuis tensor N satisfy the identities

JoppositeJopposite (9.3.14) DJXJ = −J DXJ

and

NbianchiNbianchi (9.3.15) (X,N(Y,Z)) + (Y,N(Z,X)) + (Z,N(X,Y )) = 0.

Proof. By (1.1.5), if dω = 0, DJ and N are related by

DJNDJN (9.3.16) ((DXJ)Y,Z) = 2(JX,N(Y, Z)),

for any vector fields X,Y, Z. It readily follows that DJ satisfy (9.3.14) —which is the opposite of the identity (1.1.6) which characterizes hermitianstructures — and thatN satisfies the Bianchi identity (9.3.15). Conversely, if


(9.3.14) is satisfied, from (1.1.3) we get N(X,Y ) = 12J((DY J)X−(DXJ)Y );

then, (9.3.15) and (9.3.14) again implies ((DJXJ)Z, Y ) + ((DZJ)Y, JX) +((DY J)JX,Z) = dω(JX,Z, Y ) = 0, for any vector fields X,Y, Z. We thenget dω = 0.

As a direct corollary, we obtain

prop11 Proposition 9.3.3. For any almost Kahler structure (g, J, ω) the canon-ical Cauchy-Riemann operator ∂TM and the Chern connection ∇ are givenby

CRACCRAC (9.3.17) ∂TMX Z =1

2(DXZ + JDJXZ)− 1

2J(DXJ)Z,

and

nablaACnablaAC (9.3.18) ∇XZ = DXZ −1

2J(DXJ)Z.

If T∇ denotes the torsion of ∇, we then have

torsionACtorsionAC (9.3.19) T∇ = N.

Proof. (9.3.17) readily follows (9.3.11) and (9.3.14) and (9.3.18) is alsoan immediate consequence of (9.3.13) and (9.3.14). From (9.3.18), we infer:(T∇(X,Y ) = −1

2(JDXJ)Y + 12(DY J)X; by (1.1.3), this expression is equal

to N(X,Y ) whenever (9.3.14) is satisfied.

remhermconn Remark 9.3.2. Lemma 9.3.1 can be made more precise by consider-ing the type decomposition of the (real) 3-form dω, which splits as dω =

(dω)(2,1)+(1,2) +(dω)(3,0)+(0,3). It can then be checked that (dω)(2,1)+(1,2) = 0

if and only if DJ satisfies (9.3.14), whereas (dω)(3,0)+(0,3) = 0 if and only ifN satisfies the Bianchi identity (9.3.15).

Now, Proposition 9.3.3 has been derived by using (9.3.14) only, not

(9.3.15): it then holds with the weaker assumption that (dω)(2,1)+(1,2) = 0.This class of almost hermitian structures — one of the 16 classes appearingin the Gray-Hervella classification [92] — has been called (2, 1)-symplecticby S. Salamon [171].

For a general almost hermitian structure, the hermitian connection de-fined by the rhs of (9.3.18) is the connection obtained by orthogonal pro-jection of the Levi-Civita connection D into the space of hermitian connec-tions and has therefore been called the first canonical hermitian connectionin [136]. Proposition 9.3.3 can then be rephrased by saying that for anyalmost Kahler structure, actually for any (2, 1)-symplectic almost hermitianstructure, the first and second hermitian connections coincide. The converseis also true: an almost hermitian structure is (2, 1)-symplectic if and onlyif the first and second hermitian connections coincide. More generally, a 1-parameter family of hermitian connections, denoted by ∇t in [89], is canon-ically attached to any almost hermitian structure, where the first canonicalhermitian connection is ∇0 and the second canonical hermitian connection,or Chern connection, is ∇1; it can then be shown that ∇0 = ∇1 if and onlyif ∇t = ∇0 for all t, if and only if the structure is (2, 1)-symplectic, cf. [89]for details.

From (9.3.19), we infer that the Chern connection of an almost-Kahlerstructure — more generally of a (2, 1)-symplectic almost-hermitian structure

9.4. THE HERMITIAN SCALAR CURVATURE 233

— is the (unique) connection which preserves the metric g whose torsion isequal to the Nijenhuis tensor. For a general almost-hermitian structure thisconnection is not a hermitian connection, i.e. does not preserves J . It iseasily checked that it does preserve J if and only if the almost-hermitianstructure is (2, 1)-symplectic and is then equal to the Chern connection.Indeed, in general, a connection, DT , which preserves g and whose torsionis T is uniquely defined and is determined by

(DTXZ, Y ) = (DXZ, Y ) +

1

2(T (X,Z), Y )− (T (Z, Y ), X)− (T (Y,X), Z)),

where D denotes the Levi-Civita connection of g; if T = N , we infer thatDT preserves J is and only if

((DXJ)Z, Y ) = −(N(X,Z), JY ) + (N(Z, Y ), JX) + (N(Y,X), JZ);

by (1.1.7), this condition is equivalent to (9.3.14).

9.4. The hermitian scalar curvaturesshermscalar

For a general almost hermitian structure (g, J, ω), the Chern connec-tion induces a hermitian connection on the anti-canonical bundle K∗M =Λm((TM, J)) — cf. Section 1.19. Since K∗M is a complex line bundle,the curvature of the induced hermitian connection is of the form iρ∇, forsome (closed) real 2-form ρ∇, called the hermitian Ricci form. The hermit-ian scalar curvature, s∇, is then defined as the normalized trace of ρ∇ withrespect to ω, namely by

(9.4.1) s∇ = 2Λρ∇.

The normalization is chosen in order that s∇ coincide with the riemannianscalar curvature s in the Kahler case (ρ∇ then coincides with the Ricci formρ defined in Section 1.19).

In order to relate the hermitian scalar curvature s∇ to the scalar cur-vature s, at least in the almost-Kahler setting, it is convenient to introducethe 2-form R(ω), image of the Kahler form by the (riemannian) curvatureoperator, often called the ∗-Ricci form. We similarly introduce the ∗-scalarcurvature s∗, defined by

(9.4.2) s∗ = 2Λ(R(ω)).

Again the normalization is chosen so that s∗ coincides with s in the Kahlercase, when R(ω) itself coincides with the Ricci form ρ. In general however,R(ω) is not closed, nor even J-invariant.

propriccis Proposition 9.4.1 (V. Apostolov-T. Draghici [8]). For any almost Kahlerstructure (g, J, ω), the ∗-Ricci form R(ω), the hermitian Ricci form ρ∇ andthe riemannian Ricci tensor r are related by

rhonablaRomegarhonablaRomega (9.4.3) ρ∇X,Y = R(ω)X,Y −1

4tr(JDXJ DY J),

and

RomegaricciRomegaricci (9.4.4) R(ω)X,Y =1

2(r(JX, Y )− r(X, JY )) +

1

2((D∗DJ)X,Y ).


The hermitian scalar curvature s∇, the ∗-scalar curvature s∗ and the rie-mannian scalar cuvature s are then related by

scalarsscalars (9.4.5) s∇ = s+1

2|DJ |2 = s∗ − 1

2|DJ |2 =

1

2(s+ s∗).

Proof. For any almost hermitian structure (g, J, ω), any hermitian con-nection ∇ is of the form ∇ = D + η, where η is a 1-form with values in thebundle A(TM) of skew-symmetric endomorphisms of TM and such that[ηX , J ] = −DJ for any vector field X. The curvature R∇ is then given by

RetaReta (9.4.6) R∇X,Y = RX,Y − (dDη)X,Y − [ηX , ηY ] = RX,Y − (d∇η)X,Y + [ηX , ηY ]

(the operators d∇ and dD, where ∇ and D actually stands for the inducedconnection on A(TM), have been defined in (1.6.2); the general expression(9.4.6) for the curvature of ∇ = D + η, or D = ∇ − η, is then an easyconsequence of (9.4.6)).

The corresponding Ricci form, ρ∇ — defined like in the case that ∇ isthe Chern connection, i.e. by ρ∇ = 1

2tr(−J R∇) — is then given by

(9.4.7) ρ∇X,Y = R(ω)X,Y +1

2(d(tr(J η)))X,Y −

1

2tr(J [ηX , ηY ]).

When ∇ is the first canonical hermitian connection, as defined in Remark9.3.2, we have ηX = −1

2JDXJ , which is trace-free, and the above identityreduces to

(9.4.8) ρ∇X,Y = R(ω)X,Y −1

4tr(JDXJ DY J).

This holds for any almost hermitian structure when ρ∇ stands for the Ricciform of the first canonical hermitian connection. In the almost Kahler case,the latter is equal to the Chern connection and we thus get (9.4.3). We nowhave:

R(ω)X,Y =1

2

n∑i=1

(RX,Y ei, Jei)

= −1

2

n∑i=1

((RY,eiX,Jei) + (Rei,XY, Jei))

=1

2

n∑i=1

((JRY,eiX, ei) + (JRei,XY, ei))

=1

2

n∑i=1

((RY,eiJX, ei) + (Rei,XJY, ei))

− 1

2

n∑i=1

(((RY,eiJ)X, ei) + ((Rei,XJ)Y, ei))

computecompute (9.4.9)

where ei, i = 1, . . . , n, stands for any auxiliary local orthonormal frame.From the tautological commutation formula

commutationJcommutationJ (9.4.10) [RX,Y , J ] = RX,Y J = (D2J)Y,X − (D2J)X,Y ,

9.4. THE HERMITIAN SCALAR CURVATURE 235

which holds in the general almost hermitian case, we then infer:

R(ω)X,Y =1

2(r(JX, Y )− r(X, JY ))

+1

2

n∑i=1

(((D2ei,XJ)Y, ei)− ((D2

ei,Y J)X, ei))

− 1

2

n∑i=1

(((D2X,eiJ)Y, ei)− ((D2

Y,eiJ)X, ei)).

compute-biscompute-bis (9.4.11)

Now, since (g, J, ω) is almost Kahler, we have

divJdivJ (9.4.12) δJ = −n∑i=1

(DeiJ)(ei) = 0,

and the Bianchi identity

bianchiJbianchiJ (9.4.13) ((DXJ)Y,Z) + ((DY J)Z,X) + ((DZJ)X,Y ) = 0.

From (9.4.12) we infer that the last line of the rhs of (9.4.11) vanishes andfrom (9.4.13) we infer that

∑ni=1(((D2

ei,XJ)Y, ei)− ((D2

ei,YJ)X, ei)) is equal

to −∑n

i=1((D2ei,eiJ)X,Y ) = ((D∗DJ)X,Y ); we thus get (9.4.4). Then,

(9.4.5) readily follows from (9.4.4) and (9.4.3) (note however that we usedthe convention |DJ |2 = |Dω|2 = −1

2

∑ni=1 tr(DeiJ DeiJ), in accordance

with our convention that |J |2 = |ω|2 = −12tr(J J), cf. Section 1.3).

Remark 9.4.1. In [9], (9.4.4) is deduced from the following generalWeitzenbock-Bochner formula

WbochnerWbochner (9.4.14) ∆ψ = D∗Dψ + 2R(ψ)− (r(Ψ·, ·)− r(·,Ψ·)),

which holds for any 2-form ψ and any riemannian metric g, when D standsfor the levi-Civita connection, ∆ for the riemannian Laplacian and Ψ forthe skew-symmetric endomorphism defined by g(ΦX,Y ) = ψ(X,Y ). In thealmost Kahler case, the Kahler form ω is harmonic and (9.4.4) is then adirect application of (9.4.14).

Note that, conversely, the proof of Proposition 9.4.1 can be readily ex-tended to give a proof of (9.4.14) as well.

remcalabiherm Remark 9.4.2. In the almost Kahler case, the hermitian scalar cur-vature is a natural substitute of the riemannian scalar curvature for manypurposes. In particular, the total hermitian scalar curvature S∇ :=

∫M s∇ vg

is constant on ACω0 and the critical points in ACω0 of the hermitian Calabifunctional

hermcalabihermcalabi (9.4.15) C∇(J) :=

∫M

(s∇)2 vg,

are again the J such that s∇ is the momentum of a hamiltonian Killingvector fields for the corresponding metric g. These facts are mentioned in[9] and will be proven in the next section as a consequence of the Mohsenformula.


9.5. The Mohsen formula

We now restrict our attention to the case when the almost complexstructure J belongs to the space ACω0 where ω0 is some fixed symplecticform on M , so that (g, J, ω0) is an almost Kahler with g = ω0(·, J ·). Wedenote by ∇J the corresponding Chern connection on (TM, J) and by ∇K∗Jthe induced connection on the corresponding anti-canonical bundle K∗J ; notethat K∗J is a complex vector subbundle of ΛmTM ⊗C dependent of J . Our

main goal is to compute the first variation of ∇K∗J when J moves in ACω0

in the following sense. For any J in ACω0 , any J in TJACω0 is the tangentvector to the curve

(9.5.1) Jt := exp(ta) J exp(−ta),

where a = 12JJ is a well-defined section of End(TM,ω0), cf. Section 9.2. For

simplicity, we shall denote exp(ta) by γt; then, γt is an isomorphism from(TMJ) to (TM, Jt) and induces an isomorphism, still denoted by γt, fromK∗J to K∗Jt . Since γt pointwise preserves ω0, it is actually an isomorphism

of hermitian line bundles so that the connection ∇Jt := γ−1∇Jt γt is ahermitian connection on K∗J . Choose any local section ξ of K∗J of norm 1with respect to the induced hermitian inner product. Then, ξt := γt ξ is aunit local section of K∗Jt and we then have

nablaJtnablaJt (9.5.2) ∇K∗Jt ξt = αt ⊗ Jtξt,

where αt is a real (local) 1-form, called the connection 1-form of ∇K∗Jt with

respect to ξt. The hermitian Ricci form ρ∇Jt and the hermitian scalar cur-

vature s∇Jt of Jt are then locally given by

rhohermrhoherm (9.5.3) ρ∇Jt

= −dα, s∇Jt

= −2Λdα.

By using the connection ∇Jt defined above, (9.5.2) can be rewritten as

(9.5.4) ∇Jtξ = αt ⊗ J ξ

so that αt now appears as the connection 1-form of the (moving) connection

∇Jt with respect to the (fixed) local frame ξ of the fixed complex line bundleK∗J . It follows that α := d

dt |t=0αt is independent of ξ and is then the local

restriction of a globally defined (real) 1-form on M , still denoted by α, called

the first variation along J of the Chern connection of the anti-canonicalbundle. The next proposition and its proof are taken from [150]

propmohsen Proposition 9.5.1 (Mohsen formula). For any J in ACω0, the first vari-

ation along J of the Chern connection of the anti-canonical bundle is givenby

mohsenmohsen (9.5.5) α =1

2(δJ)[,

where the codifferential δ is relative to the metric g = ω0(·, J ·) and the vector

field δJ is then defined by

deltaJdotdeltaJdot (9.5.6) δJ = −n∑i=1

(DeiJ)(ei),

9.5. THE MOHSEN FORMULA 237

for any auxiliary orthonormal frame e1, . . . , en, where D stands for theLevi-Civita connection of g.

Proof. A hermitian orthonormal (local) frame for (TM, J) is a systemof m local vector fields Z1, . . . , Zm such that h(Zj , Zk) := 1

2(g(Zj , Zk) −iω0(Zj , Zk)) = δjk, i.e. g(Zj , Zk) = 2δjk and ω0(Zj , Zk) = 0, for any j, k =1, . . . ,m (recall that the (real) dimension of M is n = 2m). Fix such a localframe; the corresponding αt has then the following expression:

αt(X) = −im∑j=1

h(∇JtXZj , Zj)

= −im∑j=1

(h(∇Jt,(0,1)X Zj , Zj)− h(Zj , ∇Jt,(0,1)

X Zj)

= −im∑j=1

(h(γt∂(TM,Jt)X γ−1

t Zj , Zj)− h(Zj , γt∂(TM,Jt)X γ−1

t Zj)

= −ω0(γt∂(TM,J)X γ−1

t Zj , Zj).

(9.5.7)

By using (9.3.2) and the identity Jtγt = γtJ , we infer

αt =1

4

m∑j=1

(ω0(γ−1t (LJtγtZjJt)(X), Zj) + ω0(γ−1

t Jt(LγtZjJt)(X), Zj)

=1

4

m∑j=1

(ω0(γ−1t (LγtJZjJt)(X), Zj) + ω0(Jγ−1

t (LγtZjJt)(X), Zj)

= −1

2

n∑k=1

g(γ−1t (LγtekJt)X, ek)

(9.5.8)

where ek, k = 1, . . . , n = 2m stands for the g-orthonormal frame1√2Z1, . . . , Zm, JZ1, . . . , JZm. Recall that α is obtained by taking the

derivative of the above expression with respect to t at t = 0. For that itis convenient to previously transform this expression by replacing the Liederivative by the Levi-Civita connection D of the (fixed) metric g = ω0(·, J ·)


by using (1.23.4), so that:

αt = −1

2

n∑k=1

g(γ−1t (DγtekJt)(X), ek)

− 1

2

n∑k=1

g(γ−1t ([Jt, D(γtek)](X), ek)

= −1

2

n∑k=1

g((DekJt)(X), ek)

− 1

2

n∑k=1

g(Jγ−1t DX(γtek), ek)

− 1

2

n∑k=1

g(γ−1t DJtX(γtek), ek).

(9.5.9)

We thus get

α = −1

2

n∑k=1

g((Dek J)(X), ek)

+1

2

n∑k=1

g(J(DXa)ek, ek)

+1

2

n∑k=1

(g((DJXa)ek, ek)− g(DaXek, ek)).

(9.5.10)

Since J is symmetric, the first term of the rhs in the above expression isequal to 1

2g(δJ ,X), whereas the other terms are 0: this is because ek is ofnorm 1, so that g(DaXek, ek)) = 0, and a and Ja are trace-free, as they bothanticommute to J . We thus get (9.5.5).

As a direct consequence of the Mohsen formula, we get

propvarherm Proposition 9.5.2. For any almost complex structure J in ACω0, thederivative of the hermitian Ricci form ρ∇ and of the hermitian scalar cur-vature s∇ along any J in TJACω0 is given by

varhermvarherm (9.5.11) ρ∇ = −1

2dδ(J)[, s∇ = −Λd(δJ)[ = −δJ(δJ)[.

Proof. From (9.5.3), we get

varhermalphavarhermalpha (9.5.12) ρ∇ = −dα, s∇ = −2Λdα.

The given expression of ρ∇ and the first given expression of s∇ are then adirect consequence of the Mohsen formula (9.5.5). The second given expres-sion of s∇ in (9.5.11) follows from the Kahler identity [Λ, d] = −δc, which,we recall, holds in the almost Kahler case, cf. Section 1.14.

Remark 9.5.1. If J belongs to Cω0 then (g, J, ω0) is Kahler. From

(9.5.6), we infer JδJ = δ(JJ) and the above formula for s∇ can then beread as

varskaehlervarskaehler (9.5.13) s∇ = −δ(δ(JJ))[ = δδg,

9.6. HAMILTONIAN ACTION OF Ham(M,ω0) ON ACω0 239

where g denotes the derivative of gt = ω0(·, Jt·), hence given by g(X,Y ) =

−g(JJX, Y ). Since the hermitian scalar curvature of a Kahler manifoldcoincides with the riemannian scalar curvature this formula should appear asa special case of the general formula (5.4.4) of the variation of the riemannian

scalar curvature, at least when J belongs to the subspace TJCω0 and this

is actually the case. Indeed, if J belongs to TJCω0 , even to TJACω0 , thecorresponding variation g of the metric is trace free and J-antilinear, henceorthogonal to the (riemannian) Ricci tensor r, which is J-invariant as (g, J)is Kahler. The rhs of (5.4.4) then reduces to the rhs of (9.5.13).

From (9.5.11), we infer that the Ricci forms, ρ∇ and ρ∇0 , of any twoelements, J and J0, of ACω0 are related by

(9.5.14) ρ∇ − ρ∇0 = −1

2d(

∫ 1

0(δgt Jt)

[gt dt),

for any curve Jt which joins J0 to J in ACω0 . In particular, ρ∇ and ρ∇0

belong to the same de Rham class in H2dR(M,R). This fits with the fact that

all almost-complex structures in ACω0 belong to a same deformation class— as they are sections of a bundle with contractible fibers, cf. Section 9.2— hence have the same Chern class (called the Chern class of the symplecticmanifold (M,ω0)).

We also infer that the total hermitian scalar curvature S∇ =∫M s∇ vg

is constant on ACω0 . Indeed, by (9.5.11), we get,

(9.5.15) S∇ − S∇0 = −∫M

Λd (

∫ 1

0(δgt Jt)

[gt dt) vg = 0

Morever, we have the following almost Kahler generalization of Theorem3.2.1, already mentioned in [9]:

propdercalabi Proposition 9.5.3. An element J of ACω0 is a critical point for thehermitian Calabi functional (9.4.15) defined on ACω0 if and only if the hamil-tonian vector field K∇ := gradω0

s∇ = Jgradgs∇ is Killing with respect to

the metric g = ω0(·, J ·).

Proof. From (9.5.11) we get

(9.5.16) dC∇J (J) = 2

∫Ms∇δc(δJ)[gt vg = 2 〈Dgradω0

s∇, J〉,

for any J in TJACω0 . Recall that the latter is the space of g-symmetric,J-anti-invariant endomorphisms of TM . It follows that J is a critical pointif and only if the symmetric, J-anti-invariant part of Dgradω0

s∇ vanishesidentically. By (9.2.2), for any hamiltonian vector field Z, the symmetricpart of DZ is equal to −2JLZJ , hence already J-invariant. We infer that Jis critical if and only if Dgradω0

s∇ is skew-symmetric, if and only if gradω0s∇

is a (hamiltonian) Killing vector field.

9.6. Hamiltonian action of Ham(M,ω0) on ACω0sshamiltonian

The following fact has been first observed by A. Fujiki [84] in the Kahlercase — cf. Remark 9.2.1 — and by S. Donaldson [73] in the general almost-kahler case.


thmdonaldson Theorem 9.6.1. For any compact symplectic manifold (M,ω0), the ac-tion of Ham0(M,ω0) on the space ACω0 of all ω0-compatible almost-structureis hamiltonian, with momentum µ given by

momentumgeneralmomentumgeneral (9.6.1) µZ(J) = −∫Mfs∇g vg = −

∫Mfs∇g

ωm0m!

,

for any Z = gradω0f in ham, where s∇g denotes the hermitian scalar curva-

ture of the almost Kahler structure (g = ω0(·, J ·), J, ω0).

Proof. Since the action of Sp(M,ω0) on ACω0 preserves κ and ACω0

is contractible, we know that for any Z in sp, the induced vector field Z

admits a momentum, µZ , defined, up to an additive constant, by

(9.6.2) ιZκ = −dµZ .

By (9.2.2) ZJ can be written as

(9.6.3) ZJ = −LZJ = −J (DZ + (DZ)∗),

for any Z in sp and any J in ACω0 . From (9.2.11) we then infer

kappamohsenbiskappamohsenbis (9.6.4) κ(ZJ , J) = 〈DZ, J〉 = 〈Z, δJ〉.

where 〈·, ·〉 denotes the global inner product with respect to g (recall that J

is symmetric with respect to g, so that 〈J , DZ〉 = 〈J , (DZ)∗〉). This holdsfor any Z in sp. In the case when Z belongs to the Lie subalgebra ham,then Z = gradω0

fZ = JgradgfZ , where fZ denotes the momentum of Z

with respect to ω0 normalized by∫M fZωm0 = 0 (note that fZ only depends

on Z and ω0). By using the expression of the variation of hermitian scalarcurvature s∇ given by (9.5.11), (9.6.4) can then be rewritten as:

(9.6.5) κ(ZJ , J) = 〈dcfZ , (δJ)[〉 = 〈fZ , δc(δJ)[〉 =

∫MfZ s∇(J)

ωm0m!

.

Since fZ only depends on Z and ω0, we thus get

kappadmukappadmu (9.6.6) κ(ZJ , J) = −dµZJ (J),

by setting:

muZmuZ (9.6.7) µZ(J) := −∫MfZs∇

ωm0m!

.

It remains to check that µZ defined by (9.6.7) determines a momentum mapfor the action of Ham(M,ω0), i.e. an equivariant map from M to the dualof the Lie algebra ham, meaning that

equivarianceequivariance (9.6.8) µΦ·J(Φ · Z) = µZ(J),

for any Z in ham, any J in ACω0 and any Φ in Ham(M,ω0). This is becauseΦ transports (g, J, ω0) to (Φ · g,Φ · J,Φ · ω0 = ω0), hence the anti-canonicalbundle of J with its Chern connection ∇K∗J to the anti-canonical bundle ofΦ ·J with its Chern connection ∇K∗Φ·J = Φ ·∇K∗J . Similarly fΦ·Z = Φ ·fZ , sothat the equivariance condition (9.6.8) immediately follows from (9.6.7).

9.6. HAMILTONIAN ACTION OF Ham(M,ω0) ON ACω0 241

In view of Section 9.8 below, Theorem 9.6.1 can be given a more generalformulation by fixing a connected, compact subgroup, G, of Ham0(M,ω0)and by replacing ACω0 by the space, ACGω0

, of G-invariant elements of ACω0 .Denote by HamG(M,ω0) the (identity component of the) normalizer of G inHam0(M,ω0), and by hamG its Lie algebra. Then, hamG is the Lie algebraof elements X of ham such that [X,Z] belongs to g whenever Z belongs to g.The quotient HamG(M,ω0)/G acts on ACGω0

and preserves the Fujiki Kahler

structure. Denote by PGω0the space of momenta — including constants —

of elements of the Lie algebra, g, of G in hamG, and by ΠGω0

the orthogo-

nal projection from C∞(M) to ΠGω0

with respect to the symplectic volume

form ωm0 . For any J in ACGω0, we then define the reduced hermitian scalar

curvature s∇,Gg by

reduced-scalhermreduced-scalherm (9.6.9) s∇g = s∇,Gg + ΠGω0

(s∇J ).

we then have

cordonaldson Corollary 9.6.1. The action of HamG0 (M,ω0)/G on ACGω0

is hamil-tonian; its momentum µG is determined by

momentummodifiedmomentummodified (9.6.10) µZG(J) = −∫Mfs∇,Gg

ωm0m!

,

for any Z = gradω0f in hamG/g and for any J in ACGω0

, where g = ω0(·.J ·)stands for the associated riemannian metric.

Proof. This is a direct consequence of Theorem 9.6.1: the dual of theLie algebra hamG/g is the subspace of elements of (hamG)∗ which vanish ong. Then, (9.6.10) readily follows from (9.6.1).

rem-mohsen Remark 9.6.1. By Mohsen formula (9.5.5), (9.6.4) can be rewritten as

kappamohsenkappamohsen (9.6.11) κ(ZJ , J) = 2

∫Mα(Z) vg.

For any chosen J0 inACω0 and any Z in sp, the momentum, µZ,J0 , of Z whichvanishes at J0 is obtained at J by integrating (9.6.11) along any curve inACω0 which joins J0 to J . By chosing the “geodesic” Jt = exp(ta)J0exp(−ta),

where a is defined as in Lemma 9.2.2, we infer that µZ,J0 is given by

(9.6.12) µZ,J0(J) = −2

∫ 1

0(

∫Mαt(Z)vg)dt = −2

∫MψJ0,J(Z)vg,

where ψJ0,J is the real 1-form on M defined by

(9.6.13) exp(−a)∇K∗J exp(a)−∇K∗J0 = ψJ0,J ⊗ J0.

In the case that Z = gradω0fZ belongs to ham, we infer as before that

(9.6.14) µZ,J0(J) = −〈s∇, fZ〉+ 〈s∇0, fZ〉

where s∇0

stands for the hermitian scalar curvature of (g0 = ω0(·, J0·), J0, ω0).

The set of momenta µZ,J0 does not form an equivariant momentum map byitself, as the chosen J0 is not invariant, but in this case we nevertheless get

an equivariant momentum map, equal to µZ , by simply substracting the

constant 〈s∇0, fZ〉 from each individual momentum µZ,J0 . It is doubtful


however that we can arrange to obtain an equivariant momentum map forthe action of whole group Sp(M,ω0) on ACω0 .

remvestislav Remark 9.6.2. It has been pointed out to me by V. Apostolov, the in-terpretation of the hermitian scalar curvature as a momentum map shedsa new light on Proposition 9.5.3 because of the following general fact. Let(M, g, J, ω) be a Kahler manilfold acted on in a hamiltonian way by a Liegroup G, whose Lie algebra is denoted by g. Let µ : M → g∗ be the cor-responding momentum map. By definition µ is G-equivariant and satisfies〈dµx(X), a〉 = −ωx(a, X) = gx(JX, a), for any a in g, where a stands for theinduced vector field on M and 〈·, ·〉 her denotes the natural pairing betweeng and its dual g∗. Suppose that g admits a G-equivariant inner product,which realizes g as a subspace of the dual g∗, and suppose moreover thatthe image of µ belongs to g. We then define the square norm of µ by|µ|2(x) := 〈µ(x), µ(x)〉, and the differential of |µ|2 at x is then given by

(9.6.15) d|µ|2(X) = 2〈dµx(X), µ(x)〉 = gx(JX, µ(x))

for any X in TxM . We conclude that x is a critical point of the function|µ|2 if and only if the condition

criticalcritical (9.6.16) µ(x)(x) = 0

is satisfies (cf. e.g. [117]). The above argument holds in an infinite-dimensional setting, in particular in the case that M is ACω0 , equippedwith its natural Kahler structure (g,J, κ), and G here stands for the “Liegroup” Ham(M,ω0). According to Theorem 9.6.1 this action is hamilton-ian, with momentum map µ(J) = s∇ viewed as an element of ham∗ via theduality 〈s∇, f〉 =

∫M s∇f µg for any f in ham when the latter is identified

with the space of all (smooth) real functions f such that∫M fωm0 = 0. Now,

the pairing (f1, f2) :=∫M f1f2 vg induces an invariant inner product on ham.

By (tacitly) replacing s∇ by s∇− S∇

Vω0— this is innocuous as S∇ is constant

on ACω0 — µ(J) can then be considered as an element of the Lie algebraham and the hermitian Calabi functional is then identified with the squarenorm |µ|2 as defined above. The critical points in ACω0 are then character-

ized by (9.6.16). In the current case, we have that µ(J)(J) = −Lgradω0s∇J ,

so that condition (9.6.16) simply expresses the fact that gradω0s∇ preserves

J . Since gradω0s∇ already preserves ω0, this amounts to saying that it is a

Killing vector field with respect to g. We thus recover Proposition 9.5.3.

9.7. A symplectic Futaki invariantsssymplecticfutaki

Let G be a compact Lie group acting in an effective, hamiltonian wayon a connected, compact symplectic manifold (M,ω0). As before, denote byACGω0

the space of G-invariant ω0-compatible almost-complex structures onM .

For any a in the Lie algebra g of G, denote by Za the induced vectorfield on M and by fa the momentum of Za relatively to ω0, so that Za =gradω0

fa. We assume that fa is normalized by∫M fa ωm0 = 0; it is the

uniquely defined.

9.8. THE MABUCHI K-ENERGY AS A KAHLER POTENTIAL 243

For any J in ACω0 , denote, as usual, by g = ω0(·, J ·) the correspondingriemannian metric, by ∇ the corresponding Chern connection and by s∇ thecorresponding hermitian scalar curvature. We then have

propsymplecticfutaki Proposition 9.7.1. For any a in g, the integral∫M fa s∇ ω0

m! is inde-

pendent of J on any connected component of ACGω0.

Proof. By (9.6.1), we have

(9.7.1)

∫Mfa s∇

ω0

m!= −µZa(J),

where Za denotes the induced vector field on ACGω0, defined by ZaJ = −LZaJ ,

and µZa

denotes the momentum of Za relatively to the canonical symplecticform κ of ACω0 . For any J in ACω0 and any J in TJACω0 , the variation of∫M fa s∇ ω0

m! along J is then equal to −dµZa(J) = κ(ZaJ , J), cf. (9.6.6). If

J belongs to ACGω0, we have that Zaj = 0, for any a in g. It follows that

for any smooth curve Jt in ACGω0, with Chern connection ∇t, µZ

a(Jt), hence∫

M fa s∇tωm0m! , is constant in t.

For any chosen connected component, VGω0say, of ACGω0

we then associatethe linear map FVGω0

: g→ R defined by

(9.7.2) FVGω0(a) =

∫Mfa s∇

ωm0m!

,

for any a in g and any J in VGω0, of hermitian scalar curvature s∇. By

Proposition 9.7.1, the rhs is independent of the choice of J in UGω0, hence

determines a symplectic Futaki invariant relative to VGω0.

In particular, a necessay condition that the chosen connected componentVGω0

contains an almost-complex structure J whose hermitian scalar curva-ture is constant is that the symplectic Futaki invariant FVGω0

be identicallyzero on g.

9.8. The Mabuchi K-energy as a Kahler potentialssmabuchipotential

In this section, we show that, under specific conditions, the MabuchiK-energy can be defined on the space Jω0 and interpreted as a Kahlerpotential. We then infer a new, simple derivation of the scalar curvature asa momentum map for the action of Ham0(M,ω0) on Jω0 .

proppotential Proposition 9.8.1. If the Futaki character FΩ is trivial, the MabuchiK-energy can be viewed as a H(M,ω0)-invariant function defined on Jω0. Ifmoreover H1(M,R) = 0, then the Mabuchi K-energy is a Kahler potentialof the natural Kahler structure of Jω0 defined by (9.2.9)-(9.2.10)-(9.2.11).

Proof. Recall that the Mabuchi K-energy E has been defined in Section4.10 by σ = −dEΩ, where σ is the (closed) 1-form onMΩ defined by σ(f) =∫M fsgvg. The 1-form σ is Aut0(M,J0)-invariant and, by (4.12.2), is basic

with respect to the action of Aut0(M,J0) if and only if the Futaki characterFΩ is trivial, cf. Remark 4.12.1. It follows that if FΩ ≡ 0, σ can be viewed as


a 1-form defined on the quotient MΩ/H(M,J0), hence on Jω0/Sp0(M,ω0)via the correspondence (9.1.1). The 1-form σ defined by

(9.8.1) σ = p∗σ,

can then be viewed as a basic 1-form on Jω0 with respect to the actionof Sp0(M,ω0). Similarly, if FΩ ≡ 0 the Mabuchi K-energy EΩ is con-stant on each orbit of H(M,J0)-invariant — hence defined on the quotientMΩ/H(M,J0) — and we then define

(9.8.2) EJ = p∗EΩ,

as a Sp0(M,ω0)-invariant function on Jω0 , with

(9.8.3) σ = −dEJ .We then set

(9.8.4) τ := −Jσ = dJEJ .The 1-form τ defined that way on Jω0 is still Sp0(M,ω0)-invariant, althoughno longer basic. It remains to check that

dtaudtau (9.8.5) dτ = κ|Jω0

— where, we recall, κ is the Kahler form defined by (9.2.11) — whenH1(M,R) = 0, i.e. when sp = ham. To compute dτ , it is convenientto use the vector fields induced by the action of Sp0(M,ω0) on Jω0 , namely

those vector fields Z given by (9.2.1)-(9.2.2) for any Z in sp. By (9.1.10)

these vector fields Z, together with the vector fields JZ, generate the tangentbundle of Jω0 . By (9.1.5), we have that

(9.8.6) τ(JZ) = −σ(Z) = 0,

for any Z = gradω0f in ham, whereas τ(Z) is given by

cruxcrux (9.8.7) τJ(Z) = −∫Mfsg vg,

for any J in Jω0 , where g = ω0(·, J ·) denotes the riemannian metric deter-

mined by the pair (J, ω0), sg the scalar curvature of g and vg =ωm0m! the

volume form of g (we again emphasize that vg is actually independent of Jin Jω0). Indeed, if J = Φ · J0 and if g = Φ∗g denotes the metric defined by(9.1.7), i.e. the metric determined by the Kahler pair (J0, ω = Φ∗ω0), byProposition 9.1.1 we have that

τJ(Z) = σJ(JZ) = σ(dp(−JLZJ))

= −σ(Φ∗f)

= −∫M

(Φ∗f) sgωm

m!

= −∫Mf sg

ωm0m!

.

We now assume that H1dR(M,R) = 0, i.e. that sp = ham. In order to

check (9.8.5), it is then sufficient to compute dτ(Z1, Z2) and dτ(Z1,JZ2),for any two hamiltonian vector fields Z1 = gradω0

f1 and Z2 = gradω0f2.

Since τ is Sp0(M,ω0)-invariant, we have that dτ(Z1, Z2) = τ([Z1, Z2]) =

9.8. THE MABUCHI K-ENERGY AS A KAHLER POTENTIAL 245

−τ( [Z1, Z2]), where the bracket [Z1, Z2] is again a hamiltonian vector fieldwhose momentum is the Poisson bracket f1, f2 = Λ(df1 ∧ df2) = δJ(f1df2)(the first expression is the same as (1.2.10 ; the second one follows by usingthe identity (1.14.1)). By (9.8.7), we then have

dτJ(Z1, Z2) =

∫Mf1, f2sgvg

= −〈LKf1, f2〉= 2〈δδD−dJf1, f2〉= 2〈D−dJf1, D

−df2〉

(9.8.8)

by setting K = gradω0sg = Jgradgsg and by using (1.23.15) (here, the inner

products and the operator D− are relative to the Kahler structure (g, J, ω0)).On the other hand, by using (1.23.7) we get

κ(Z1, Z2) =1

2〈LJZ1J,LZ2J〉

= 2〈D−dJf1, D−df2〉

(9.8.9)

which is then equal to dτJ(Z1, Z2). Now, each Z preserves J, so that

[Z1,JZ2] = J[Z1, Z2]; we infer

(9.8.10) dτ(Z1,JZ2) = −JZ2 · τ(Z1) = JZ2 ·∫Mf1sgvg.

In the integral, f1 and vg =ωm0m! are independent of J , whereas the variation

g of the metric g = ω0(·, J ·) along JZ is given by

(9.8.11) g = ω(·, J ·) = −g(JJ ·, ·) = −g(LZ2J ·, ·) = 2D−df2.

Now g is J-invariant, hence traceless and orthogonal to the Ricci tensor ofthe Kahler structure; from (5.4.4), we then infer sg = δδg = 2δδD−df2 andwe thus get

(9.8.12) dτ(Z1,JZ2) = 2〈f1, δδD−df2〉 = 2〈D−df1, D

−df2〉

whereas, by using (1.23.7) again we get(9.8.13)

κ(Z1,JZ2) =1

2〈LJZ1J,LJZ2J〉 = 2〈D−df1, D

−df2〉 = dτ(Z1,JZ2).

By Theorem 9.6.1 we know that the hermitian scalar curvature can beinterpreted as a momentum for the natural action of the group Ham0(M,ω0)on ACω0 , hence a fortiori for the action of Ham0(M,ω0) on Jω0 (on whichthe hermitian curvature coincides with the riemannian one). By Proposition9.8.1, we see that the Kahler structure of Jω0 is itself encoded by the scalarcurvature, via the Mabuchi K-energy. This observation in turn allows foran alternative, direct interpretation of the scalar curvature as a momentummap for the action of Ham0(M,ω0) on Jω0 :

propmoment Proposition 9.8.2. With the above hypotheses, namely FΩ ≡ 0 andH1(M,R) = 0, the action of Sp0(M,ω0) — which then coincides with


Ham0(M,ω0) — on Jω0 is hamiltonian with respect to the natural symplecticform κ|Jω0

, with momentum µ given by

(9.8.14) µZ(J) = τ(Z(J)) = −∫MfZsg

ωm0m!

,

for any Z = gradω0fZ in ham, where sg is the scalar curvature of the metric

g = ω0(·, J ·) dtermined by J .

Proof. By Proposition 9.8.1, κ|Jω0= dτ = ddJE , where E and τ = dJE

are Sp0(M,ω0)-invariant. For any Z in ham we then have

(9.8.15) 0 = LZτ = d(τ(Z)) + ιZdτ.

From this identity — which is in fact quite general for Kahler manifoldswhose Kahler form admits an invariant primitive — we infer that τ(Z),

whose value is given by (9.8.7), is a momentum of Z on Jω0 .

9.9. The relative casessrelative

In this section, we fix a connected compact subgroup G of the re-duced automorphism group Hred(M,J0) and we assume that the backgroundKahler structure (g0, J0, ω0) is G-invariant. In particular, G is a subgroupof Ham(M,ω0), hence of Sp0(M,ω0).

Recall — cf. Section 4.14 — that the identity component of the normal-izer, resp. the centralizer, of G in H(M,J0) as been denoted by HG(M,J0),resp. HG(M,J0). Similarly, we denote by DiffG(M), resp. DiffG(M), theidentity component of the normalizer, resp. centralizer, of G in Diff0(M),and by SpG(M,ω0), resp. SpG(M,ω0), the identity component of the nor-malizer, resp. centralizer, of G in Sp(M,ω0). The Lie algebras are denotedaccordingly: spG, spG, etc. We then have

lemmaidaction-bis Lemma 9.9.1. Let G be any connected, compact subgroup of Hred(M,J0)and assume that ω0 is G-invariant, so that G is a subgroup of Ham(M,ω0) ⊂Sp0(M,ω0). Then

(9.9.1) SpG(M,ω0)/G = SpG(M,ω0)/Z(G),

by setting Z(G) = G ∩ SpG(M,ω0).

Proof. Since SpG(M,ω0)/Z(G) is a subgroup of SpG(M,ω0), and bothgroups are connected, it is sufficient to check that

(9.9.2) spG = spG + g.

The argument is quite similar to the one in the proof of Lemma 4.14.2 andwe refer to it for more detail. Let Z = ZH + gradω0

fZ be any element ofsp, where ZH is the dual of a harmonic 1-form with rspect to g0. Then, Zbelongs to spG if and only if [X,Z] = LXZ = LXZH + gradω0

(LXfZ) =

gradω0(LXfZ) belongs to g for any X in g. This holds if and only if LXfZ

belongs to the space, PGg0, of Killing potentials of elements of g with respect

to g0. This implies that γ∗fZ − fZ belongs to PGg0for any γ in G. By

integrating over G, we get fZ = fF + f , where fZ is G-invariant and fbelongs to PGg0

. It follows that Z = (ZH + gradω0fZ) + gradω0

f belongs to

9.9. THE RELATIVE CASE 247

spG + g. We then get spG ⊂ spG + g; the opposite inclusion spG + g ⊂ spGis obvious.

We define J Gω0as the space of those elements J in J Gω0

which satisfy thefollowing two properties:

(1) J is G-invariant.

(2) J can be connected in Jω0 by a curve Jt := Φt · J0, with J1 = J ,where Φt is a smooth curve in DiffG(M), with Φ0 = Id (notice thatΦ1 is then weel-defined up to right composition by an element ofHG(M,J0)).

Then, J Gω0is acted on by SpG(M,ω0) as well as by SpG(M,ω0). By

Lemma 9.9.1, these two actions coincide.

propidG-relative Proposition 9.9.1. (i) There exists an SpG(M,ω0)-invariant map, pG,from J Gω0

to the quotient MGΩ/HG(M,J0) and an HG(M,J0)-invariant map,

qG, from MGΩ to the quotient J Gω0

/SpG(M,ω0), which are inverse to eachother, hence induce a natural identification

identificationbisidentificationbis (9.9.3) J Gω0/SpG(M,ω0)

p

qMG

Ω/AutG(M,J0)

The maps pG and qG are defined as follows. For any J in J Gω0there

exists Φ in Diff0(M), uniquely defined up to left composition by an elementof H(M,J0), such that J = Φ · J0 and we then set

pGpG (9.9.4) p(J) = Φ∗ω0 = Φ−1 · ω0 mod Aut0(M,J0).

Similarly, if ω belongs to MΩ, by Moser lemma there exists Φ in Diff0(M),uniquely defined up to right composition by an element of Sp0(M,ω0), suchthat ω = Φ∗ω0 and we then set

qGqG (9.9.5) q(ω) = Φ · J0 mod Sp0(M,ω0).

(ii) For any J in Jω0 the tangent space TJJω0 is described by

TJGTJG (9.9.6) TJJω0 = A = −LZJ | Z ∈ sp + Jsp = sp⊕ J ham.

Via this identification, the differential of p at J = Φ · J0 is given by

dpGdpG (9.9.7) dp(−LZJ) = Φ∗(LZω0) = ddc(Φ∗f)

for each Z = X − JY in sp ⊕ J ham, where f denotes the normalized mo-mentum of the hamiltonian vector field Y , hence is determined by

ZGZG (9.9.8) LZω0 = ddJf

and∫M fωm0 = 0 (here dc stands for the operator J0dJ

−10 , also denoted by

dJ0, whereas dJ := JdJ−1).(iii) Jω0 is a complex submanifold, hence a Kahler submanifold of Cω0.

Recall that the relative Futaki character FGΩ relative to G and a fixedKahler class Ω has been defined in Section 4.14. Propositions 9.8.1 and 9.8.2have then the following counterparts:


proppotentialbis Proposition 9.9.2. If the relative Futaki character FGΩ is identicallyzero, the relative K-energy EGΩ can be viewed as a SpG(M,ω0)-invariantfunction defined on J Gω0

. If moreover H1(M,R) = 0, then the relative K-energy is a HamG(M,ω0)-invariant Kahler potential of the natural Kahlerstructure of J Gω0

.

Proof. The proof is quite similar to the proof of Proposition 9.8.1; inparticular, the condition FGΩ ≡ 0 is equivalent to the fact that σ is basic withrespect to the action of HG(M,J), i.e. that the relative Mabuchi K-energyEGΩ is constant along the orbits of HG(M,J). If this condition is satisfied,EGΩ can then be viewed as a SpG(M,ω0)-invariant function, call it EGJ , on

J Gω0. Similarly, σG can be viewed as a basic 1-form defined on J Gω0

, then

denoted by σGJ , with σGJ = −dEGJ . We then set τG = −J σGJ = dJEGJ . itremains to prove that

(9.9.9) κ|JGω0= dτG = ddJEGJ ,

when we assume that H1(M,R) = 0, i.e. that SpG(M,ω0) = HamG(M,ω0).This is checked as in the proof of Proposition 9.8.1, provided we take

into account that for any X = gradω0hX in hamG, the momentum hX is

G-invariant (cf. the observation in the proof of Proposition 4.14.2); thisimplies that in expressions like LKhX , the vector field K = gradgsg can be

replaced by K := gradgsGg ; the rest of the computation is identical.

propmomentbis Proposition 9.9.3. For any maximal compact subgroup G of Hred(M,J)and with the above hypotheses, namely FGΩ ≡ 0 and H1(M,R) = 0, theaction of SpG(M,ω0) — which then coincides with HamG(M,ω0) — on Jω0

is hamiltonian with respect to the natural symplectic form κ|JGω0, with mo-

mentum µG given by

(9.9.10) µZG(J) = τ(Z(J)) = −∫MfZsg

ωm0m!

,

for any Z = gradω0fZ in hamG, where sg denotes the reduced scalar curva-

ture relative to G of the metric g = ω0(·, J ·) determined by J .

Proof. Same argument as for Proposition 9.8.2.

CHAPTER 10

Extremal Kahler metrics on Hirzebruch-like ruledsurfaces

shfkg3

In the same paper [45] where E. Calabi introduced extremal Kahlermetrics, he also constructed extremal Kahler metrics — of non constantscalar curvature — on a family a (compact) complex manifolds, which, as ithad been known for a long time by using Theorems 3.6.2 and 3.6.1, admitno Kahler metric of constant scalar curvature, a fortiori no Kahler-Einsteinmetric. These manifolds are the Hirzebruch surfaces, already consideredin Section 6.5 and, more generally, the (complex) ruled manifolds of theform P(1⊕L) for any non trivial holomorphic line bundle L over a complexprojective space Pm. Here, P(1⊕L) stands for the completion of L obtainedby adding an infinity section or, equivalently, the projective line bundleassociated to the rank 2 holomorphic vector bundle 1 ⊕ L, where 1 standsfor the product complex line bundle (more detail in Section 10.2).

In this chapter, we recall Calabi’s construction for Hirzebruch surfaces— for simplicity, we omit the case when m > 1, which is quite similar— and, following [194], [195], we show how Calabi’s method extends toHirzebruch-like ruled surfaces over Riemann surfaces of all genera, with twomain differences however:

(1) Hirzebruch surfaces admit a large — non reductive — group ofautomorphisms, whose maximal compact subgroups — isomorphicto U(2)/µ` if L is of (negative) degree −` — acts with genericorbits of codimension 1. According to Theorem 3.5.1, the issueof constructing extremal metrics can then be reduced a priori tosolving an ordinary differential equation. For Hirzebruch-like ruledsurfaces of genus greater than 0, the maximal compact subgroupsof the group of automorphisms are isomorphic to the circle S1 andCalabi’s method only applies to a subclass of Kahler metrics, which,following [4], [5] we name admissible, cf. Section 10.3.

(2) Unlike the genus 0 case — and also the genus 1 case, as observedby A. Hwang in [107] and D. Guan in [96] — Calabi’s method ingenera greater than 1 provides (admissible) extremal Kahler met-rics on a part only of the Kahler cone, a fact first discovered by C.Tønnesen-Friedman in [194], [195], cf, Theorem 10.4.1.

Following [6], the existence/non-existence issue of — non necesarily ad-missible — extremal Kahler on Hirzebruch-like ruled surfaces in the part ofthe Kahler cone which cannot be solved by using Calabi’s approach is even-tually settled — cf. Theorem 10.9.1 — by using Chen-Tian Theorem 4.15.2.This requires the explicit computation of the Futaki invariant relative to thenatural C∗-action on each Hirzebruch-like rules surface and of the (relative)

249

250 10. HIRZEBRUCH-LIKE RULED SURFACES

K-energy on the space of all (admissible) Kahler metrics in a given Kahlerclass, which happens to be quite similar to Donaldson’s computation in [77],cf. Sections 10.7, 10.8 and 10.9.

This in turn happens to be a special case of similar Calabi-like con-structions and existence/non-existence theorems which appear in [5], [6] inthe broader context of admissible projective bundles and admissible Kahlermetrics.

10.1. Complex ruled surfacesssruled

A (complex, geometrically) ruled surface of genus g is a complex surfaceof the form P(E), where E is a rank 2 complex holomorphic vector bundleover a (connected) compact Riemann surface Σ of genus g and P(E) thendenotes the corresponding projectivized bundle, i.e. the holomorphic bundlewhose fiber at any point y of Σ is the complex projective line P(Ey), whereEy deontes the fibre of E at y.

We denote by OE(−1) the relative tautological line bundle over P(E),whose fiber at any point x of P(E), view as a complex line in Ey for some yin Σ, is x itself, and by OE(1) the dual of OE(−1).

Notice that P(E) remains unchanged if E is replaced by E ⊗ L, for anyholomorphic line bundle L, while OE(−1) and OE(1) do depend on E.

In the sequel of this chapter, P(E) will be denoted by M , whereas thenatural projection from M to Σ will be denoted by π.

We denote by γE the first Chern class of OE(1) in H2(M,Z) and by γΣ

the (positive) generator of H2(Σ,Z) = Z (here, positive refers to the orien-tation of Σ determined by its complex structure). It is a simple consequenceof the Leray-Hirsch theorem — cf. e.g. [36] — that H∗(M,Z) is a freemodule over H∗(Σ,Z) generated by 1 and γE . In particular, H2(M,Z) hasno torsion, hence can be considered as a lattice in H2(M,R), and the pairγE , π

∗γΣ is then a R-basis of H2(M,R).Moreover, both γE and π∗γΣ are of type (1, 1): this is because γE is

the Chern class of a holomorphic line bundle over M , whereas H2(Σ,R)coincides with H1,1(Σ,R) as Σ is of complex dimension 1. We then haveH2(M,R) = H1,1(M,R).

The ring structure of H∗(M,Z) is determined by the ring structure ofH∗(Σ,Z) and the relation

chern-define-ruledchern-define-ruled (10.1.1) γE · γE + γE · π∗c1(E) = 0,

where c1(E) = deg(E) γΣ is the first Chern class of E, cf. [36]).The table for the cup product — or intersection form — in H2(M,Z),

with values in H4(M,Z) = Z is then

tabletable (10.1.2) γE · γE = −deg(E), γE · π∗γΣ = +1, π∗γΣ · π∗γΣ = 0;

the third identity is clear, as H4(Σ,Z) = 0; the second one follows fromγΣ being Poincare dual of a point in H0(Σ,Z) — so that π∗γΣ is Poincaredual of the element of H2(M,Z), say F , represented by any fiber of π —whereas

∫F γE = 1, as the restriction of γE to any fiber π−1(x) is the Chern

class of the dual tautological line bundle over π−1(x), which is of degree 1;the first identity then follows from (10.1.1).

10.1. COMPLEX RULED SURFACES 251

It is convenient to substitute to γE — which depends upon the choice ofE in the expression of M = P(E) — the element ε of H2(M,Q) defined by

epsilonepsilon (10.1.3) ε := γE +degE

2π∗γΣ,

which only depends on M . As a matter of fact, ε is the Chern class ofthe positive square root of the vertical tangent bundle on P(E) cf. [83].Indeed, since c1(E) = c1(Λ2E), 2ε is the Chern class of the holomorphic linebundle OE(2) ⊗ π∗Λ2E, whose fiber at any point x of M over y — viewedas a complex line in Ey — is the tensor product x∗ ⊗ x∗ ⊗ Λ2Ey; this iscanonically identified with x∗ ⊗ (Ey/x), hence with Tx(P(Ey)), cf. Remark6.2.5.

For any Riemann surface Σ, the first Chern class of the (holomorphic)tangent bundle of Σ is χ(Σ) γΣ = 2(1 − g) γΣ; from the foregoing, we theninfer that the first Chern class of M in H2(M,Z) is given by

chernMchernM (10.1.4) c1(M) = 2ε+ 2(1− g)π∗γΣ.

The pair ε, π∗γΣ provides an alternative basis of H2(M,R), with thefollowing table for the product:

table-bistable-bis (10.1.5) ε · ε = 0, ε · π∗γΣ = 1, π∗γΣ · π∗γΣ = 0.

Following [83], define t(E) by

tEtE (10.1.6) t(E) = max(deg(F ))− deg(E)

2,

where F runs over the space of all proper holomorphic subline bundles of E.Then, E is said to be stable if t(E) < 0; semi-stable if t(E) ≤ 0; unstable

if t(E) > 0; polystable — or quasistable — if E is either stable or can bewritten as E = L1 ⊕ L2, where L1, L2 are hololomorphic line bundles overΣ of the same degree. In general, if E is decomposable, i.e. of the formE = L1⊕L2 for any two holomorphic line bundles, with deg(L1) ≥ deg(L2),we easily check that

tELtEL (10.1.7) t(E) =1

2(deg(L1)− deg(L2));

E is then unstable, unless it is polystable. Notice that t(E) is unchangedwhen E is replaced by E ⊗ L; the above conditions then qualify the ruledcomplex surface M = P(E).

A celebrated theorem of M. S. Narasimhan and C. S. Seshadri [153]asserts that E is polystable if and only if P(E) arises from an representation,say ρ : Γ → PU(2), of the fundamental group, Γ = π1(Σ), of Σ to theprojectivized unitary group PU(2), meaning that M = P(E) can be realizedas

NSNS (10.1.8) M = (Σ× P1)/Γ,

where Σ denotes the universal covering of Σ and Γ acts on the product Σ×P1

via its covering action on Σ and via the representation ρ on P1. Recall — cf.Proposition 6.2.2 — that PU(2) is the identity component of the isometrygroup of the complex projective line P1 equipped with a standard Fubini-Study metric. It follows that the action of Γ on the product Σ × P1 isisometric whenever P1 is equipped with a standard Fubini-Study metric – of


any (positive) constant sectional curvature — and Σ with a metric arisingfrom a Kahler metric of constant sectional curvature on Σ, equal to a positivemultiple of the Euler characteristic χ(Σ) = 2(1− g)) of Σ). Denote by ωFSthe Kahler form of the standard Fubini-Study of P1 of constant sectionalcurvature +2 and by ωΣ the pull back on Σ of the Kahler form of a Kahlermetric on Σ of constant sectional curvature equal to χ(Σ); by the Gauss-Bonnet formula 1, the volume of (P1, ωFS) and the volume of (Σ, ωΣ) areboth equal to 2π. For any two positive real numbers A,B, AωFS + BωΣ

is then the pull-back of the Kahler form of a Kahler metric on M , whoseKahler class in H2(M,R) is equal to

positiveconepositivecone (10.1.10) Ω(A,B) = 2π(Aε+B π∗γΣ),

and whose (constant) scalar curvature is equal to

constantsconstants (10.1.11) s =4

A+

4(1− g)

B.

The Kahler cone of M is contained in the positive cone generated by εand π∗γΣ, i.e. the set of elements Ω of H2(M,R) defined by (10.1.10) for anytwo positive numbers: indeed, if Ω is a Kahler class,

∫F Ω = 2πA

∫F γE =

2πA must be positive, as well as∫M Ω2, which, by (10.1.5), is equal to

8π2AB, cf. [83]. When M is polystable, it readily follows from the abovethat the Kahler cone is then exactly the positive cone generated by ε andπ∗γΣ and that any Kahler class Ω(A,B) contains the Kahler form of a metricwhich is locally the product of two Riemann surfaces of constant sectional

curvature, equal to 2A and 2(1−g)

B ; in particular, this metric is of constantscalar curvature given by (10.1.11).

For a general complex ruled surface M = P(E), the Kahler cone of Mis given by

thmfujiki Theorem 10.1.1 (A. Fujiki [83] Proposition 1). The Kahler cone of acomplex ruled surface M = P(E) is the cone of elements of H2(M,R) givenby (10.1.10) for any two positive real numbers A,B satisfying the additionalcondition:

(10.1.12)B

A> max(0, t(E)),

where t(E) is defined by (10.1.6).

For the proof, we refer the reader to Fujiki’s original paper [83].

10.2. Hirzebruch-like ruled surfacessshirzebruchlike

The ruled complex surfaces of main interest in this chapter are thoseM = P(E) such that E is unstable and decomposable, i.e. E = L1 ⊕ L2,with deg(L1) > deg(L2). By substituting E ⊗ L∗1, we can assume thatE = 1 ⊕ L, where, we recall, 1 stands for the product bundle M × C and

1Recall that the Gauss-Bonnet for a compact (real) riemannian surface Σ of genus greads:

GBFGBF (10.1.9)

∫M

kg vg = 2π χ(Σ) = 4π (1− g),

for any riemannian metric g, of (sectional) curvature kg, cf. e.g. [99].

10.2. HIRZEBRUCH-LIKE RULED SURFACES 253

L denotes a holomorphic line bundle of negative degree deg(L) = −`, with` > 0.

If g = 0, i.e. if Σ ∼= P1, these are the Hirzebruch surfaces F` already con-sidered in Section 6.5. In particular, as a complex surface, F1 is isomorphicto the blow-up of the complex projective plane P2 along one point. Morever,except for the product P1 × P1 = P(1 ⊕ 1), all complex ruled over P1 areof this type, i.e. are Hirzebruch surfaces: this is because any holomorphicvector bundle over the complex projectif line is decomposable — cf. e.g.[24, Proposition III.15] — and any holomorphic line bundle of degree d isisomorphic to O(d).

In general, a ruled surfaceM of the above kind will be called a Hirzebruch-like ruled surface of genus g and of degree `, meaning that M = P(1 ⊕ L),where L is a holomorphic line bundle of negative degree −` over a (compact)Riemann surface Σ of genus g 2.

Any Hirzebruch-like ruled surface M = P(1⊕L) can be regarded as thecompactification of L obtained by adding a point at infinity to each fiber Ly,namely the complex line Ly itself, viewed as a point in P(Ey) = P(C⊕ Ly),cf. Remark 6.5.1. The union of these points at infinity is the infinity section,denoted by Σ∞, whereas the zero section of L, viewed as a (holomorphic)section of P(1⊕L) via the inclusion L ⊂ P(1⊕L), is denoted by Σ0 (here andhenceforth we identify the zero section σ0 : y 7→ C and the infinity sectionσ∞ : y 7→ Ly of P(E) with their images, Σ0 and Σ∞, in M = P(1⊕ L): Σ0,resp. Σ∞, is then a copy of Σ and its normal bundle in M is holomorphicallyisomorphic to the restriction of π∗(L), resp. π∗(L∗)).

Denote by M0 the open set of M obtained by removing the zero sectionΣ0 and the infinity section Σ∞; we then have

MdecMdec (10.2.1) M = Σ0 ∪M0 ∪ Σ∞.

remassociated Remark 10.2.1. The open setM0 is equivalently defined asM0 = L\Σ0,i.e. the set of non-zero elements of L, and can then be regarded as the totalspace of the C∗-principal bundle of “frames” of L. As a complex manifold,M = P(1 ⊕ L) itself can then viewed as the associated bundle3 M0 ×C∗ P2

determined by the diagonal C∗-action: ζ · (z1, z2) = (ζz1, z2) on C2 and theinduced action on P1). If Q denotes the set of elements of norm 1 withrespect to h in L, viewed as a S1-principal bundle over Σ, M can be likewiseregarded as the associated bundle Q ×S1 P1 with respect to the standardS1-action on P1 = S2.

2When g > 1, these are the pseudo-Hirzebruch surfaces studied by C. Tønnesen-Friedman in [194], [195].

3 In general, for anyG-principal bundle P over some manifoldN , and for any (smooth)associatedG-action on some manifold F , the associated bundle M = P×GF is defined as the quotientP ×F/G for the (right) G-action on P ×F defined by γ · (s, u) = (sγ, γ−1 ·u), for any γ inG, any s in P , any u in F . If P is equipped with a connection, determined either by a G-equivariant g-valued connection 1-form η or by the corresponding G-invariant horizontaldistribution H = kerη, then, for any G-space F , the associated bundle P ×G F comesnaturally equipped with a connection, whose horizontal distribution is induced by theG-invariant distribution, say H agian, on P × F defined as follows: for any pair (s, u) inP × F , H(s,u) := Hs ⊗ 0.


Elements of H2(M,Z), in particular the generators γE and π∗γΣ, will beconveniently represented by their Poincare duals in H2(M,Z): π∗γΣ is thusidentified with the class, F , of any fiber of π (see Section 10.1), whereas γEcan be represented by the (homology class of the) infinity section Σ∞: this isbecause the (holomorphic) projection of E = 1⊕L to the trivial line bundle1 = Σ× C along L determines a holomorphic section of OE(1), whose zerodivisor is Σ∞; it follows that Σ∞ is Poincare dual of c1(OE(1)) = γE , cf.footnote of page 154. Similarly, the (holomorphic) projection of E = 1⊕L toL along the trivial line bundle 1 = Σ×C determines a holomorphic sectionof OE(1) ⊗ π∗L, whose zero divisor is Σ0; Σ0 is then the Poincare dual ofc1(OE(1)⊗π∗L) = c1(OE(1))+π∗c1(L); since c1(L) = −` γΣ, we eventuallyget:

(10.2.2) Σ0 = Σ∞ − ` F

and the table (10.1.2) then becomes

Σ∞ · Σ∞ = `, Σ∞ · F = 1, F · F = 0,

Σ0 · Σ0 = −`, Σ0 · F = 1, Σ0 · Σ∞ = 0.table-sigmainftytable-sigmainfty (10.2.3)

The class ε defined in Section 10.1 can then be written as

(10.2.4) ε =1

2(Σ0 + Σ∞).

Since ` > 0, (the Poincare duals of) Σ0 and Σ∞ form a basis of H2(M,R)(but not a set of generators of H2(M,Z), unless ` = 1). We then have:

propkaehlercone Proposition 10.2.1. For any Hirzebruch-like ruled surface M = P(E),the Kahler cone of M is the set of classes Ω in H2(M,R) of the form

kaehler-conekaehler-cone (10.2.5) Ωa,b = 2π(−aΣ0 + bΣ∞),

for any two real numbers a, b satisfying the condition

abab (10.2.6) 0 < a < b.

Moreover,Ωa,b2π belongs to H2(M,Z) if and only if a, b both belong to Z[1

` ]and the difference b− a is a (positive) integer; we then have

(10.2.7)Ωa,b

2π= (b− a) γE + a` π∗γΣ,

meaning thatΩa,b2π is the Chern class of the holomorphic line bundle

OE(b− a)⊗ a` π∗(L−a).

Proof. For E = 1 ⊕ L, with deg(L) = −` < 0, we readily infer from(10.1.7) that t(E) = `

2 , whereas, from the above tables we easily deducethat the real numbers a, b in (10.2.5) are related to the real numbers A,Bin (10.1.10) by: a = B

` −A2 , b = B

` + A2 ; Proposition 10.2.1 is then a direct

consequence of Theorem 10.1.1. We here provide a more specific argument.We first show that the condition 0 < a < b is necessary by using the generalfact that the integral of a Kahler form on any (compact) complex submani-fold of positive dimension p is equal to its riemannian volume multiplied byp!, hence is positive. By integrationg Ω given by (10.2.5 over F , Σ0, Σ∞ andassuming that Ω belongs to the Kahler cone, we get

∫F Ω = 2π(b− a) > 0,

10.2. HIRZEBRUCH-LIKE RULED SURFACES 255∫Σ0

Ω = 2π`a > 0,∫

Σ∞Ω = 2π`b > 0; this evidently implies 0 < a < b. Con-

versely, for any pair of real numbers a, b satisfying ) < a < b, there exists aKahler form on M whose Kahler class is given by (10.2.5), cf. Proposition10.3.3 below. In terms of γE and π∗γΣ, which generate H2(M,Z), we havethat Σ∞ ∼= γE and Σ0

∼= γE − `π∗γΣ, so that (10.2.5) can be rewritten as

kaehler-cone-intkaehler-cone-int (10.2.8) Ωa,b = 2π(a` π∗γΣ + (b− a) γE).

The last statement of the proposition readily follows from (10.2.8) (if a = r` ,

where r is a positive integer, L−a stands for (L1` )−r, where L

1` denotes any

`-th root of L, whose existence is guaranteed since deg(L) = −` ; the Chernclass is then independent of the choice of the root).

The group of automorphisms of complex ruled surfaces has been stud-ied by M. Maruyama in [147], from which we extract the following piecesof information for H(M) — the identity component of the automorphismgroup of M — in the special case when M = P(1 ⊕ L) is a Hirzebruch-likeruled surface of genus g and degree ` (cf. also [194] for the case when ofHirzebruch-like ruled surfaces of genus greater than 1).

We first observe that the natural action of C∗ on L extends to a (holo-morphic) C∗-action on M = P(1⊕L) as follows: any point y of M in π−1(x)is a complex line of C⊕Lx, generated by, say, the pair (z, u), where z belongsto C and u belongs to Lx; for any ζ in C∗, ζ · y is then the complex lineof C ⊕ Lx generated by the pair (z, ζ u). This action clearly preserves thefibration π from M to Σ, pointwise fixes Σ0 and Σ∞ and is free — as wellas the induced action of S1 ⊂ C∗ — on the open set M0 = L \ Σ0.

In the sequel of this chapter, the generator of the natural S1-action willbe denoted by T .

Denote by H0(Σ, L∗) the complex vector space of holomorphic sectionsof the dual bundle L∗. Then, H0(Σ, L∗) can be viewed as a subgroup ofH(M). For any element, σ, of H0(Σ, L∗), the corresponding automorphismof M , say σ, is defined as follows: for any y in π−1(x), generated by thepair (z, u) as above, σ · y is defined as the complex line of C⊕Lx generatedby the pair (z+ σ(u), u). This action preserve each fiber of π and pointwisepreserves Σ0, but does not preserve Σ∞. Notice that the inverse of σ is −σand that ζ σ ζ−1 = ζ−1σ.

By putting these two actions together, we get a subgroup of H(M) whichis the semi-direct product C∗nH0(Σ, L∗) for the linear action ζ ·σ = ζ−1σ ofC∗ on H0(Σ, L∗) (cf. footnote 3 of page 162 for the definition of semi-directproducts).

Denote by HΣ(M) the subgroup of H(M) whose elements, say γ, satisfyπ γ = π. Then, H(M) is described by the following proposition:

propauthirzebruch Proposition 10.2.2. For any Hirzebruch-like ruled surface M = P(1⊕L) of any genus g and any (positive) degree `, we have

(10.2.9) HΣ(M) = C∗ nH0(Σ, L∗).

Moreover:


(i) If g > 0, we have

HHSigmaHHSigma (10.2.10) H(M) = HΣ(M) = C∗ nH0(Σ, L∗).

(ii) If g = 0, i.e. if M = F` for some `, we have the following exactsequence:

Hgenus0Hgenus0 (10.2.11) 1→ C∗ nH0(P1,O(`))→ H(F`)→ PG`(2,C)→ 1

and the isomorphism

isohirzebruchisohirzebruch (10.2.12) H(F`) = G`(2,C)/µ` nH0(P1,O(`)),

for the action of G`(2,C)/µ` on H0(P1,O(`)) induced by the natural actionof G`(2,C) on O(−1). In this isomorphism, G`(2,C)/µ` is the subgroup ofelements of H(F`) wich (globally) preserve Σ0 — as do all elements of H(M)— and Σ∞.

In all cases, we have that

HH0HH0 (10.2.13) H(M) = Hred(M),

where, we recall, Hred(M) denotes the reduced automorphism group definedin Section 2.4.

Proof. The first assertion follows from [147], Theorem 1, Theorem 2,Lemma 6. Then, (i) follows from [147] Lemma 7 and the exact sequence(10.2.11) in (ii) from [147] Lemma 8 (recall that PG`(2,C) is the group ofautomorphisms of P1 — cf. Proposition 6.1.2 — whereas H0(P1,O(`)) =`(C2)∗, which is of (complex) dimension `+ 1, cf. Proposition 6.1.1). Thelinear group G`(2,C) acts on O(−1) via the natural isomorphism O(−1) \Σ0 = C2\0; this action covers the natural action of PG`(2,C) = G`(2,C)/C∗on P1 readily extends to F1 = P(1 ⊕ O(−1)); moreover, the (holomorphic,linear) action of G`(2,C) on L = O(−1) induces a (holomorphic, linear)action on L∗ = O(1), hence on H0(P1,O(1)), which extends the aboveaction of C∗; the semi-direct product G`(2,C) nH0(P1,O(1)) is then con-tained in H(F1) and fits into the same exact sequence (10.2.11); it followsthat G`(2,C) n H0(P1,O(1)) coincides with H(F1) (this isomorphism hadalready be otained in Section 6.4), via the identification of F1 with theblow-up of P2 along a point). Now, the above action of G`(2,C) on O(−1)induces an action on O(−`) which is effective on the quotient G`(2,C); thesame reasoning then gives (10.2.12). All elements of H(F`) globally preserveΣ0, as Σ0 is the only complex curve of negative self-intersection (this alsofollows from the isomorphism (10.2.12), since all elements of G`(2,C) andH0(P1,O(`)) preserve Σ0); on the other hand, all elements of G`(2,C) pre-serve Σ∞, whereas no element of H0(P1,O(`)) do preserve Σ∞, except 0.The last assertion (10.2.13) is clear as the Jacobi map from M to Alb(M) —see Section 2.4 — factors through π: if g = 0, the Albanese torus is trivial,whereas, if g > 0, the Jacobi map is trivial because of (10.2.10).

remautFk Remark 10.2.2. If M = F`, the open set M0 appearing in (10.2.1)is biholomorphically isomorphic to (C2 \ 0)/µ`; it is preserved by theG`(2,C)/µ` part of H(F`), whose action is then the natural action ofG`(2,C)/µ`on (C2 \ 0)/µ`.

As an direct consequence of Proposition 10.2.2 we get

10.3. ADMISSIBLE KAHLER METRICS 257

propmaxhirzebruch Proposition 10.2.3. For any Hirzebruch-like ruled surface M of genusg and degree `, the maximal compact subgroups of H(M) = Hred(M) areconjugate to

(i) S1, if g > 0;(ii) U(2)/µ`, if g = 0.

Proof. In any semi-direct prodct of the form GnV , where V is a vectorspace and G is a (connected) Lie group acting linearly on V — cf. footnote3 of page 162 — the maximal compact subgroups of G n V are clearly themaximal compact subgroups of G. Proposition 10.2.3 then readily followsfrom Proposition 10.2.2.

10.3. Admissible Kahler metricsssadmissible

We now equip L with a hermitian inner product, h, in such a way thatthe opposite of the curvature form of the corresponding Chern connectionbe the Kahler form, say ωL, of a metric of constant curvature κ. Since∫M ωL = 2π `, it follows from the Gauss-Bonnet formula (10.1.9) that

kappa-unikappa-uni (10.3.1) κ =χ(Σ)

`=

2(1− g)

`.

The norm function determined on L by h is denoted by r; we then have

r(u) = (h(u, u)12 for any u in L. On M0, r is positive and we then set:

(10.3.2) t = log r.

The 1-form dct — or, rather, its restriction to the space, Q, of unit elementsof (L, h), viewed as a S1-principal bundle over Σ — is then the connection1-form of the Chern connection determined by h; on M0, we then have:

dctTdctT (10.3.3) dct(T ) = 1,

where, we recall. T denotes the generator of the action of S1 on M inducedby the natural S1-action on L, cf. Section 10.2, whereas

ddctomegaLddctomegaL (10.3.4) ddct = π∗ωL,

cf. (1.7.3). Moreover, e2t dt ∧ dct = dr ∧ dcr restricts to the volume formdetermined by h on each fiber of L \ Σ0 (here and henceforth, the twisteddifferential dc is relative to the natural complex structure of M , which willbe denoted by J). In particular, ddct smoothly extend to M , e2tdt ∧ dctsmoothly extends to Σ0 and e−2tdt ∧ dct smoothly extends to Σ∞.

When g = 0, i.e. when M = F` is a (genuine) Hirzebruch surface, allmaximal compact subgroup of H(M) = Hred(M) are conjugate to U(2)/µ`,cf. Proposition 10.2.3, which acts transitively on Σ0 and Σ∞ and acts withcohomogeneity 1 in M0, with orbits the distance spheres t = constant, allisomorphic to S3/µ`. By Theorem 3.5.1, any extremal Kahler metric on F`— if any — can be made U(2)/µ`-invariant by the action of an element ofH(M). We then have:

propdemailly Proposition 10.3.1. Any U(2)/µ`-invariant Kahler metric defined onM0 = (C2 \ 0)/µ` admits a U(2)/µ`-invariant globally defined Kahlerpotential, unique up to an additive constant.


Proof. Without loss of generality, we can assume ` = 1, as C2 \ 0 isa covering of (C2 \ 0)/µ`. Let ω be the Kahler formm of a U(2)-invariantKahler metric defined on M0 = C2 \ 0. Since M0 is simply-connected, wehave ω = dα, where α is a real 1-form, which can be chosen U(2)-invariantas well (if not, replace α by α :=

∫U(2) γ

∗αdγ, where γ stands for any bi-

invariant measure on the compact Lie group U(2)). Since ω is of type (1, 1),we have that ∂α0,1 = 0; it then remains to show that α0,1 = ∂F , for someU(2)-invariant complex function; if so, ImF would be the required U(2)-invariant Kahler potential. On U , the U(2)-orbits are the distance spheresof radius r, r > 0, for the standard euclidean structure of C2. For eachSr, denote by Vr the U(2)-invariant tubular neighbourhood of Sr defined

by Vr = u ∈ U | r2

2 < |u|2 < 32r

2. Then, Vr is contained in the polydisk

D2r × D2r ⊂ C2, which is a Stein manifold. It follows that α0,1 is ∂-exacton D2r × D2r, hence, a fortiori, on Vr; for each r, we then get Fr, whichcan be chosen U(2)-invariant, such that α0,1

|Vr = ∂Fr. Moreover, since U(2)-

invariant holomorphic functions are necessarily constant (easy verification),the the Fr’s differ by a constant. Since the Fr’s are entirely determinedby their restrictions on any radial half-line, ray D = R>0, we eventuallyget on open covering of R>0, parametrized by r, and, on each open interval

Ir = (√

12r,√

32r), a complex function, still denoted Fr, in such a way that

Fr−Fr′ is constant for any r, r′ such that Ir∩Ir′ 6=. Since, H1(R>0,C) = 0,we infer that there exists F globally defined on R>0 such that Fr = F|Vron each Vr; this, F can be be viewed as a U(2)-invariant function definedon U and we then have α0,1 = ∂F . If ϕ,ϕ′ are two U(2)-invariant Kahlerpotential defined on U . then ϕ′−ϕ belongs to the kernel of ddc, hence is thereal part of holomorphic function; this is U(2)-invariant, hence constant.

Definition 16. A Kahler metric (g, J, ω) defined on a Hirzebruch-likeruled surface M = P(1⊕L) is called admissible if its restriction to M0 admitsa globally defined Kahler potential, F = F (t), which only depends on t; onM0, we then have

omega-uniomega-uni (10.3.5) ω = ωψ := ψ ddct+ ψ′ dt ∧ dct,

by setting: ψ(t) = F ′(t), and ψ′(t) denotes the derivative of ψ with respectto t.

Notice that a 2-form ωψ of the form (10.3.5) is the Kahler form of aKahler metric, gψ say, defined on M0 if and and only if ψ and ψ′ are bothpositive

By Theorem 3.5.1, Proposition 10.2.3 and Proposition 10.3.1, any ex-tremal Kahler metric on a Hirzebruch surface F` is isomorphic, via an ele-ment of H(F`), to an admissible metric gψ.

If M = P(1⊕L) is a Hirzebruch-like ruled surface of genus greater than 0,then the maximal subgroups of H(M) are conjugate to S1 — cf. Proposition10.2.3 — and we can no longer pretend a priori that all extremal Kahlermetrics on M can be made admissible by the action of H(M). In this case,we agree to search a priori for extremal Kahler metrics among admissibleKahler metrics. Notice that the latter are obviously S1-invariant.


For any admissible Kahler metric on M0, the (positive, increasing) func-tion ψ has then the following geometric interpretation:

lemmacalabimomentum Lemma 10.3.1. Let gψ be any admissible Kahler metric on M0, whoseKahler form ωψ is given by (10.3.5) for some positive, increasing function ψdefined on (−∞,+∞). Then, ωψ is S1-invariant for the natural S1-actionon M0 ⊂M and ψ is a momentum of this action with respect to ωψ, meaningthat:

calabimomentumcalabimomentum (10.3.6) ιTωψ = −dψ,where T denotes the generator of the S1-action.

Moreover, for any (positive) value c of the momentum ψ the correspond-ing Kahler quotient ψ−1(c)/S1 is Σ, equipped with the Kahler structure c ωL,of constant sectional curvature equal to κ

c .

Proof. The S1-action on M is induced by the standard S1-action on L,which preserves the complex structure of L, hence of M , and any hermitian(fiberwise) inner product, in particular h, the radial function r and t = log ron M0, corresponding Chern connection and its curvature form π∗ωL: ωψ isthus preserved by this action. Moreover, from (10.3.3), (10.3.4) and dt(T ) =0, we infer: ιTdd

ct = 0 and ιT (dt ∧ dct) = −dt; (10.3.6) follows readily.For any c > 0, the level set ψ−1(c) is a distance sphere in M0 and the

restriction of ωψ to this sphere is equal to c ddct = c π∗ωL = π∗(c ωL).

Admissible Kahler metrics which are defined on the whole of M are thencharacterized as follows:

propcompactification Proposition 10.3.2. Let gψ be any admissible Kahler metric on M0,whose Kahler form ωψ is given by (10.3.5) for a positive, increasing functionψ. Then, gψ extends to a Kahler metric to Σ0∪M0 if and only if the followingcondition is satisfied:

(A0) Near t = −∞, ψ(t) = Φ0(e2t), where Φ0 is a smooth positive func-tion defined in some neighbourhood of 0 in R≥0, with Φ0(0) = a > 0and Φ′0(0) > 0.

Similarly, gψ extends to M0 ∪ Σ∞ if and only if:

(A∞) Near t = +∞, ψ(t) = Φ∞(e−2t), where Φ∞ is a smooth positivefunction defined in some neighbourhood of 0 in R≥0, with Φ∞(0) =b > a > 0 and Φ′∞(0) < 0.

Then, gψ extends to a Kahler metric on M if and only conditions (A0)and (A∞) are both satisfied. If so, we have in particular:

limpsi-unilimpsi-uni (10.3.7) limt→−∞

ψ(t) = a, limt→+∞

ψ(t) = b,

with 0 < a < b, whereas ψ′(t) and all successive derivatives tend to 0 whent tends to ±∞ in such a way that

limpsi1-unilimpsi1-uni (10.3.8) limt→−∞

ψ′(t)

ψ(t)− a= 2, lim

t→+∞

ψ′(t)

ψ(t)− b= −2,

and

limpsi2-unilimpsi2-uni (10.3.9) limt→−∞

ψ(j+1)(t)

ψ(j)(t)= 2, lim

t→+∞

ψ(j+1)(t)

ψ(j)(t)= −2,


where, for each integer j ≥ 1, ψ(j) denotes the j-th derivative of ψ. More-over, the Kahler class [ωψ] is then Ωa,b given by (10.2.5).

Proof. The Kahler metric smoothly extends to Σ0 if and only if it issmoothy defined on the total space of L = M \ Σ∞. By setting ψ(t) =Φ0(e2t), ωψ can be written as ωψ = Φ0(r2)π∗ωs2 + 2Φ′0(r2)dr ∧ dcr, whereboth π∗ωS2 = ddct and dr ∧ dcr are smoothly defined on L. We concludethat ωψ is smoothly defined on Λ if and only if Φ0 and Φ′0 are smooth andpositive on R≥0. In particular a := limt→−∞ ψ(t) = Φ0(0) is positive aswell as limt→−∞ ψ

′(t)e−t = Φ′0(0). The second part of the proposition isobtained by exchanging the roles of Σ0 and Σ∞, by observing that M canalso be identified to P(1 ⊕ L∗), i.e. as the natural compactification of thedual line bundle L∗ over Σ obtained by adding a “infinity section”: in thispicture, the latter is Σ0, whereas Σ∞ becomes the zero section of L∗. (notethat the two pictures are isomorphic via the transformation t → −t; thistransformation changes the orientation and transforms J into the complexstructure J obtained by reversing on the fibres of π and keeping J on thehorizontal distribution determined by the Chern connection). Since

∫Σ ωL =

2π`, we have that∫

Σ0ωψ = 2π`a and

∫Σ∞

ωψ = 2π`b. In view of (10.2.3),this implies the last statement of Proposition 10.3.2.

Positive, increasing smooth functions ψ = ψ(t) defined on the real line(−∞,+∞) which satisfy the boundary conditions (A0)-(A∞) near t = ±∞for some pair of real numbers a, b will be referred to as admissible momentumfunctions relative to the pair a, b (then, a, b necessarily satisfy (10.2.6)).

Notice that if ψ is an admissible momentum function relative to a, bas above, for any real number c, the function ψc(t) := ψ(t + c) is again anadmissible momentum function relative to a, b as well and the correspondingadmissible Kahler metric, gψc , is then the image of gψ by the (holomorphic)automorphism of M induced by the multiplication by ec in L. To avoid thisambiguity, it may be convenient to normalize ψ by fixing ψ(0) in (a, b), e.g.

ψ(0) = a+b2 or, in view of Remark 10.4.2, ψ(0) =

√ab.

For any pair a, b of real numbers satisfying (10.2.6), we denote byMadmΩa,b

the space of admissible Kahler metrics of Kahler class Ωa,b. In view of

Proposition 10.3.2,MadmΩa,b

is naturally identified with the space of admissible

momentum functions relative to a, b. Moreover:

propab Proposition 10.3.3. For any pair a, b of real numbers satisfying (10.2.6),the space Madm

Ωa,bis non-empty.

Proof. The function ψ0 defined by

psi0psi0 (10.3.10) ψ0(t) =a+ be2t

1 + e2t,

evidently satisfies the required conditions.

rempsi0 Remark 10.3.1. The Kahler metric, gψ0 , determined by the standardadmissible momentum function ψ0 will be called the standard admissibleKahler metric in the Kahler class Ωa,b. A Kahler potential for the Kahler


class ωψ0 is the function F0 = F0(t) defined by

F0(t) = at+(b− a)

2log (1 + e2t)

= bt+(b− a)

2log (1 + e−2t)

=b

2log (1 + e2t)− a

2log (1 + e−2t).

standardpotentialstandardpotential (10.3.11)

In restriction to any fiber π−1(y) of π, for any point y in Σ, we have thatddct|π−1(y) = 0: a Kahler potential for the restriction of gψ0 to π−1(y) is then(b−a)

2 log (1 + e2t) = (b−a)2 log (1 + r2). We recognize a Kahler potential for

the Fubini-Study metric of sectional curvature c = 2(b−a) of the projective line

P(C⊕Lx) determined by the hermitian inner product hx on the affine openset P(C⊕Lx)\[Lx], cf. (6.2.5) in Section 6.2. In view of this observation andof Remark 10.2.1, the standard admissible Kahler metric can be constructedaccording to the following recipe:

(1) Realize M as the associated bundle M = Q×S1 P1 — cf. footote 3on page 253 — relatively to the standard S1-action on the complexprojective line P1, where, we recall, Q denotes the set of elementsof L of norm 1, viewed as a S1-principal bundle over Σ. SinceQ×S1P1 is tautologically identified with (L\Σ0)×C∗P1, cf. Remark10.2.1, the manifold M constructed that way comes equipped witha natural complex structure.

itemFS (2) Equip the complex projective line P1 with the standard Fubini-Study Kahler structure (gFS,c, J0, ωFS,c) of sectional curvature c =

2(b−a) . Since gFS,c is S1-invariant, each fiber π−1(y) = P(C⊕Ly) is

therefore endowed with a Kahler metric, namely the Fubini-Studymetric of sectional curvature c = 2

(b−a) determined by the hermitian

inner product of C ⊕ Ly induced by the hermitian inner product,hy, of Ly.

itemrecipe (3) By using the horizontal distribution, say H∇, on M = Q ×S1 P1

induced by the Chern connection ∇, cf. footnote 3 of page 253,for any y in Σ and any point x in the fibre π−1(y) = P(C ⊕ Ly),decompose the tangent space TxM into the direct sum

TdecTdec (10.3.12) TxM = Tx(P(C⊕ Ly))⊕H∇y ,

where H∇y is naturally identified with TyΣ, via the differential π∗ ofπ. The standard admissible Kahler metric gψ0 is then determinedby the following conditions:• the two summands in (10.3.12) are gψ0-orthogonal.

• the restriction of gψ0 to the horizontal part H∇x = TyΣ is equalto ψ0(x) gL(y), where, we recall, gL denotes the metric of Σ ofconstant sectional curvature κ.• the restriction of gψ0 to each fiber coincides with the natural

Fubini-Study metric gFS,c, with c = 2(b−a) , cf. (2).

More generally, any admissible Kahler metric gψ on M can be constructedaccording to the same recipe by only replacing ψ0 by ψ, in particular, the


Fubini-Sudy metric gFS,c on P1 with the S1-invariant Kahler metric

metric-CP1metric-CP1 (10.3.13) gP1

ψ :=ψ′(t)

ψ′0(t)gFS,c,

in (2) and (3). In (10.3.13), t = log r is viewed as a function defined onC∗ = P1 \ (x0 ∪ x∞), where x0 = (0 : 1), x∞ = (1 : 0) denote the two fixedpoints of the natural S1-action: ζ · (u1 : u2) = (ζu1 : u2), and r denotesthe standard distance to the origin in C = P1 \ x∞, so that r((u1 : u2)) =|u1||u2| ; then, ψ′ = ψ′(t) and ψ′0 = ψ′0(t) can both be viewed as defined on

P1 \ (x0 ∪ x∞); in view of the boundary conditions of Proposition 10.3.2,ψψ0

= ψ(t)ψ0(t) then extends to a positive, smotth function on P1. When ψ runs

over the space of admissible momentum functions relative to the pair a, b,

then ψ′

ψ′0can be any positive, S1-invariant function, on P1, with the only

constraint that∫P1

ψ′

ψ′0ωFS,c =

∫P1 ωFS,c = 2π(b− a).

Via the above construction, admissible Kahler metrics gψ on M are

entirely encoded by the Kahler metric gP1

ψ on the standard projective line

P1, equipped with its standard complex structure and its standard S1-action.

On C = P1 \ (x0 ∪ x∞), the Kahler form of gP1

ψ is ωP1

ψ = ψ′(t)dt ∧ dct. In

particular, ψ = ψ(t), now viewed as a function on P1, is a momentum relative

to ωP1

ψ of the natural S1-action, which maps P1 to the closed interval [a, b].

By using (1.19.6), we easily deduce that the scalar curvature of gP1

ψ has thefollowing expression:

scal-CP1scal-CP1 (10.3.14) sP1

ψ = −(logψ′(t))′′

ψ′(t).

Let ψ be any admissible momentum function relative to the pair a, b.Since ψ is an increasing function of t, t can in turn be expressed as anincreasing function of ψ, and we then set

(10.3.15) ψ′(t) = Θ(ψ),

for some positive smooth function Θ = Θ(x) defined for any x in the openinterval (a, b). Following [109], Θ will be called the momentum profile ofthe metric gψ and has the following geometric interpretation: the inducedriemannian metric, say gψ,∆, on (a, b) via the momentum map ψ : M0 →(a, b) is given by

(10.3.16) gψ,∆ =dx⊗ dx

Θ(x)

(here and henceforth, the natural parameter of the open interval (a, b) isdenoted by x). Alternatively, Θ(x) is equal to the square norm of T at anypoint of ψ−1(x) in M0.

In particular, Θ can also be regarded as the momentum profile of the

Kahler structure (gP1

ψ , ωP1

ψ ) on P1 via the momentum map ψ : P1 → [a, b], cf.Remark 10.3.1. In terms of the momentum profile Θ, the scalar curvature


sP1

ψ of gP1

ψ , given by (10.3.14), when viewed as a function defined on a, b] via

ψ has then the following expression (compare with (10.4.9) below):

s-psi-CP1s-psi-CP1 (10.3.17) sP1

ψ (x) = −Θ′′(x).

Momentum profiles associated to admissible momentum functions willbe referred to as admissible momentum profiles relative to the pair a, b.

The momentum profile, Θ0, of the standard admissible metric ωψ0 inRemark 10.3.1, called the standard admissible momentum profile relative toa, b, is easily checked to be given by

(10.3.18) Θ0(x) =2

(b− a)(x− a)(b− x).

By using (10.3.17), we retrieve the fact that the scalar curvature of thecorresponding Kahler metric on P1 is constant, equal to 4

b−a .

Notice that the momentum profile Θ remains unchanged when ψ is re-placed with ψc for any real number c.

Admissible momentum profiles Θ are characterized by the followingproposition:

proppsiF Proposition 10.3.4. By replacing the momentum function ψ with themomentum profile Θ , the space Madm

Ωa,bis identified, up to the action of R>0

on M , with the space of smooth fonctions Θ = Θ(x) defined on the closedinterval [a, b] and satisfying the following properties

(B0) Θ is positive on the open interval (a, b);(B1) Θ(a) = Θ(b) = 0;(B2) Θ′(a) = 2, Θ′(b) = −2.

Proof. (B0) follows from ψ′ being positive on (−∞,+∞); (B1) followsfrom the fact that ψ′(t) → 0, when t → ±∞; (B2) follows from (10.3.8).Conversely, let Θ = Θ(x) be a smooth function defined on [a, b], satisfying(B0)-(B1)-(B2) and let t = t(x) be defined on (a, b) by

(10.3.19)dt

dx=

1

Θ(x);

t(x) is then well-defined — up to translation by a constant — and increasingon (a, b). Near x = a, Θ(x) = 2(x − a)Θ0(x), where Θ0 is positive, withΘ0(a) = 1; t is then of the form t(x) = 1

2 log (x− a) + f0(x), where f0 issmooth at x = a. Likewise, near x = b, Θ(x) = (b− x)Θ∞(x), where Θ∞ ispositive, with Θ∞(b) = 1 and t is then of the form t(x) = −1

2 log (b− x) +f∞(x), where f∞ is smooth at x = b. This determines t as an increasingreal function of x, smoothly defined on the open interval (a, b), tending to−∞ like 1

2 log (x− a) when x tends to a and towards +∞ like −12 log (b− x)

when x tends to b. This in turn defines x as a well defined increasingfunction of t on the interval (−∞,+∞), tending towards a when t tendsto −∞ and to b when t tends to +∞, and smooth as a function of e±2t inthe neighbourhood of t = ∓∞, hence satisfying conditions (A0) − (A∞) ofProposition 10.3.2.


remmomentum-uni Remark 10.3.2. By choosing the momentum ψ as an independent vari-able, instead of t — this can be done, as ψ is an increasing function of t —ωψ can be rewritten as

(10.3.20) ωψ = ψ π∗ωL + dψ ∧ η,

where we have written η instead of dct to emphasize that dct is here viewedas the connection 1-form of the Chern connection 1-form of (L, h), with noexplicit reference to t any longer. The corresponding metric, gψ, can thenbe written as

(10.3.21) gψ = ψ π∗gL +1

Θ(ψ)dψ ⊗ dψ + Θ(ψ) η ⊗ η,

where Θ = Θ(ψ) is the momentum profile, defined on the interval (a, b).This change of viewpoint is best understood by introducing the symplecticpotential, G = G(ψ), determined, up to a additive constant, by

(10.3.22)dG

dψ= t.

We then have: d2Gdψ2 = 1

Θ , so that the induced metric on the image of the

momentum map ψ can be rewritten as d2Gdψ2 dψ ⊗ dψ. Then, ψ = dF

dt and

t = dGdψ appear as dual variables and the Kahler potential F = F (t) and the

symplectic potential G = G(ψ) are linked together by the following partialLegendre transformation:

legendrelegendre (10.3.23) F +G− tψ = 0.

remcalabichern Remark 10.3.3. In terms of (the Poincare dual of) Σ0 and Σ∞, thefirst Chern class c1(M) of a Hirzebruch-like ruled surface, whose generalexpression for a complex ruled surface is given by (10.1.4), has the followingexpression

calabicherncalabichern (10.3.24) c1(M) = (1− κ) Σ0 + (1 + κ) Σ∞.

An alternative derivation of (10.3.24) uses the fact that c1(M) is representedby 1

2π [ρ] — cf. Section 1.19 — where ρ stands for the Ricci form of anyKahler metric, in particular any admissible Kahler metric on M . Let gψbe any such metric and recall that, on the open set M0, its Ricci formhas the form ρ = ddcw, where w = w(t) is given by (10.4.4) in Lemma10.4.2. We then have c1(M) = (−xΣ0 + yΣ∞), where x, resp. y, is the

limit of w′(t) = κ − 12ψ′

ψ −12ψ′′

ψ′ when t tends to −∞, resp. +∞. From the

boundary conditions (A0)-(A∞), in particular from (10.3.8)-(10.3.9), we get

that limt→±∞ψ′

ψ = 0, whereas limt→±∞ψ′′

ψ′ = ∓2. We thus get (10.3.24)again.

By (10.3.24), c1(M) belongs to the Kahler cone — so that M is Fano— if and only if

∫Σ0c1(M),

∫Σ∞

c1(M) and∫F c1(M) are all positive; by

(10.3.24) this happens if and only if κ > 1, hence κ = 2 and this occurs ifand only if g = 0 and ` = 1, i.e. M = F1. This, of course, is coherent withthe classification of Fano surfaces given in Section 6.6.

10.4. EXTREMAL ADMISSIBLE KAHLER METRICS 265

10.4. Extremal admissible Kahler metricsssextremaladm

From now on we assume that ωψ is the Kahler form of an admissibleKahler metric gψ defined on a Hirzebruch-like ruled surface M = P(1 ⊕ L)of any genus g and any degree ` > 0, for some admissible momentum ψ.

The geometry of these Kahler metrics is easily described. We start withthe following easy lemma:

lemmasimple Lemma 10.4.1. For any function f = f(t) on M which factorizes by t,the square norm |df |2 and the laplacian ∆f of f with respect to gψ are givenby

calabidfcalabidf (10.4.1) |df |2 =(f ′)2

ψ′,

and

calabideltafcalabideltaf (10.4.2) ∆f = −Λddcf = −(f ′

ψ+f ′′

ψ′

),

where f ′, f ′′ denote the first and second derivative of f with respect to t.

Proof. Since f is a function of t only, we have that df = f ′ dt, whereaswe readily infer from (10.3.5) that |dt|2 = 1

ψ′ : this gives (10.4.1). Similarly,

we have that ddcf = f ′ ddct+f ′′ dt∧dct; since Λddcf =2ddcf∧ωψωψ∧ωψ , we readily

get (10.4.2).

lemmacalabigeom-uni Lemma 10.4.2. On M0, the Ricci form, ρ = ρψ, and the scalar curva-ture, s = sψ, of ωψ have the following expressions:

rho-unirho-uni (10.4.3) ρ = ddcw = w′ddct+ w′′ dt ∧ dct,where w = w(t) is defined by

ricci-potential-uniricci-potential-uni (10.4.4) w(t) = κt− 1

2logψ(t)− 1

2logψ′(t),

and

s-unis-uni (10.4.5) s =2(w′ψ)′

ψψ′.

Proof. The Ricci form ρ = ρψ is computed on M0 by using the generalformula (1.19.6), namely

rho-gen-unirho-gen-uni (10.4.6) ρ = −1

2ddc log

vgv0,

where vg is the volume form of g and v0 the volume form of the standardflat Kahler metric determined by any (local) holomorphic coordinate system.From (10.3.5), we readily infer

v-univ-uni (10.4.7) vg = 2ψψ′ ddct ∧ dt ∧ dct,on M0. In order to compute v0, it is here convenient to choose holomorphiccoordinates of the form (z, λ), where z stands for any (local) holomorphiccoordinate defined on some open subset, U , of Σ, whereas λ depends onthe choice of a (local) nowhere vanishing holomorphic section, s, of L de-fined on U and is then defined by: ξ = λ(ξ) s(π(ξ)), for any ξ in M0,where π denotes the natural (holomorphic) projection from M to Σ; then,


v0 = ( i2dz ∧ dz) ( i2dλ∧ dλ). The first factor i2dz ∧ dz) is the pull-back of the

Kahler form ω0 of a flat Kahler metric defined on U , which is then of the formω0 = φ−2 ωL, for some positive function φ defined on U ; by using (1.19.6)again, φ is determined by: ρωL = κωL = ρω0 − ddc log φ = −ddc log φ. Itfollows that i

2dz ∧ dz = |Φ(z)|2 e2κtddct, for some nowhere vanishing holo-

morphic function Φ of z, whereas i2dλ ∧ dλ = |λ|2 dt ∧ dct, up to additional

terms involving the derivative of |s|, hence dz and dz, which disappear whenmultiplied by i

2dz ∧ dz; we then eventually get:

v0-univ0-uni (10.4.8) v0 = |Φ(z)|2|λ|2e2κtddct ∧ dt ∧ dct,

where Φ is holomorphic and nowhere vanishing. By plugging (10.4.7) and(10.4.8) into (10.4.6), we get (10.4.3). We then infer (10.4.5) from (10.4.2).

It is convenient to express the scalar curvature s as a function of ψ byusing the momentum profile Θ; we then have (cf. [109, Theorem A]):

Lemma 10.4.3. As a function of ψ, the scalar curvature of ωψ has thefollowing expression:

s-psis-psi (10.4.9) s =2κ− (ψΘ(ψ))′′

ψ,

where (ψΘ(ψ))′′ denotes the second derivative iwith respect to ψ.

Proof. Easy consequence of (10.4.5) (notice that Θ′(ψ) = ψ′′

ψ′ ; more

generally, any derivative with respect to ψ is equal to the derivative withrespect to t divided by ψ′ = Θ(ψ)).

propcalabiconstruction Proposition 10.4.1 (E. Calabi [45]). For any Hirzebruch-like ruled sur-face, an admissible Kahler metric on M0 is extremal if and only if the scalarcurvature s is of the form

affineaffine (10.4.10) s = αψ + β,

for real constants α, β. This happens if and only if there exist real constantsγ, δ, such that

calabiequation-thetacalabiequation-theta (10.4.11) Θ(ψ) =P (ψ)

ψ,

or, equvalently,

calabiequationcalabiequation (10.4.12) ψψ′ = P (ψ),

where P is the polynomial defined by

calabipolynomialcalabipolynomial (10.4.13) P (x) = − α

12x4 − β

6x3 + κx2 − γ

6x− δ

12.

Proof. Recall that a Kahler metric is extremal if and only if its scalarcurvature s is a Killing potential, i.e. the (unnormalized) momentum of ahamiltonian Killing vector field; by Lemma 10.3.1 ψ is a Killing potential;since both s and ψ only depends on t, the corresponding symplectic gradientare proportional to each other; they are then both Killing if and only if they


are multiple of each other by a constant; we thus get (10.4.10). By puttingtogether (10.4.9) and (10.4.10), we get the following differential equation

(10.4.14) (ψΘ(ψ))′′ = 2κ− αψ2 − βψ,which obviously integrates into ψΘ(ψ) = P (ψ), where P is the polynomialdefined by (10.4.13) and γ, δ are any two (real) constants.

Proposition 10.4.1 achieves the first step of Calabi’s construction, namelythe description of all extremal admissible Kahler metrics on M0, dependingupon 4 real parameters. The second and final step, namely the constructionof (admissible) extremal Kahler metrics on the whole of M , is summarizedby the following theorem: 10.3.2. We

thmcalabigen Theorem 10.4.1. Let M = P(1 ⊕ L) be any Hirzebruch-like ruled sur-faces of genus g and degree `. Then,

(i) If g = 0 or g = 1, any Kahler class Ωa,b contains an essentiallyunique admissible extremal Kahler metric.

(ii) If g > 1, denote by xκ the unique real root greater than one of theequation

boundbound (10.4.15) (κx2 + 4x− κ)2 − 4x ((2− κ)x+ (2 + κ))2 = 0,

where κ is given by (10.3.1). Then, a Kahler class Ωa,b contains an admis-sible extremal Kahler metric if and only if

christinaboundchristinabound (10.4.16)b

a< xκ

and it is then essentially unique.

(iii) In all cases, the scalar curvature of these admissible extremal Kahlermetrics is non-constant.

Proof. First notice that the expression “essentially unique” in (i) and(ii) means that ψ is well-defined up to translation of the parameter t, i.e.that gψ is uniquely defined up to the action of R>0 on M .

By Proposition 10.4.1, an admissible Kahler metric on M0 is extremal

if and only if Θ(ψ) = P (ψ)ψ , where P is the polynomial defined by (10.4.13).

Moreover, this metric is the restriction of a — still extremal — Kahlermetric on M if and only if the momentum profile Θ satsfies Conditions(B0) − (B1) − (B2) of Proposition 10.3.4. Now, Condition (B1) impliesP (a) = P (b) = 0, whereas Condition (B2) then implies P ′(a) = 2a andP ′(b) = −2b. We thus get the following constraints on the polynomial P :

P (a) = 0, P (b) = 0,

P ′(a) = 2a, P ′(b) = −2b,

P ′′(0) = 2κ

Pcond-uniPcond-uni (10.4.17)

(the additional fifth constraint simply means that the coefficient of x2 inP is κ). The polynomial P is actually entirely determined by Conditions(10.4.17), hence by the chosen Hirzebruch-like ruled surfaceM , via the pa-rameter κ, and by a, b, i.e. by the chosen Kahler class Ωa,b. It will be denotedby PΩa,b and called the Calabi polynomial or, following [6], the extremal poly-nomial, of Ωa,b. Notice that PΩa,b is defined for any Kahler class Ωa,b on any


Hirzebruch-like ruled surface, independently of the actual existence/non-existence of an extremal metric within Ωa,b. A simple computation showsthat PΩa,b has the following explicit expression:

Pab-uniPab-uni (10.4.18)

PΩa,b(x) = (x−a)(b−x)((2(a+b)−κ(b−a))x2+(4ab+κ(b2−a2))x+ab(2(a+b)−κ(b−a)))(b−a)(a2+4ab+b2)

.

Equivalently, the (normalized) coefficients α, β, γ, δ appearing in (10.4.13)are given by

α =12(2(a+ b)− κ(b− a))

(b− a)(a2 + 4ab+ b2),

β =12(−(a2 + b2) + κ(b2 − a2))

(b− a)(a2 + 4ab+ b2),

γ = ab β =12ab(−(a2 + b2) + κ(b2 − a2))

(b− a)(a2 + 4ab+ b2),

δ = a2b2 α =12a2b2(2(a+ b)− κ(b− a))

(b− a)(a2 + 4ab+ b2).

coef-unicoef-uni (10.4.19)

Conversely, Θ(ψ) :=PΩa,b

ψ clearly satisfies Conditions (B1)-(B2) and satisfies

Condition (B0) — hence determines an (extremal) admissible Kahler metricby Proposition 10.3.4 — if and only if the extremal polynomial PΩa,b(ψ)is positive or, equivalently, the degree 2 polynomial p defined by p(ψ) :=((2 + κ)a + (2 − κ)b)ψ2 + (κ(b2 − a2) + 4ab)ψ + ((2 + κ)a + (2 − κ) b)ab)remains positive for any ψ in (a, b).

If g = 0, then 0 < κ ≤ 2 and p(ψ) is clearly positive for any ψ in (a, b);if g = 1, then κ = 0 and we get the same conclusion: this proves Part (i) ofTheorem 10.4.1.

We now assume g > 1, so that κ < 0. By putting x = ba , the discriminant

of p is equal to a4fκ(x), with fκ(x) = ((κx2+4x−κ)2−4x((2−κ)x+(2+κ))2).Notice that (10.4.15) is the equation fκ(x) = 0. It is easily checked thatfκ(x) has a unique real zero, say xκ, greater than 1: this readily follows fromthe fact that fκ and its derivatives f ′κ, f ′′κ , f ′′′κ are all positive at infinity butnegative at x = 1. Moreover, xκ is certainly greater than the unique zerosgreater than 1 of f ′κ, f ′′κ , f ′′′κ , in particular than the one of f ′′′κ , which is equalto 1− 6

κ + 4κ2 ; we then have

xkappaxkappa (10.4.20) xκ > 1− 6

κ+

4

κ2

(recall that we are considering the case when κ < 0). If 1 < x < xκ,the discriminant of p is negative and PΩa,b(ψ) is then positive whenever ψbelongs to (a, b): we then get an extremal Kahler metric within the Kahlerclass Ωa,b. If x ≥ xκ, p has two real roots, whose product is equal to ab

and whose sum is equal to (−κx2−4x+κ)(2−κ)x+(2+κ) ; because of (10.4.20), the latter is

positive; moreover, the two roots of p are then positive; since their product isab, one of them certainly belongs to (a, b): the corresponding Kahler metricis then singular along the hypersurface determined by this value of ψ. Thisproves Part (ii) of Theorem 10.4.1.


By (10.4.19), α has the sign of 2(b + a) − κ(b − a) and is then is thenpositive in all cases, as κ ≤ 2. The scalar curvature s = αψ + β — cf.(10.4.10) — is then non-constant; this proves Part (iii) of Theorem 10.4.1.

Theorem 10.4.1 is due to E. Calabi for g = 0 and to C. Tønnesen-Friedman [194], [195] for g > 1; for g = 1, it can be viewed as a specialcase of theorems by A. Hwang [107] and by D. Guan in [95]. It can berephrased as follows:

thmcalabigen-bis Theorem 10.4.2. For any Hirzebruch-like ruled surface M = P(1 ⊕ L)of genus g and of degree `, let Ωa,b be a Kahler class on M . Then, Ωa,b

contains an admissible extremal Kahler metric if and only if its extremalpolynomial PΩa,b is positive on the open interval (a, b). If g = 0 or g = 1,this condition is satisfied for all Kahler classes. If g > 1, this condition issatisfied if and only if b

a < xκ, where xκ is the unique real solution greaterthan 1 of the equation (10.4.15).

remchristina Remark 10.4.1. Theorems 10.4.1-10.4.2 leaves open the question of theexistence/non-existence of non admissible extremal Kahler metrics on Hirzebruch-like ruled surfaces of genus g > 1, cf. [195]. A full answer of this questionwill be given below in Section 10.9.

remduality Remark 10.4.2. For any Hirzebruch-like ruled surfaceM = P(1⊕L), thechosen hermitian inner product h on L determines a bundle isomorphismfrom M to P(L∗ ⊕ 1); if τh denotes the hermitian duality from L to L∗,this map is defined by x = C (z, u) 7→ C (τh(u), u), for any z in C, any u inLy and any y in Σ. By composing this map with the natural identificationP(1⊕ L) = P(L∗ ⊕ 1), obtained by tensoring any x in P(1⊕ Ly) by L∗y, weget an involution, ι, in M , which is an anti-holomorphic diffeomorphism oneach fiber π−1(y) of π. This involution exchanges Σ0 and Σ∞, preserves M0

and, on M0 = L \ Σ0, is simply the transformation u 7→ ur2(u)

. The image

of the natural complex structure J of M by ι is the complex structure, sayJ , which coincides with −J on each fiber of π and with J on the horizontaldistribution determined by the Chern connection ∇. In particular, J andJ determine opposite orientations. If gψ is an admissible Kahler metric on

M0, then the pair (gψ, J) determines a hermitian structure whose Kahlerform, ωψ, has the following expression on M0:

omegatildeomegatilde (10.4.21) ωψ = ψ(t)ddct− ψ′(t) dt ∧ dct,

(where dc is still relative to J). Then, ωψ is no longer closed: we have instead

lee-deflee-def (10.4.22) dωψ = 2ψ′dt ∧ ddct = −2θ ∧ ωψ,

where θ — the so-called Lee form of the hermitian structure (g, J) — isgiven by

leelee (10.4.23) θ = −d logψ.

It follows that ωψ := ψ−2 ωψ is closed and, together with J , determines aKahler structure, whose riemannian metric is ψ−2 gψ. The image by ι of


the Kahler structure (ψ−2 gψ, J , ψ−2 ωψ) is the admissible Kahler structure

(gφ, J, ωφ), where φ = φ(t) is defined by

(10.4.24) φ(t) =1

ψ(−t).

Moreover, if gψ is extremal, i.e., by Proposition 10.4.1, if ψ satisfies (10.4.12)for the polynomial P defined by (10.4.13), then φ satisfies φφ′ = P ∗(φ) forthe “dual polynomial” P ∗ defined by

polynomial-dualpolynomial-dual (10.4.25) P ∗(x) = − δ

12x4 − γ

6x3 + κx2 − β

6x− α

12,

hence is extremal as well. Recall — cf. Section 5.5 — that an oriented 4-dimensional riemannian manifold is called selfdual, resp. antiselfdual, if theselfdual Weyl tensor W+, resp. the antiselfdual Weyl tensor W−, vanishesidentically and that a Kahler manifold of (real) dimension 4 is antiselfdualif and only if its scalar curvature vanishes identically. In particular, anyantiselfdual Kahler surface is extremal. In the present case, when gψ isan extremal admissible metric associated to the polynomial P , its scalarcurvature is sgψ = αψ+β and gψ is then antiselfdual if and only if α = β = 0.Similarly, sgφ = δφ + γ, and gφ is antiselfdual if and only if γ = δ = 0.

Now, ι∗gφ = ψ−2gψ; since ι reverse the orientation and W± are conformallyinvariant, we infer that gψ is selfdual if and only if γ = δ = 0. If gψ extendsto M , then ψ satisfes Conditions (A0)− (A∞) of Proposition 10.3.2 for thepair a, b and it is easily checked that φ then satisfies (A0)−(A∞) for the pair1b ,

1a . Then, gφ extends to M and [ωφ] = Ω 1

b, 1a

= 1ab Ωa,b (this is coherent

with the obvious observation that, in formulae (10.4.19), the transformationa 7→ 1

b , b 7→1a induces the transformation α 7→ δ, β 7→ γ, γ 7→ β, δ 7→ α).

From the uniqueness part of Theorem 10.4.1, we infer that φ(t) = 1abψ(t+c),

for some real number c, i.e. that ψ(t + c)ψ(−t) = ab. The momentumψ = ψ(t) of any extremal admissible Kahler metric on M , when normalized

by ψ(0) =√ab, then satisfies

(10.4.26) ψ(t)ψ(−t) = ab.

10.5. Bach-flat Kahler metrics on Hirzebruch-like rules surfaces:the Page metric

sspageThe Bach tensor of a riemannian manifold has been introduced in Chap-

ter 5. A Kahler manifold whose Bach tensor vanishes identically will becalled Bach-flat. According to Theorem 5.4.1, a Kahler manifold of (real)dimension 4 is Bach-flat if and only if it is extremal and the riemannianmetric g := s−2 g is Einstein on the open set where the scalar curvature s isnon-zero. This section is devoted to the search of Bach-flat Kahler metricsamong extremal admissible Kahler metrics on Hirzebruch-like ruled surfaces.

lemmabachflat Lemma 10.5.1. An extremal admissible Kahler metric on M0 determinedby a polynomial P as in Proposition 10.4.1 is Bach-flat if and only if the(normalized) coefficients α, β, γ, δ of P satisfy the condition

bachflatbachflat (10.5.1) αδ − βγ = 0.

10.5. THE PAGE METRIC 271

Proof. All Kahler metrics described by Proposition 10.4.1 are extremal:in view of Theorem 5.5.1 such a metric is then Bach-flat if and only theadditional condition (5.4.15) is satisfied, i.e. if and only if ddcs+ s ρ is pro-portional to the Kahler form ω. Since the scalar curvature s is a function oft, we have that ddcs = s′(t) ddct+s′′(t) dt∧dct, whereas the Ricci form ρ andthe Kahler form ω are given by (10.4.3) and (10.3.5) respectively. Condition(5.4.15) then reads

calabibachflatcalabibachflat (10.5.2)s′ + sw′

ψ=s′′ + sw′′

ψ′.

Recall that s = αψ + β, that w(t) = 2t− 12 logψ − 1

2 logψ′ and that ψψ′ =P (ψ), as the metric is extremal, where P is given by (10.4.13), cf. (10.4.12).Notice that from ψψ′ = P (ψ), we get

psisecondpsisecond (10.5.3) ψ′′ =P ′(ψ)ψ′

ψ− (ψ′)2

ψ=

(P ′(ψ)

ψ− P (ψ)

ψ2

)ψ′.

We then get s′ = αψ′ = αP (ψ)ψ , s′′

ψ′ = αψ′′

ψ′ = α(P ′(ψ)ψ − P (ψ)

ψ2

), w′ = κ −

12P ′(ψ)ψ , w′′

ψ′ = −12P ′′(ψ)ψ + 1

2P ′(ψ)ψ2 . By substituting in (10.5.2), this condition

now reads

calabibachflatbiscalabibachflatbis (10.5.4) 2αP (ψ)− (2αψ + β)P ′(ψ) +ψ

2(αψ + β)P ′′(ψ) + κ(αψ + β) = 0.

It turns out that all coefficients of the lhs of (10.5.4) — a polynomial ofdegree at most 4 — equal zero, except for the constant term, which is equalto −1

6(αδ − βγ).

lemmascalarcurvature Lemma 10.5.2. For any extremal e Kahler metric g on M0, associatedto the polynomial (10.4.13), the scalar curvature s, of the riemannian metricg := s−2 g — on the open set where the scalar curvature, s, of g is not zero— is given by

scalarcurvaturescalarcurvature (10.5.5) s = β3 + 12καβ + α2γ +α(αδ − βγ)

ψ.

In particular, s is constant if and only if either α = 0, meaning that s itselfis constant, or αδ − βγ = 0, meaning that g is Bach-flat by Lemma 10.5.1.

Proof. In general, for any two conformally related riemannian metricsg and g = φ−2 g, on a n-dimensional manifold, the scalar curvatures, s, of gand s of g are related to each other by

sconformalsconformal (10.5.6) s = φ2 s− 2(n− 1)φ∆φ− n(n− 1) |dφ|2,cf. e.g. [28, Chapter 2]. In the current case, where φ = |s| and n = 4, wethus get

sconformal-4sconformal-4 (10.5.7) s = s3 − 6s∆s− 12 |ds|2,From (10.4.1) and (10.4.2), and from (10.4.10)-(10.4.12), we infer

(10.5.8) |ds|2 = α2P (ψ)

ψ, ∆s = −αP

′(ψ)

ψ.

Then, (10.5.5) follows readily, as well as the last statement of Lemma 10.5.2.


Remark 10.5.1. The last statement of Lemma 10.5.2 is actually a quitegeneral fact for extremal Kahler surfaces, in view of the following proposi-tion, cf. [67]:

propderdzinski Proposition 10.5.1. For any extremal Kahler complex surface (M, g, J),denote by M0 the open set where the scalar curvature, s, does not vanish,and by M1 ⊂M0 the open set where s is not constant, i.e. where ds does notvanish. On M0, denote by s the scalar curvature of the conformal metricg = s−2g, by r its Ricci tensor, by r0 the trace-free part of r. Then, ds andds are related by

derdzinskiderdzinski (10.5.9) ds = −12s r0(ds).

At any point of M1, we then have r0 = 0 if and only if ds = 0. In particular,on M1, g is Bach-flat if and only if s is constant.

Proof. By definition of an extremal Kahler metric, ds is the g-dualof a (real) holomorphic vector field. By Lemma 1.23.4, we then have thatr(ds) = 1

2∆ds = 12d(∆s). Equivalently, r0(ds) = 1

2d(∆s) − 14sds. By using

(5.4.16) and (10.5.7), we then infer (on M0):

s r0(ds) =1

2s∆s− 1

4s2ds+ 2 (Dds, ds) +

1

2δs ds

=1

2d(s∆s)− 1

12ds3 + d(|ds|2)

= − 1

12ds.

We thus get (10.5.9). Now, r0 is trace-free and, as an operator, commuteswith J : since n = 4, this implies that at any point of M , r0 is zero oran isomorphism. The second assertion then readily follows from (10.5.9).Finally, since g is extremal, it is Bach-flat on M0 if and only if r0 = 0; inview of the above, this happens on M1 if and only if s is constant.

thmbachflat Theorem 10.5.1. Let M = P(1⊕ L) be a Hirzebruch-like ruled surfaceof genus g and degree `. Then:

(i) If g = 0 and ` = 1, i.e. if M is the first Hirzebruch surface F1 —equivalently M is the blow-up of P2 at a point — then M admits a Bach-flatKahler metric, say g, unique up to scaling and the action of H(M); if Ωa,b

is the corresponding Kahler class, then x := ba is the unique real root greater

than 1 of the equation

bachflat-pagebachflat-page (10.5.10) (x2 − 3)2 − 16x = 0.

Moreover, the scalar cuvature, s, of g is everywhere positive and the rie-mannian metric g := s−2 g is then an Einstein metric, of positive scalarcurvature, globally defined on F1, namely the Page metric, cf. Remark10.5.2.

(ii) If g = 0 and ` = 2, i.e. if M is the second Hirzebruch surface F2, thescalar curvature of all extremal admissible Kahler metrics on M is positiveeverywhere but M admits no Bach-flat Kahler metric.

(iii) If g = 0 and ` > 2, i.e. if M is the Hirzebruch surface F`, with` > 2, then M admits a Bach-flat Kahler metric, say g, unique up to scaling

10.5. THE PAGE METRIC 273

and the action of H(M); if Ωa,b is the corresponding Kahler class, then

x := ba is the unique real root greater than 1 of the equation

bachflat-page-ellbachflat-page-ell (10.5.11) ((2− `)x2 − (2 + `))2 − 4x((`− 1)x+ (`+ 1))2 = 0.

(iv) If g > 0, then M admits an admisssible Bach-flat Kahler metric,say g, unique up to scaling and the action of homotheties on M . If Ωa,b is

the corresponding Kahler class, then x := ba is the unique real root greater

than 1 of the equation

bachflat-kappabachflat-kappa (10.5.12) ((κ− 1)x2 − (κ+ 1))2 − x ((2− κ)x+ (κ+ 2))2 = 0,

where κ is defined by 10.3.1.(v) In Cases (iii) and (iv), the scalar curvature, s, of g vanishes on the

distance sphere, S√ab, determined by

(10.5.13) ψ =√ab

and M \ S√ab is the union of two connected open sets, namely U−, where

ψ <√ab, and U+, where ψ >

√ab. Denote by g−, resp. g+, the re-

striction of s−2 g to U−, resp. U+; then, g− and g+ are both complete,Einstein riemannian metric of negative scalar curvature; in particular, bothare asymptotically hyperbolic (up to a common rescaling).

Proof. From Lemma 10.5.1 and from formulae (10.4.19) we readilyinfer that an extremal admissible Kahler metric on M whose Kahler classis Ωa,b is Bach-flat if and only if x := b

a is a root of the equation (10.5.12),which is also the equation

(10.5.14) pκ(x) := (1−κ)2x4− (2−κ)2x3− 6x2− (2 +κ)2x+ (1 +κ)2 = 0,

hence a polynomial equation of degree 4, except when κ = 1, i.e. wheng = 0 and ` = 2.

If κ = 1, i.e. if g = 0 and ` = 2, the equation reduces to p1 := 4 −x(x+ 3)2 = 0, which has no real solution greater than 1: indeed, for x ≥ 1,p′1(x) = −3(x + 1)(x + 3) is negative, so that p1(x) is decreasing, withp1(1) = −12 < 0. Morever, by (10.4.19) with κ = 1, the restriction of the

scalar curvature s = αψ+β on Σ0 is equal to 12a(a+b)(b−a)(a2+4ab+b2)

, hence positive;

since α > 0 — cf. the proof of Theoerm 10.4.1 (iii) — we infer that s ispositive everywhere. This achieves the proof of (ii).

We can then assume that κ 6= 1, i.e. that pκ is a polynomial of degree4 with positive leading coefficient, whose successive derivatives are given byp′κ(x) = 4(1− κ)2x3 − 3(2− κ)2x2 − 12x− (2 + κ)2, p′′κ(x) = 12(1− κ)2x2 −6(2 − κ)2x − 12, p′′′κ (x) = 24(1 − κ)2x − 6(2 − κ)2. In particular, all thesederivatives tend to +∞ when x tends to +∞, whereas pκ(1) = −12 < 0,p′κ(1) = −24 < 0, p′′κ(1) = 6(κ2 − 4), p′′′κ (1) = 6κ(3κ− 4). It readily followsthat pκ has a unique real root greater than 1, say xκ.

If κ = 2, i.e. if g = 0 and ` = 1, (10.5.12) specializes to (10.5.10), whoseunique solution greater than 1 satisfies x2 > 3, as p2(

√3) < 0; it then follows

form (10.4.19) that all coefficients α, β, γ, δ are positive, so that s = αψ+ βand s = β3 + 24αβ + α2γ are both positive. This achieves the proof of (i).


If 0 < κ < 1, i.e. if g = 0 and ` > 2, any extremal Kahler, a fortiori,any Bach-flat Kahler metric can be made admissible by a suitable action ofH(M), whereas (10.5.12) specializes to (10.5.11); this proves (iii).

If κ ≤ 0, i.e. if g > 0, there is no guarantee a priori that a Bach-flat metric can be made admissible by the action of H(M) = C∗. On theother hand, the above reasoning shows that an extremal admissible Kahlermetric on M admits an admissible is Bach-flat if and only if its Kahlerclass is Ωa,b, whith x := b

a = xκ. In view of Theorem 10.4.1, it remainsto prove that, if κ < 0, i.e. if g > 1, xκ < xκ; for that, it is sufficientto check that fκ(xκ) is negative, where fκ is the function which appears inthe proof of Theorem 10.4.1; since xκ is a root of (10.5.12), we have thatfκ(xκ) = h(xκ), by setting h(x) := (κx2−4x−κ)2−4((1−κ)x2 +(1+κ))2 =−(x−1)((2−κ)x−(2+κ))((2−3κ)x2 +4x+2+3κ); since κ is now assumedto be negative, it is a simple matter to check that h(x) is negative for anyx ≥ 1. This completes the proof of (iv).

In order to prove (v), we first observe, by using formulae (10.4.19), thatthe Bach-flat condition αδ − βγ = 0 on M can be rewritten as

bachflat-compactbachflat-compact (10.5.15) β2 = abα2,

as δ = a2b2 α and γ = ab β, and that β is negative for any values of a, b —0 < a < b — if κ < 1, whereas α is always positive, as we already observed;it follows that if κ < 1, the zero locus of the scalar curvature s = αψ + β ofany admissible Bach-flat metric gψ is determined by ψ = −β

α =√ab; denote

it by S√ab. Since ds = αdψ does not vanish on M0, in particular on S√ab,

s can be chosen as a defining function of S√ab, so that g := s−2 g — whichis Einstein as g is Bach-flat — is complete on each of the two connectedcomponents, U− and U+, of M \ S√ab. In particular, the (constant) scalarcurvature s of g is negative. This can also be checked directly by using(10.5.5), which, in the Bach-flat case, reduces to

(10.5.16) s = β3 + 12καβ + α2γ.

By using (10.4.19), γ = ab β and (10.5.15), we thus get

(10.5.17) s = βα(2abα+ 12κ) = αβ12((a+ b)2(4ab+ κ(b2 − a2))

(b− a)(a2 + 4ab+ b2),

which has the opposite sign of κx2κ + 4xκ− κ since αβ is negative. To check

that this expression is positive, it is sufficent to show that xκ is smaller thanthe positive root, say x0, of the equation κx2+4x−κ = 0. By eliminating κ =4x0

1−x20

in pκ(x0) we obtain pκ(x0) =(x2

0+4x0+1)2(x0−1)2

(x0+1)2 , which is positive; this,

shows that x0 > xκ as required. Finally, g+ and g− are both asymptoticallyof (negative) constant sectional curvature, i.e., up to a common rescaling,both asymptotically hyperbolic; this is because both are Einstein, whereasthe norm, |W |, of their Weyl tensor relatively to g±, is related to the norm,

relative to g, of the Weyl tensor, W , of g, by |W | = s2 |W |, which tends to

0 when ψ tends to√ab as |W | is bounded.

rempage Remark 10.5.2. The Einstein metric g has been firstly discovered andconstructed — in a different setting — by D. Page in [158], and is thususually called the Page metric. For a long time, it has remained the only

10.6. WEAKLY SELFDUAL KAHLER METRICS 275

known example of a hermitian, non-Kahler, Einstein metric defined on acompact (smooth) complex surface until the discovery of the Chen–LeBrun–Weber metric on the blow-up of P2 at two points, cf. Remark 5.5.2.

remalphadelta Remark 10.5.3. The condition αδ−βγ = 0, as a condition that the cor-responding Calabi extremal Kahler metric be conformally Einstein appearsin [49].

10.6. Weakly selfdual Kahler metrics and hamiltonian 2-formsssWBF

In general, a Kahler manifold (M, g, J, ω) of any (real) dimension n =2m > 2 is said to be Bochner-flat if its Bochner tensor, WK — see AppendixA — vanishes identically, and weakly Bochner-flat if WK is co-closed withrespect to the Levi-Civita connection D of g, meaning that

deltaWKdeltaWK (10.6.1) δDWK = 0,

where δD denotes the codifferential with respect to D applied to W− whenthe latter is regarded as a Λ2M -valued 2-form. It then follows from thegeneral Bianchi identity δDR = 0 applied to the riemannian curvature R—cf. 1.18.4 — and from (A.2.6) that (10.6.1) is equivalent to the followingcondition for the Ricci form ρ:

DricciDricci (10.6.2) DXρ =1

4(m+ 1)(ds ∧ JX[ − dcs ∧X[ + 2 ds(X)ω),

for any vector field X, cf. [4] for details. At this point, it is convenient tointroduce the normalized Ricci form, ρ, defined by

(10.6.3) ρ = ρ− s

2(m+ 1)ω,

and to observe that the Ricci form ρ satisfies (10.6.2) if and only if thenormalized Ricci form ρ is a solution to the following equation

hamiltonianformhamiltonianform (10.6.4) DXϕ =1

2(dtrϕ ∧ JX[ − dctrϕ ∧X[),

for any vector field X: here, the unknown ϕ is a real J-invariant 2-form and,we recall, trϕ stands for the trace of ϕ with respect to ω, i.e. trϕ = (ϕ, ω);equivalently, trϕ is the trace of the associated hermitian operator Φ, cf.Section 1.12. In general, a solution to (10.6.4) is called a hamiltonian form.The above discussion can be summarized by the following statement:

proptautologic Proposition 10.6.1. A Kahler manifold is weakly Bochner-flat if andonly if its normalized Ricci form ρ is a hamiltonian 2-form.

Hamiltonian 2-forms have nice properties — this is why we substituted ρto ρ and Equation (10.6.4) to Equation (10.6.2) — in particular the followingone. For any J-invariant 2-form ϕ, denote by σr(ϕ), r = 1, . . . ,m, the r-thelementary symmetric function of the roots of the corresponding hermitianoperator Φ: in particular, σ1(ϕ) = trϕ, whereas σm(ϕ) is the pfaffian, pfϕ,

of ϕ, alternatively defined by pf(ϕ) = ϕ∧m

ω∧m , cf. Section 1.12. We then have:

propmomenta Proposition 10.6.2 ([4] Proposition 3). If ϕ is a hamiltonian 2-form,the σr(ϕ), r = 1, . . . ,m, are pairwise Poisson communting Killing potentials.


For any hamiltonian 2-form ϕ, the 2-form ϕ := ϕ + σ1(ϕ)ϕ is closed(easy consequence of (10.6.4)): it will be referred to as the associated closed2-from of ϕ and is characterized by the equation

closedhamiltonianclosedhamiltonian (10.6.5) DX ϕ =1

2(m+ 1)(dtrϕ ∧ JX[ − dctrϕ ∧X[ + 2dtrϕ ω).

If the Kahler metric is weakly Bochner-flat the normalized Ricci form ρis a Hamiltonian 2-form and σ1(ρ) = s

2(m+1) is then a Killing potential. We

then have:

propWBFextremal Proposition 10.6.3. Weakly Bochner-flat Kahler metrics — a fortioriBochner-flat Kahler metrics — are extremal.

When M is a Kahler complex surface, the Bochner tensor coincides withthe anti-selfdual Weyl tensor W−, cf. Section A.4: Bochner-flat Kahlermetrics are then selfdual Kahler metrics and weakly Bochner-flat Kahlermetrics are then simply called weakly selfdual. By Proposition 10.6.3 weaklyselfdual — a fortiori selfdual — Kahler complex surfaces are then extremal.

The order of a hamiltonian 2-form ϕ is defined via the following

proporder Proposition 10.6.4 ([3] Proposition 13). Let ϕ be a hamiltonian 2-formdefined on a (connected) Kahler manifold (M, g, J, ω), σr(ϕ), r = 1, . . . ,m,the associated Killing potential as in Proposition 10.6.2, Kr = gradωσr(ϕ) =Jgradgσr(ϕ) the corresponding (hamiltonian) Killing vector fields. Then,there exists an integer `, 0 ≤ ` ≤ m, such that K1, . . . ,K` are independenton a dense open set M0 of M .

Hamiltonian 2-forms of order 0 are simply parallel 2-forms; in particular,the Kahler form ω is a hamiltonian 2-form of order 0. In the opposite case,when ϕ is a hamiltonian 2-form of maximal order m, the tangent bundle TMis trivialized over M0 by the 2m vector fields K1, . . . ,Km, JK1, . . . , JKm andM is then a toric Kahler manifold of a very specific type, called orthotoric,whose local structure is described in [3, Section 3.4] (the main specific fea-ture of orthotoric Kahler structure of complex dimension m is that theylocally depend on m real functions of one real variable, while a general toricKahler metric locall depends on one function of m real variables). In theintermediate cases, when 0 < ` < m, M0 is foliated by totally geodesicorthotoric complex submanifolds of complex dimension `, [4, Proposition 8and Theorem 1].

Hamiltonian 2-forms appear in many contexts of the Kahler geometry,in particular in the general Calabi-like constructions described in [3], [5],[6], [7], hence also in the framework of the Hirzebruch-like ruled surfacesconsidered in this chapter.

A Hamiltonian 2-form, ϕ, defined on a Hirzebruch-like ruled M equippedwith an admissible Kahler metric ωψ, is called admissible if the restrictioncorresponding closed 2-form ϕ = ϕ+σ1(ϕ)ωψ on M0 is of the form ϕ = ddcv,where v = v(t) is a real function which factors through t. We then have:

propcalabihamiltonian Proposition 10.6.5. Let M = P(1⊕ L) be a Hirzebruch-like ruled sur-faceof genus g and degree ` and let gψ be an admissible Kahler metric onM . Then, the space of admissible Hamiltonian 2-forms with respect to gψ


is generated by the Kahler form ωψ and by the Hamiltonian 2-form ϕ0 oforder 1 which, on M0, is defined by

(10.6.6) ϕ0 = ψψ′ dt ∧ dct = ψdψ ∧ dct.

Proof. Let ϕ be any admissible hamiltonian 2-form with respect to gψand let ϕ = ϕ + σ1(ϕ)ωψ be the associated closed 2-form. We then havethat ϕ = v′ddct + v′′dt ∧ dct. The remaining part of the argument thenrequires somme identities given by the following lemma, where D denotesthe Levi-Civita connection of the metrique gψ.

Lemma 10.6.1. For any admissible Kahler metric (gψ, ωψ) on M0, wehave

DX(dt ∧ dct) =1

2ψ(dt ∧ JX[ +X[ ∧ dct)

− (ψ′

ψ+ψ′′

ψ′) dt(X) dt ∧ dct,

DXdtdctDXdtdct (10.6.7)

DX(ddct) = − ψ′

2ψ2(dt ∧ JX[ +X[ ∧ dct)

+ dt(X)((ψ′

ψ

)2

dt ∧ dct− ψ′

ψddct

),

DXddctDXddct (10.6.8)

for any vector field X on U .

Proof. The dual 1-form, T [, of the vector field on M0 generated by thenatural S1-action is given by T [ = ψ′ dct. Since T is a Killing vector field,we have that DT [ = 1

2dT[. We then infer

DdctDdct (10.6.9) Ddct =1

2ddct− 1

2

ψ′′

ψ′(dt⊗ dct+ dct⊗ dt),

i.e.

DXdctDXdct (10.6.10) DXdct =

1

2ιXdd

ct− 1

2

ψ′′

ψ′(dt(X) dct+ dct(X)dt),

for any vector field X. Since DJ = 0, we also get

DXdtDXdt (10.6.11) DXdt =1

2ιJXdd

ct− 1

2

ψ′′

ψ′(dt(X) dt+ dct(X)dct).

From (10.6.10) and (10.6.11) we readily obtain (10.6.7). By using Dωψ = 0and (10.6.7) we now readily obtain (10.6.8).

From (10.6.7) and (10.6.8), we infer the following expression for thecovariant derivative of ϕ = v′ ddct+ v′′ dt ∧ dct:

DX ϕ = (v′′

2ψ− v′ψ′

2ψ2)(dt ∧ JX[ − dct ∧X[)

+ (v′′ − v′ψ′

ψ) dt(X) ddct

+ (v′′′ − v′′(ψ′

ψ+ψ′′

ψ′) + v′(

ψ′

ψ)2) dt(X) dt ∧ dct.

DXtildevarphiDXtildevarphi (10.6.12)


Now, ϕ is hamiltonian if and only if ϕ satisfies (10.6.5), with m = 2. Inparticular, we must have

v′′

ψ− v′ψ′

ψ2=v′′′

ψ′− v′′( 1

ψ+

ψ′′

(ψ′)2) + v′

ψ′

ψ2,

which is clearly solved by

vprimevprime (10.6.13) v′ = Aψ2 +Bψ,

for some real constants A,B. By substituting this value of v′ in (10.6.12),we get

(10.6.14) DX ϕ =ψ′

2(dt ∧ JX[ − dct ∧X[ + 2 dt(X)ωψ).

On the other hand, the trace of ϕ = v′ddct + v′′ dt ∧ dct with respect toωψ = ψ ddct+ ψ′ dt ∧ dct is given by

(10.6.15) trϕ =v′

ψ+v′′

ψ′

(cf. Lemma 10.4.1). In the current situation when v′ is given by (10.6.13)we then get

(10.6.16) trϕ = 3Aψ + 2B.

It follows that ϕ is a hamiltonian 2-form if and only if ϕ = ddcv, withv′(t) = Aψ2(t) + Bψ(t), for any constants A,B. Equivalently, ϕ = ϕ −trϕ3 ωψ = Aψψ′ dt∧ dct+ B

3 ωψ. This completes the proof of the first part ofProposition 10.6.5. The second part directly follows from (10.4.12).

lemmanormalizedricci Lemma 10.6.2. For any extremal admissible Kahler metric gψ, the nor-malized Ricci form has the following expression

normalizedriccibisnormalizedriccibis (10.6.17) ρ =α

6P (ψ) dt ∧ dct+

β

12ωψ +

γ

12ωψ,

where, we recall, ωψ := 1ψ dd

ct− ψ′

ψ2 dt∧dct is the (antiselfdual) Kahler form

introduced in Remark 10.4.2.In particular, ρ is a hamiltonian 2-form — equivalently, gψ is weakly

selfdual — if and only if

(10.6.18) γ = 0.

Proof. The Ricci form ρ of any admissible Kahler metric (g, ωψ) on

M0 is of the form ρ = w′ ddct + w′′ dt ∧ dct, with w′(t) = κ − 12ψ′

ψ −12ψ′′

ψ′ .

If the Kahler metric is extremal, we have that ψψ′ = P (ψ), where P is

the polynomial defined by (10.4.13) and we then have w′ = κ − 12P ′(ψ)ψ ,

w′′ = 12(P

′(ψ)ψ2 − P ′′(ψ)

ψ ). We then get

rhoexplicitrhoexplicit (10.6.19) ρ = (α

6ψ +

β

4+

γ

12ψ2)ψ ddct+ (

α

3ψ +

β

4− γ

12ψ2)ψ′ dt ∧ dct,

hence

rhotildeexplicitrhotildeexplicit (10.6.20) ρ = (β

12+

γ

12ψ2)ψ ddct+ (

α

6ψ +

β

12− γ

12ψ2)ψ′ dt ∧ dct,


which is the same as (10.6.17). By Proposition 10.6.5, ψψ′ dt ∧ dct and ωψare both hamiltonian 2-forms, while ωψ is not: the rhs of (10.6.17) is thena hamiltonian 2-form if and only if γ = 0.

Remark 10.6.1. It easily follows from (10.6.17) that the pfaffian of thenormalized Ricci form ρ of any Calabi extremal metric is given by

pfaff-riccipfaff-ricci (10.6.21) pf(ρ) =1

144

(β (2αψ + β) + γ (

2α

ψ− γ

ψ4)

).

Then, pf(ρ) is a Killing potential if and only if it is an affine function ofψ, hence if and only if γ = 0 or, by Lemma 10.6.2, if and only if ρ is ahamiltonian 2-form.

remalter Remark 10.6.2. We here present a more direct argument for the laststatement of Lemma 10.6.2. For that, it is convenient to simultaneouslyconsider the (admissible, extremal) Kahler structure (g = gψ, J, ωψ) and thedual Kahler structure (g = ψ−2g, J , ωψ) introduced in Remark 10.4.2. Theanti-selfdual Weyl tensor W− of g relatively to the orientation determinedby J is then the selfdual Weyl tensor of g — which is also the selfdualWeyl tensor of g by conformal invariance — relatively to the orientationdetermined by J . Now, the selfdual Weyl tensor of any Kahler complexsurface is given by (5.5.22) and we thus get the following expression forW−:

calabiW-calabiW- (10.6.22) W− =κg8ωψ ⊗ J −

κg12

Π−

where ωψ is the Kahler form (10.4.21) of the hermitian pair (g, J) introducedin Remark 10.4.2, and where we have set

calabikappacalabikappa (10.6.23) κg :=sgψ2

=δ

ψ3+

γ

ψ2.

Here, sg denotes the scalar curvature of g = ψ−2g, which is given by

sbargsbarg (10.6.24) sg =δ

ψ+ γ;

this identity in turn can be obtained either from g = ι∗gφ — cf. Remark

10.4.2 — so that sg = ι∗sgφ = ι∗(δφ+γ) = δ+ δψ , or by using the general for-

mula (10.5.6), which here specializes to sg = ψ2s−6ψ∆ψ−12|dψ|2, whereas,

by (10.4.1) and (10.4.2), ∆ψ = (−(ψ′

ψ + ψ′′

ψ′

)= −P ′(ψ)

ψ and |dψ|2 = ψ′ =

P (ψ)ψ ; we then get sg = αψ3 + βψ2 + 6P ′(ψ)− 12P (ψ)

ψ , which clearly implies

(10.6.24) (κg is actually the so-called conformal scalar curvature of the her-mitian pair (g, J), i.e. the scalar curvature of the canonical Weyl connectionattached to (gψ, J), cf. e.g. [10]). In the rhs of (10.6.22), Π− must beinterpreted as follows: for any pair X,Y of vector fields, Π−(X,Y ) is theanti-selfdual part of X∧Y , viewed as an endomorphism of the tangent spacevia the metric g; then, Π− is clearly parallel with respect to the Levi-Civitaconnection, D, of g. For convenience, we introduce auxiliary local sections,ω2, ω3, of Λ−M in such a way that ω1 := ωψ, ω2, ω3 form a (local, normalized)direct orthonormal frame of Λ−M and we denote by I1 = J , I2, I3 the cor-responding almost-complex structures with respect to g, so that I1I2 = I3,


I2I3 = I1, I2I3 = I1 and Π− = 12(ω1⊗ I1 +ω2⊗ I2 +ω3⊗ I3). We then have

the following general fact, cf. e.g. [10]:

Lemma 10.6.3. The covariant derivative of ωψ with respect to the Levi-Civita connection, D, of g is given by

DomegaDomega (10.6.25) Dωψ = −I3θ ⊗ ω2 + I2θ ⊗ ω3

where θ denotes the Lee form of the hermitian structure (g, J , ωψ).

Proof. It is equivalent to prove that the covariant derivative of J = I1

is given by DJ = −I3θ⊗ I2 + I2θ⊗ I3. Since the norm of J is constant, DJis necessarily of the form α ⊗ I2 + β ⊗ I3, for some (smooth) real 1-formsα, β. Moreover, since J is integrable, we have that DJX J = JDX J , for anyvector field X, cf. Proposition 1.1.2. We then have that α(X) = β(JX),i.e. β = I1α, so that DJ = −I3θ

′ ⊗ I2 + I2θ′ ⊗ I3 for some real 1 -form

θ′. By contracting this expression with respect to g, we infer δωψ = 2Jθ′.Since ωψ is selfdual with respect to the orientation induced by J , we inferthat dωψ = −2θ′ ∧ ωψ. This identifies θ′ with the Lee form θ defined by(10.4.22).

From (10.6.25), we easily infer

DW− =dκg12⊗ ω1 ⊗ I1 −

dκg24⊗ ω2 ⊗ I2 −

dκg24⊗ ω3 ⊗ I3

− κg8I3θ ⊗ ω2 ⊗ I1 +

κg8I2θ ⊗ ω3 ⊗ I1

− κg8I3θ ⊗ ω1 ⊗ I2 +

κg8I2θ ⊗ ω1 ⊗ I3,

(10.6.26)

and, upon contracting the above,

δDW− = − 1

12I1(dκg − 3κg θ)⊗ I1

+1

24I2(dκg − 3κg θ)⊗ I2 +

1

24I3(dκg − 3κg θ)⊗ I3;

deltaW-deltaW- (10.6.27)

It follows that the Kahler pair (g, J) is weakly seldual if and only if the 1-form dκg− 3κg θ is identically zero. From (10.4.23) and (10.6.23) we readilyinfer:

(10.6.28) dκg − 3κg θ = γdψ

ψ3,

which is evidently identically zero if and only if γ = 0.

When considering weakly selfdual Kahler metrics on Hirzebruch-likeruled surfaces, we get

thmwsd Theorem 10.6.1. Let M = P(1⊕L) be a Hirzebruch-like ruled surfaceoggenus g and degree `.

(i) If g = 0 and ` = 1, i.e. if M is the first Hirzebruch surface F1,then M admits a weakly selfdual Kahler metric, unique up to scaling andthe action of H(M), its Kahler class is Ωa,b with

wsd-abwsd-ab (10.6.29)b

a=√

3,


and the momentum function of the admissible represntative in Ωa,b is given,

up to translation 4 of t, by

wsd-psiwsd-psi (10.6.30) ψ(t) =√ab

(a+ b e2t

b+ a e2t

) 12

,

whereas the momentum profile is given by

wsd-Thetawsd-Theta (10.6.31) Θ(x) =(x2 − a2)(3a2 − x2)

2a2 x,

for x in the interval [a, b =√

3 a]. In particular, the associated Calabipolynomial is then

wsd-Pwsd-P (10.6.32) P (x) =1

a2(x2 − a2)(3a2 − x2),

whose normalized coefficients are

wsd-coefwsd-coef (10.6.33) α =6

a2, β = γ = 0, δ = 18a2.

The normalized Ricci form is then the hamiltonian 2-form

wsd-ricciwsd-ricci (10.6.34) ρ =1

2a2(ψ2 − a2)(3a2 − ψ2) dt ∧ dct

and the pfaffian pf(ρ) of ρ is equal to 0.(ii) If g = 0 and ` > 1, i.e. if M = F` with ` > 1, M admits no weakly

selfdual Kahler metric.(iii) If g > 0, then M admits no admissible weakly selfdual Kahler met-

ric.

Proof. By Lemma 10.6.2, an admissible Kahler metric gψ is weaklyselfdual if and only if γ = 0; from (10.4.19) this can only occur if κ = 1,hence g = 0 and ` = 1, or M = F1, and if b2 = 3a2. The remaining normal-ized coefficients of the Calabi polynomial P (ψ) are then given by (10.6.33)and P (ψ) itself by (10.6.32). Since, P (ψ) is a quadratic function of ψ2,the differential equation ψψ′ = P (ψ) is easily solved by the function ψ(t)given by (10.6.30), up to translation of t and (10.6.31) is easily deduced

from (10.6.30) or, more simply, by the fact that Θ(x) = P (x)x for any ex-

tremal admissible Kahler metric. Finally, (10.6.34) is a direct consequenceof (10.6.17), with β = γ = 0 and pf(ρ) readily follows from (10.6.34) or from(10.6.21).

Remark 10.6.3. An analytic argument for the existence of weakly self-dual Kahler metrics on F1 appears in [111].

It turns out that statement (iii) in Theorem 10.6.1 can be replaced by:“If g > 0, then M admits no weakly selfdual Kahler metric (admissible ornot)” This follows either from Theorem 10.9.1, or from the following one (aweaker version of Theorem 10.6.2 previously appeared in [110]):

4Here t is chosen in such a way that ψ(0) =√ab, hence ψ(t)ψ(−t) = ab, cf. Remark

10.4.2.


thmWBF Theorem 10.6.2 ([3] Theorem 5). A (smoth) compact weakly selfdualKahler complex surface is either of constant scalar curvature — hence Kahler-Eisntein or, locally, the product of two Riemann surfaces of constant Gausscurvature — or isomorphic to a Calabi weakly selfdual Kahler metric on thefirst Hirzebruch surface F1.

For the proof, we refer the reader to [3]. This result in turn can becompared with the following theorem by B.-Y. Chen [54], cf. also [67,Theorem1]:

Theorem 10.6.3. A (smooth) compact selfdual Kahler complex surfaceis either of constant holomorphic sectional curvature or locally the productof two Kahler Riemann surfaces of opposite constant curvature

and its generalization by R. Bryant [41] (first stated — with an incom-plete argument however — in [113]; a revised version appears in [114]; analternative argument can be found in [5]):

Theorem 10.6.4. A (smooth) compact Bochner flat Kahler manifold iseither of constant holomorphic sectional curvature or locally the product oftwo Kahler manifolds of opposite constant holomorphic sectional curvature.

For the proof we refer the reader to the original papers.

Remark 10.6.4. In all results mentioned above it is essential to considersmooth Kahler manifolds, as there are many more compact examples if weallow for Kahler metrics with orbifold singularities. In particular it is shownin [41] — cf. also [63] — that each weighted projective space admits aBochner flat Kahler metric, for any (relatively prime) weights a0, a1, . . . , am,whose scalar curvature is non-constant except in the standard case whenthe weights are 1, 1, . . . , 1 and the metric is the Fubini-Study metric. Otherexamples appear in [5].

10.7. The Futaki character of admissible ruled surfacesssfutakiuni

Define T by T := −J T , where, we recall, T denotes the generator ofthe S1-action on the Hirzebruch-like ruled surface M = P(1 ⊕ L) induced

by the natural S1-action on L. Then T is a (real) holomorphic vector field,whose real potential with respect to any admissible metric gψ is equal to themomentum of T , hence to ψ, up to an additive constant. In this section,

we compute FΩa,b(T ), where FΩa,b =FΩa,b

VΩa,bdenotes the (normalized) Futaki

character of M with respect to the Kahler class Ωa,b, cf. Section 4.12.We then have the following proposition, which is a particular case of [6,Proposition 6]:

propfutakiruled Proposition 10.7.1. For any Hirzebruch-like ruled surface M of genusg and degree `, and for any Kahler class Ωa,b, we have

futakiunifutakiuni (10.7.1) FΩa,b(T ) =2

3

(b− a)

(a+ b)2(2(a+ b)− κ (b− a)) ,

where κ is defined by (10.3.1). In particular,

futakipositivefutakipositive (10.7.2) FΩa,b(T ) > 0,

for any genus g and any degree `..

10.7. THE FUTAKI CHARACTER OF ADMISSIBLE RULED SURFACES 283

Proof. The real potential of T with respect to an admissible metricgψ is equal to the part of the momentum ψ which integrates to 0, hence to

ψ − ψ. By the very definition of FΩa,b — see Section 4.12 — we then have

futakicomputefutakicompute (10.7.3) FΩa,b(T ) =1

Va,b

∫Msgψ(ψ − ψ) vgψ =

1

Va,b

∫Mψsgψ vgψ − sgψ ψ,

for any admissible Kahler metric gψ in Ωψ, where Va,b denotes the totalvolume of M with respect to Ωa,b.

From (10.2.5), we readily infer that

volumeunivolumeuni (10.7.4) Va,b =

∫M

Ωa,b · Ωa,b

2= 2π2` (b2 − a2).

Similarly, by using (10.3.24). we infer that the total scalar curvature, Sa,b,determined by the Kahler class Ωa,b, is given by

Sa,b = 4π

∫Mc1(M) ∧ Ωa,b

= 8π2`((a+ b) + κ(b− a)).

(10.7.5)

The mean scalar curvature s :=Sa,bVa,b

is then given by

meansmeans (10.7.6) s =4 ((a+ b) + κ (b− a))

(b2 − a2).

The computation of the rhs of (10.7.3) relies on the following simplegeneral fact. Let’s call invariant a (real) function f on M which can beexpressed on M0 = L \ (Σ0 ∪ Σ∞) as a function of t. We then have

lemmaft Lemma 10.7.1. For any invariant real function f = f(t) and for anyadmissible metric g = gψ, we have

fintegralfintegral (10.7.7)

∫Mf vg = 4π2`

∫ +∞

−∞f ψ ψ′ dt.

Proof. This readily follows from

(10.7.8) vg = ψψ′ddct ∧ dt ∧ dct = ψψ′ dt ∧ η ∧ π∗ωΣ,

for any admissible metric, and the fact thatt, by (10.3.3) the contributionof dct in the integral is equal to 2π, whereas by (10.3.4), the contribution ofddct is equal to 2π`.

From (10.4.5) and by using (10.7.7), we get∫Msg ψ vg = 4π2`

∫ +∞

−∞2ψ(w′ψ)′ dt

= 4π2` 2w′ψ2|+∞−∞ − 4π2`

∫M

2w′ψψ′ dt.

We already observed in Remark 10.3.3 that limt→−∞w′(t) = κ − 1 and

limt→+∞w′(t) = κ+ 1. We then have 2w′ψ2|+∞−∞ = 2(b2− a2)κ+ 2(a2 + b2),

whereas ∫ +∞

−∞2w′ψψ′ dt =

∫ +∞

−∞(2κψψ′ − (ψ′)2 − ψψ′′) dt

= κψ2|+∞−∞ − ψψ′|+∞−∞ = κ(b2 − a2),


as limt→±∞ ψ′ = 0; we thus get

spsiintegralspsiintegral (10.7.9)

∫Msgψ vg = 4π2` ((b2 − a2)κ+ 2(a2 + b2)).

We similarly get

(10.7.10)

∫Mψ vg = 4π2`

∫ ∫ +∞

−∞ψ2ψ′ dt = 4π2`

(b3 − a3)

3,

so that, by using (10.7.4),

meanpsimeanpsi (10.7.11) ψ =2

3

(b3 − a3)

b2 − a2).

By using (10.7.6), we then have

barsbarpsibarsbarpsi (10.7.12) sgψ ψ =8

3

(b3 − a3)((a+ b) + κ (b− a)

(b2 − a2)2.

By plugging (10.7.9), (10.7.12) and (10.7.4) in (10.7.3), we eventually get(10.7.1), where the rhs is clearly positive for any genus g and any degree `(recall that ` is assumed to be positive).

Corollary 10.7.1. An Hirzebruch-like ruled surface of any genus andany degree carries no Kahler metric of constant scalar curvature.

Proof. By Theorem 4.12.1, the existence of a Kahler metric of constantscalar curvature in Ωa,b implies that the corresponding Futaki characterFΩa,b is trivial. Because of (10.7.2), this cannot happen for admissible ruledsurfaces.

remhfkg3 Remark 10.7.1. Formula (10.7.1) fits with the general formula given in

[6] for all admissible projective bundles. Note that FΩa,b(T ) can also bewritten as

futakialphafutakialpha (10.7.13) FΩa,b(T ) =(b− a)2(a2 + 4ab+ b2)

18(a+ b)2α,

meaning that FΩa,b(T ) is a positive multiple of α, which, we recall, is the(normalized) leading coefficient of the Calabi polynomial PΩa,b , as given by(10.4.19). This too fits with [6].

10.8. The extremal vector field of a Hirzebruch-like ruled surfacessextremalvf

For a Hirzebruch-like ruled surface M = P(1 ⊕ L) of genus g and de-gree `, a natural maximal compact subgroup of the automorphism groupH(M) = H0(M) is G, with G = U(2)/µ` if g = 0, and G = S1 if g > 0, cf.Propositions 10.2.2 and 10.2.3. In all cases, the center of G is the S1-actioninduced by the natural S1-action on L, whose generator is denoted by T .

Notice that admissible Kahler metrics are G-invariant, so that MadmΩ is

a subspace of MGΩ .

For any admissible metric gψ on M , the center of the corresponding

space of Killing potentials PGgψ for the Poisson bracket — cf. Section 4.13

— is then the space generated by ψ and the constant 1.It follows that, for any Kahler class Ωa,b on M , the Killing part of the

scalar curvature is of the form Aψ+B, where A,B are constants depending


on a, b, and the extremal vector field ZGΩa,b is then equal to AT . More

explicitely, we have

propZuni Proposition 10.8.1. For any Hirzebruch-like ruled surface M = P(1⊕L) of genus g and degree `, and for any Kahler class Ωa,b, let G be a maximalcompact subgroup of Hred(M,J) as above. Then, for any admissible Kahlermetric gψ in MG

Ωa,bthe Killing part of the scalar curvature of gψ is given by

PiGsPiGs (10.8.1) ΠGg (sgψ) = αψ + β =

2κ− P ′′Ωa,b(ψ)

ψ,

where α, β are the first two normalized coefficients of the Calabi polynomial

PΩa,b, whose explicit expression is given by (10.4.18), and κ = 2(1−g)` is the

coefficient of ψ2 in PΩa,b, cf. (10.4.17). Moreover, the extremal vector fieldis given by

(10.8.2) ZΩa,b = αT.

Proof. We already know that ΠGg (sgψ) = Aψ + B, for some real con-

stants A,B. These are determined by the following two relations. The firstone comes from the fact that the Futaki character can be computed by onlyconsiderig the Killing part of sgψ ; we then get

FΩa,b(T ) =1

Va,b

∫Msgψ(ψ − ψ) vg

=1

Va,b

∫M

ΠGg (sgψ) (ψ − ψ) vg

=A

Va,b

∫Mψ(ψ − ψ) vg

(10.8.3)

where the last integral can be computed as we already did, by using (10.7.7)which readily implies

(10.8.4)

∫Mψr vg = 4π2`

(br+2 − ar+2)

r + 2,

for any real number r 6= −2, whereas

(10.8.5)

∫Mψ−2 vg = 4π2` log

b

a.

We then easily get

(10.8.6) A =18(a+ b)2

(b− a)2(a2 + 4ab+ b2)FΩa,b(T ) = α,

cf. (10.7.13). The second relation comes from the fact that∫M sgψ vg =∫

M ΠGgψ

(sgψ) vg, as the reduced scalar curvature sGgψ integrates to 0. We

then get

(10.8.7) s = Aψ +B,

where the mean values s and ψ are given by (10.7.6) and (10.7.11) re-spectively, and A = α, given by (10.4.19). We then get B = β. Fi-nally, for any admissible Kahler metric gψ in MG

Ωa,bwe have that ZGΩa,b =

gradgψ(ΠGgψ

(sgψ) = α gradgψ(ψ) = αT .


10.9. Extremal metrics on complex ruled surfacessshfkg3

In this section we compute the restriction of the relative K-energy rela-tive to G — see Section 4.14 — to the space Madm

Ωa,b⊂MG

Ωa,b, of admissible

Kahler metrics, for any chosen Kahler class Ωa,b on any Hirzebruch-like ruledsurface M = P(1⊕ L).

For that, we shall use the identification of MadmΩa,b

with the space, call it

Aa,b, of admissible momentum profiles Θ relative to the pair a, b describedby Proposition 10.3.4.

We then have the following proposition, which is a particular case of [6,Proposition 7]:

propenergy Proposition 10.9.1. For any Hirzebruch-like ruled surface M = P(1⊕L) of genus g and degree ` and for any Kahler class Ωa,b on M , the restric-

tion of the relative K-energy EGa,b to MadmΩa,b

has the following expression ( up

to an additive constant ):

energy-unienergy-uni (10.9.1) EGa,b(Θ) = 4π2`

∫ b

a(PΩa,b(x)

Θ(x)+ x log (Θ(x))) dx,

where, we recall, PΩa,b denotes the extremal polynomial of the Kahler class

Ωa,b and where elements ofMadmΩa,b

are represented by their momentum profile

Θ, viewed as a function defined on the open interval (a, b).

Proof. Fix any ω0 inMadmΩa,b

— e.g. the standard admissible metric ωψ0

of Ωa,b, cf. Remark 10.3.1 — so that any ωψ in MadmΩa,b

can be written as

ω = ω0 + ddcφ, where φ = φ(t) is normalized by Iω0(φ) = 0, cf. Section 4.1.

This normalization implies∫M f vg = 0 for any variation f := φ := d

ds |s=0φs

of φ, hence of ωψ in MadmΩa,b

. On M0 = L \ (Σ0 ∪ Σ∞), ωψ is written as

ω = ψ ddct+ψ′ dt∧ dct, with ψ = F ′0 +φ′, where F0 is a Kahler potential ofω0, still a function of t (if ω0 is the standard admissible metric, then we canchoose F0(t) = Fψ0 , where Fψ0 is given by (10.3.11)). The corresponding

variation of ψ is then ψ = f ′; similarly, ψ′ = ψ′ = f ′′, etc. On the otherhand, from the defining relation Θ(ψ) = ψ′, we infer the following expressionfor the first variation of the momentum profile Θ:

varFpsivarFpsi (10.9.2) Θ = ψ′ −Θ′(ψ) ψ,

where ψ′ denotes the derivative of ψ = f ′ with respect to t, so that ψ′ = f ′′,while Θ′ denotes the derivative of Θ with respect to x.

For any Kahler class Ωa,b, the relative K-energy EGΩa,b is determined, up

to additive constant, via its differential on MadmΩa,b

, defined by

(dEGΩa,b)ωψ(f) =

∫M

(ΠGgψ

(sgψ)− sgψ) f vg

= 4π2`

∫ +∞

−∞(ΠG

gψ(sgψ)− sgψ) f ψψ′dt.

EGderivativeEGderivative (10.9.3)

By putting together (10.4.9) and (10.8.1), we get

(10.9.4) sGgψ := sgψ −ΠGgψ

(sgψ) =(PΩa,b(ψ)− ψΘ(ψ))′′

ψ,

10.9. EXTREMAL METRICS ON COMPLEX RULED SURFACES 287

where the second derivative is relative to ψ. We then rewrite (10.9.3) as

(dEGΩa,b)ωψ(f) = 4π2`

∫ +∞

−∞(ψΘ(ψ)− PΩa,b(ψ))′′ f ψ′dt

= 4π2`

∫ b

a(xΘ(x)− PΩa,b(x))′′ f dx,

EGderivative-bisEGderivative-bis (10.9.5)

where (xΘ(x) − PΩa,b(x))′′ denotes the second derivative with respect tox. By integrating by parts two times and by observing that the functionxΘ(x)− PΩa,b(x) and its first derivative (with respect to x) both vanish atx = a and x = b, we get

EGderivative-terEGderivative-ter (10.9.6) (dEGΩa,b)ωψ(f) = 4π2`

∫ b

a(xΘ(x)− PΩa,b(x))

d2f

dx2dx.

In this expression, f = f(t) is regarded as a function of x, when t is viewedas a fonction of x whose first derivative is equal to 1

Θ(x) , cf. Proposition

10.3.4. We then have: dfdx = ψ

Θ and d2fdx2 = ψ′

Θ2 − Θ′

Θ2 ψ, where, again, the

derivatives are relative to the natural variable: t for ψ and x for Θ. From(10.9.2), we then obtain

(10.9.7)d2f

d2x=

Θ

Θ2.

By reporting this expression in (10.9.6) we get:

(10.9.8) (dEGΩa,b)ωψ(f) = 4π2`

∫ b

a

(xΘ(x)− PΩa,b(x))Θ(x)

Θ2(x)dx,

which is clearly the first variation of the rhs of (10.9.1).

Proposition 10.9.1 has the following two immediate consequences:

cor1 Corollary 10.9.1. The Calabi polynomial PΩa,b is positive on (a, b) if

and only if MadmΩa,b

contains an extremal metric. This is then an absolute

minimum of the restriction of EGΩa,b to MadmΩa,b

.

Proof. When the restriction of EGΩa,b toMG,adΩa,b

is viewed as a function

defined on Aa,b, we get from (10.9.1) the following expression of its first

derivative at any element Θ of Aa,b along any variations Θ:

(10.9.9) (dEGΩa,b)Θ(Θ) = 4π2`

∫ b

a(x−

PΩa,b(x)

Θ(x))Θ(x)

Θ(x)dx.

It follows that Θ is critical if and only if Θ(x) =PΩa,b

(x)

x . This is only possibleif PΩa,b is positive on (a, b), as Θ belongs to Aa,b. Conversely, if PΩa,b is

positive on (a, b) then Θ(x) =PΩa,b

(x)

x is well defined as an element of Aa,band the corresponding admissible Kahler metric is extremal, as Θ(ψ) = ψ′(t),whereas admissible extremal Kahler metrics gψ are characterized by theequation ψψ′ = PΩa,b(ψ); this means that this metric is critical not only for

the restriction of EGΩa,b toMG,adΩa,b

but also for EGΩa,b defined on the whole space

MGΩa,b

. This fits with the already observed fact that no admissible extremal


Kahler metric exists in MΩa,b if the corresponding Calabi polynomial PΩa,b

has zeros on the open interval (a, b). On the other hand, if PΩa,b is positive on

(a, b), then Θ0(x) :=PΩa,b

(x)

x belongs toAa,b, hence determines an admissibleextremal metric, and we then have

(10.9.10) EGΩa,b(Θ0) = 4π2`

∫ b

ax(1 + log

PΩa,b(x)

x) dx.

It is easy to check that for any other Θ in Aa,b, we have

(10.9.11) EGΩa,b(Θ)− EGΩa,b(Θ0) ≥ 0,

with equality if and only if Θ = Θ0. Indeed, EGΩa,b(Θ) − EGΩa,b(Θ0) can be

written as

(10.9.12) EGΩa,b(Θ)−EGΩa,b(Θ0) = 4π2`

∫ b

ax(PΩa,b(x)

xΘ(x)−1−log

PΩa,b(x)

xΘ(x)) dx,

where the integrand is clearly non negative and identically zero if and only

ifPΩa,b

(x)

Θ(x)x ≡ 1 (the function h(x) := x − 1 − log x is non negative on the

positive half line (0,+∞) and h(x) = 0 if and only if x = 1). It follows that

if MG,admΩa,b

contains an extremal metric, say ω0, then this metric is a global

minimum for the relative K-energy EGΩa,b on MG,admΩa,b

.

Remark 10.9.1. The last statement of Corollary 10.9.1 fits with Theo-rem 4.15.2, which says that ω0 is actually a global minimum of EGΩa,b on the

whole space MGΩa,b

.

The second corollary is a particlar case of [6, Corollary 2]:

cor2 Corollary 10.9.2. If PΩa,b is negative on a non-empty open subinterbal

(a′, b′) of (a, b), then EGΩa,b is unbounded from below on MGΩa,b

.

Proof. We actually prove the stronger result that EGΩa,b is unbounded

from below on MG,admΩa,b

. The argument reproduces the one in [6, Corollary

2]. Choose any Θ0 in Aa,b and, for any non-negative real number s, define

Θs := Θ01+s ρΘ0

, where ρ stands for any non-negative function defined on

(a, b), with support in (a′, b′); then, Θs belongs to Aa,b for any s ≥ 0, andwe have

EGΩa,b(Θs) = EGΩa,b(Θ0) + 4π2`

∫ b′

a′s ρ(x)PΩa,b(x) dx

− 4π2`

∫ b

ax log (1 + s ρ(x)Θ0(x)) dx,

(10.9.13)

where the rhs clearly tends to −∞ when s tends to +∞.

Corollary 10.9.2 together with Theorem 4.15.2 imply the following the-orem, which provides a full answer to the question asked in Remark 10.4.1.

thmhfkg3 Theorem 10.9.1 ([6] Corollary 2). Let M = P(1⊕ L) be a Hirzebruch-like ruled surface of any genus g and any degree `. Let Ωa,b be any Kahlerclass on M and PΩa,b be the corresponding Calabi polynomial. Then M

10.9. EXTREMAL METRICS ON COMPLEX RULED SURFACES 289

admits an extremal Kahler metric of Kahler class Ωa,b if and only if PΩa,b

is positive on the open interval (a, b).

Proof. We already know that M admits an admissible extremal Kahlermetric within Ωa,b whenever the extremal polynomial PΩa,b is positive on(a, b) and that this happens for all admissible ruled surfaces of genus 0 orof genus 1 and for all admissible ruled surfaces of genus g > 1 and degree`, provided x = b

a < xκ, cf. Theorem 10.4.1. If g > 1 and x > xκ, PΩa,b

is negative on some non-empty open subinterval of (a, b) and the relativeK-energy is then not bounded from below on MG

Ωa,b, as we just checked:

from Theorem 4.15.2 we then infer that M admits no extremal metric ofKahler class Ωa,b. It remains the case when g > 1 and x = xκ; in thiscase, we derive the non-existence of any extremal metric from the fact thatthe extremal Kahler cone is open, cf. Chapter 5: if there were an extremalKahler metric in a Kahler class Ωa,b with b

a = xκ there would exist extremal

Kahler metrics in all Kahler classes Ωa,b with ba in (xκ − ε, xκ + ε) for some

ε > 0; by the above, this cannot be.

APPENDIX A

Bochner tensor and Apte formulae

sapteA.1. Curvature decomposition

The curvature, R, of a general n-dimensional riemannian manifold (M, g)is defined by (1.18.1), where R appears as a 2-forms with values in the bun-dle, AM , of skew-symmetric endomorphisms of TM . We make it into a 4-linear form, still denoted byR, by putting: RX1,X2,X3,X4 = g(RX1,X2X3, X4).Then, R satisfies the following three identities:

(1) RX1,X2,X3,X4 +RX2,X1,X3,X4 = 0,(2) RX1,X2,X3,X4 +RX1,X2,X4,X3 = 0,(3) RX1,X2,X3,X4 +RX2,X3,X1,X4 +RX3,X1,X2,X4 = 0,

for all vector fields X1, X2, X3, X4. The first identity holds for the curvatureof any sort of linear connection. The second identity holds for the curvatureof any g-metrical linear connection. The third identity is the (algebraic)Bianchi identity and holds for the curvature of any torsion-free linear con-nection, cf. Section 1.18. These identities imply the following fourth one:

semi-bianchisemi-bianchi (A.1.1) RX1,X2,X3,X4 = RX3,X4,X1,X2 ,

which can be considered as a weak version of the (algebraic) Bianchi identity.The riemannian curvature R is then a section of a vector bundle whose

fiber is the subvector space, R(n), of elements of ⊗4V ∗ which satisfy theabove identities (where V stands for Rn). This vector space is called thespace of abstract riemannian curvature tensors. It can be viewed as a repre-sentation space of the full linear group G`(n,R) or as a representation spaceof the orthogonal group O(n). As a representation space of G`(n,R), R(n)is irreducible, meaning that the riemannian curvature admits no additionalgeneral symmetries. As a representation space of O(n), R(n) is still irre-ducible if n = 2, splits into two irreducible components if n = 3, into threeirreducible components if n > 3. If n = 4, R(n) splits into three irreduciblecomponents under the action of O(n), but into four irreducible componentsunder the action of the special orthogonal group SO(4); for n > 4, there isno difference between the decomposition into irreducible components underO(n) and SO(n).

By (A.1.1), R(n) is a subspace of the symmetric tensor product Λ2(n)©Λ2(n), where Λ2(n) stands for Λ2V ∗. The restriction of the Bianchi projec-tion to Λ2(n)© Λ2(n) takes values in the space Λ4(n) = Λ4V ∗ and reads:

(A.1.2) κ(φ⊗ φ) =1

6φ ∧ φ,

for each φ in Λ2(n); κ is a G`(n,R)-equivariant map from Λ2(n)© Λ2(n)onto Λ4(n), whose kernel is R(n).

291

292 A. BOCHNER TENSOR AND APTE FORMULAE

We denote by Sym(n) the space of symmetric elements of V ∗ ⊗ V ∗, bySym0(n) the subspace of trace-free elements of Sym(n).

The Ricci trace is the O(n)-equivariant map, c, from R(n) to Sym(n)defined by:

(A.1.3) c(R)X,Y =

n∑j=1

RX,ej ,Y,ej ,

for any orthonormal basis e1, . . . , en of V .For n ≥ 3, this map is onto (an isomorphism if n = 3). For n > 3,

the kernel of c is denoted by W(n) and called the space of (abstract) Weyltensors. The orthogonal complement of W(n) in R(n) coincides with theimage of the (metric) adjoint c∗ of c. In order to give an explicit expressionof c∗, it is convenient to consider elements of R(n) as (symmetric) mapsfrom Λ2(n) to itself, via the metric g. In the same way, elements of Λ2(n)and of Sym(n) may be viewed as endomorphisms of V , respectively skew-symmetric and symmetric. We then have

(A.1.4) c∗(S) : φ→ S, φ := S φ+ φ S,for any S in Sym(n) and any φ in Λ2(n).

It is easily checked that

(A.1.5) cc∗(S) = trS I + (n− 2)S,

where trS denotes the trace of S and I stands for the identity of V , viewedas an element of Sym(n). The equation cc∗(S) = r is thus solved by

(A.1.6) S =s

2n(n− 1)I +

1

(n− 2)r0,

where s denotes the trace of r and r0 stands for the trace-free part of r.We thus get the well-known decomposition of any element R of R(n),

hence of the curvature R of any n-dimensional riemannian manifold (M, g)(viewed as a symmetric endomorphism of Λ2M):

R =s

n(n− 1)I|Λ2M

+1

(n− 2)r0, ·

+W,

decRdecR (A.1.7)

where: r = c(R), s = tr(r) and W , the Weyl tensor, is defined as theorthogonal projection of R into the kernel of c.

As a AM -valued 2-form on M , the Weyl tensor W only depends uponthe conformal class of g. If the latter is flat, i.e. if g can be locally madeinto a flat metric by conformal change, W is then identically zero. If n > 3,the converse is true: W ≡ 0 implies that the conformal class of g is flat.

If n = 4, then the orthogonal Lie algebra so(4) splits, as a Lie algebra,as the sum of two copies of the orthogonal Lie algebra so(3). AccordinglyΛ2(4) ∼= so(4) splits as the orthogonal sum Λ+ ⊕ Λ−, where each Λ± isisoomorphic to so(3) (the choice of ± depends on the choice of an orientationof V = R4). If ∗ denotes the corresponding Hodge operator, then ∗ actsas a SO(4)-equivariant involution on Λ2(4) and Λ± is the eigenspace of ∗

A.2. THE BOCHNER TENSOR 293

with respect to the eigenvalue ±1. Each Λ± is an irreducible representionspace under the (adjoint) action of SO(4). As a SO(4)-representation space,Sym0(4) is isomorphic to the tensor product Λ+ ⊗ Λ−: the isomorphismΛ+ ⊗ Λ− → Sym0(4) is defined by ψ+ ⊗ ψ− → ψ+ ψ−, for any ψ± inΛ±, both viewed as (commuting) skew-symmetric endomorphisms of V . Itfollows that the r0-part of R in (A.1.7) exchanges Λ+ and Λ−. In otherwords, the r0-part of R anti-commutes with ∗. On the other hand, theWeyl tensor W commutes with ∗, hence splits as W = W+ ⊕W−, whereW+, resp. W−, acts trivially on Λ−, resp. Λ+. Both W+ and W− areconformally invariant. The algebraic splitting Λ2(4) = Λ+ ⊕ Λ− implies asimilar splitting Λ2M = Λ+M⊕Λ−M of the bundle of (real) 2-forms into theorthogonal direct sum of the bundle Λ+M of selfdual 2-forms, on which theHodge operator acts as the identity, and the bundle Λ−M of antiselfdual2-forms, on which the Hodge operator acts as minus the identity. Withrespect to this decomposition, the curvature can then be pictured as follows

STST (A.1.8) R =

(W+ + s

12IΛ+M12r0, ·

12r0, · W− + s

12IΛ−M

)where IΛ±M stands for the identity of Λ±M and where W±, which actstrivially on Λ∓M , is here considered as a (trace-free, symmetric) endomor-phisms of Λ±M , cf. [179].

A.2. The Bochner tensor

We now consider the case when n = 2m the metric g is kahlerian withrespect to some complex structure J . Then, Λ2M splits as

splitJsplitJ (A.2.1) ΛJ,+M ⊕ ΛJ,−M,

where ΛJ,±M denotes the bundle of ±-invariant (real) 2-forms on M underthe natural action of J on Λ2M (which acts as an involution) and the curva-ture R, viewed as a symmetric endomorphism of Λ2M , acts trivially on thefactor ΛJ,−M . From an algebraic point of view, the decomposition (A.2.1)corresponds to the decomposition of U(m)-representation spaces

(A.2.2) so(2m) = u(m)⊕ u(m)⊥

where U(m) denotes the unitary group of order m, u(m) = Λ1,1(m) itsLie algebra, u(m)⊥ the orthogonal of u(m) in so(2m) = Λ2(2m). Thespace K(m) of abstract kahlerian curvature tensors is then defined as thespace of elements of R(2m) which act trivially on the factor u(m)⊥ or,equivalently, as the kernel of the restriction of the Bianchi projector κ toΛ1,1(m)© Λ1,1(m), which takes its values in the space Λ2,2(m): K(m) isthen a subspace of R(2m) and both are now regarded as representationspaces of U(m). A first difference with the preceding situation is that nei-ther Λ1,1(m) nor Λ2,2(m) are irreducible under the action of U(m); we have

instead: Λ1,1(m) = Λ1,10 (m)⊕R and Λ2,2(m) = Λ2,2

0 (m)⊕Λ1,10 (m)⊕R, where

R stands for the trivial representation space.If R belongs to R(m) the components of R which appear in (A.1.7) don’t

belong to K(m); in particular, the scalar part sn(n−1) I|Λ2(2m) definitely does

not belong to K(m) if s 6= 0 (see however section A.4 below for the casewhen m = 2).


The restriction of the Ricci trace c to K(m) will be denoted by cK . Itis easily checked that cK takes its values in the subspace Sym1,1(m) of J-invariant elements of Sym(m). Moreover, as a reprentation space of U(m),Sym1,1(m) is the same as Λ1,1(m), via the map: S → σ := J S (beware,however, that this “identification” does not preserve the natural metrics ofthe two spaces; more precisely, according to our special convention for thenorm of exterior forms, see Section 1.3, we have that |J S|2 = 1

2 |S|2). Then,

a simple apllication of the algebraic Bianchi identity shows that

(A.2.3) c(R) = cK(R) = R(ω),

for any R in K(m).The map cK is a U(m)-equivariant map from K(m) onto Sym1,1(m). It

is an isomorphism if m = 1.We henceforth assume that m ≥ 2.The kernel of cK , i.e. the intersectionW(2m)∩K(m), is called the space

of abstract Bochner tensors, denoted by WK(m).The orthogonal complement of WK(m) in K(m) is the image of the

metric adjoint c∗K of cK , from Sym1,1(m) to K(m) (for the natural metric of

Sym1,1(m)). It can be checked that c∗K has the following expression:

(A.2.4) c∗K(S) : φ→ S, φ1,1+ (φ, ω)σ + (σ, φ)ω,

for each φ in Λ2(2m) and each S in Sym1,1(m) (here, σ denotes the cor-responding J-invariant 2-form; the scalar product is the scalar product inΛ1,1(m)).

The equation cKc∗K(SK) = r is now solved by

(A.2.5) SK =2

(m+ 2)r0 +

s

2m(m+ 1)g.

We thus get the following decomposition into U(m)-irreducible compo-nents of K(m), hence also the decomposition of the curvature R of anyKahler metric:

R =s

2m(m+ 1)(Π1,1 + ω ⊗ ω)

+1

(m+ 2)(r0, ·+ ω ⊗ ρ0 + ρ0 ⊗ ω)

+WK ,

decRKdecRK (A.2.6)

where Π1,1 stands for the orthogonal projection from Λ2(2m) onto Λ1,1(m),ρ0 denotes the primitive part of the Ricci form ρ, whereas WK denotes theBochner tensor, defined as the orthogonal projection of R into the space ofBochner tensors.

We observe that the first line, namely the (kahlerian) scalar part of R,is the curvature of any 2m-dimensional Kahler manifold of constant holo-morphic sectional curvature c = s

m(m+1) .

Again, the case when m = 2 deserves a special mention. Then, the(riemannian) r0-part of R appearing in (A.1.7) does belong to K(m) foreach R in K(m): indeed, any J-anti-invariant element φ of Λ2(m) belongsto Λ+; it is then sent to Λ− by the r0-part of R; on the other hand, r0, φis certainly J-anti-invariant again, as r0 is J-invariant, hence belongs to Λ+;

A.3. APTE FORMULAE 295

we conclude that φ is killed by the riemannian r0-part of R; this proves thatthe riemannian r0-part of R already belongs to K(m). On the other hand,W− also kills the J-anti-invariant part of Λ2(m), which is included in Λ+; itfollows that W− also belongs to K(m). Finally, it is easily checked that the(riemannian) scalar part of R put together with W+ is equal to the kahlerianscalar part of R. It follows that the riemannian r0-part of R coincides withits kahlerian r0-part, whereas the Bochner tensor WK coincides with W−.

As in [28], Chapter 2, we get a coarser decomposition of a kahleriancurvature tensor R by just considering R as an element of Λ1,1(m)©Λ1,1(m)

and by decomposing Λ1,1 as Λ1,1 = Λ1,10 ⊕Rω. Then, R induces a symmetric

endomorphism of Λ1,10 , denoted by B. If we denote by B0 the trace-free part

of B, we end up with the following decomposition R:

R =s

2m(m+ 1)(Π1,1 + ω ⊗ ω)

+1

m(ρ0 ⊗ ω + ω ⊗ ρ0)

+B0,

decR-K1decR-K1 (A.2.7)

or, by introducing an “orthonormal” (local) frame of Λ1,10 M , so that (ωr, ωs) =

mδrs,

R =s

2m2ω ⊗ ω

+1

m(ρ0 ⊗ ω + ω ⊗ ρ0)

+1

m

m2−1∑r=1

(s

2m(m+ 1)+ λr)ωr ⊗ ωr,

decR-K2decR-K2 (A.2.8)

where the λr denote the eigenvalues of B0.The tensor B0 coincides with W−, hence with WK , if m = 2, but in

general we have that:

(A.2.9) B0 = WK +2

m(m+ 2)(ρ0 ⊗ ω + ω ⊗ ρ0)− 1

(m+ 2)r0, ·.

It follows that

normnorm (A.2.10) |B0|2 = |WK |2 +2(m− 2)

m(m+ 2)|ρ0|2.

A.3. Apte formulae

Formula (A.2.8) turns out to be very convenient for establishing thefollowing two formulae, known as Apte formulae [12], [25, Chapitre XII][28, Chapter 2]:

apte1apte1 (A.3.1) (c21 ∪ Ωm−2)[M ] =

1

4π2

∫M

(s2

4m2− |ρ0|2

m(m− 1))ωm,

andapte2apte2 (A.3.2)

(c2 ∪Ωm−2)[M ] =1

8π2

∫M

(s2

4m(m+ 1)− 2 |ρ0|2

(m− 1)(m+ 2)+|WK|2

m(m− 1))ωm,


where, we recall, Ω = [ω] denotes the de Rham class of ω, and c1, c2 denotethe first and second Chern class of (M,J), in H2

dR(M,R) and H4dR(M,R)

respectively.Before proving (A.3.1) and (A.3.2), we recall the following facts. For any

φ in Λ1,1(m), denote by Φ = −J φ the corresponding hermitian operator,acting on V = R2m viewed as a m-dimensional complex vector space. Asin Section 1.12, denote by P (t) =

∑mk=1(−1)rσr(φ) tm−r the characteristic

polynomial of Φ. Then, each σr is a homogeneous U(m)-invariant polyno-mial of degree r on the Lie algebra Λ1,1(m) = u(m) and, by definition, thecorresponding characteristic class is the r-th Chern class. This means thatfor any Kahler structure, of curvature R, the r-th Chern class cr of (M,J),viewed as an element of H2r(M,R), is represented by the (closed) 2r-classcr(R) defined by

(A.3.3) cr(R) =1

(2π)rσr(R ∧ . . . ∧R),

(r times), with the following significance: R is a Λ1,1M -valued 2-form; theformal exterior product R ∧ . . . ∧ R has to be understood as a 2r-formwith values in the tensor product Λ1,1M ⊗ . . .⊗Λ1,1M (r times); we finallycontract by σr, viewed as a r-linear form on each fiber of Λ1,1M , in order toget a real 2r-form.

The differential Bianchi identity

(A.3.4) dDR = 0,

implies that σr(R) is closed and that its de Rham class is independent ofthe chosen Kahler metric. The factor 1

(2π)r ensures that the corresponding

element of H2r(M,R) is integral, i.e. belongs to the image of H2r(M,Z).By using (1.12.2)-(1.12.4)-(1.12.5) of Section 1.12, we are now ready to

prove (A.3.1)-(A.3.2).The LHS of (A.3.1) is equal to 1

4π2

∫M ρ ∧ ρ ∧ ωm−2. We then have:

1

4π2

∫Mρ ∧ ρ ∧ ωm−2 =

1

4π2(m− 2)!

∫M

Λ2(ρ ∧ ρ) vg

=1

4π2

1

m(m− 1)

∫M

Λ2(ρ ∧ ρ)ωm

=1

4π2

1

m(m− 1)

∫M

(s2

4− |ρ|2)ωm

=1

4π2

1

m(m− 1)

∫M

((m− 1) s2

4m− |ρ|20)ωm.

This proves (A.3.1).The LHS of (A.3.2) is equal to 1

4π2

∫M σ2(R∧R)∧ωm−2. We then have:

1

4π2

∫Mσ2(R ∧R) ∧ ωm−2 =

1

4π2(m− 2)!

∫M

Λ2(σ2(R ∧R) vg

=1

4π2

1

m(m− 1)

∫M

Λ2(σ2(R ∧R)ωm.

A.4. THE FOUR-DIMENSIONAL CASE 297

By using (1.12.4) and (A.2.8), we get

Λ2(σ2(R ∧R)) =s2

m2σ2(ω, ω)σ2(ω, ω)

+8

m2σ2(ω, ω)σ2(ρ0, ρ0)

+s2

m4(m+ 1)2

m2−1∑r=1

σ2(ωr, ωr)σ2(ωr, ωr)

+4

m2

m2−1∑r=1

σ2(ωr, ωr)σ2(ωr, ωr).

By using (1.12.5), we get

(A.3.5) Λ2(σ2(R ∧R)) =(m− 1)

4(m+ 1)s2 − 2(m− 1)

m|ρ0|2 + |B0|2.

By (A.2.10), this gives (A.3.2).

Proposition A.3.1 (S.-T. Yau [197]). A compact kahlerian manifoldwith c1(M) = 0 and c2(M) = 0 admits a (unique) flat Kahler metric ineach Kahler class.

Proof. Since c1(M) = 0, it follows from Calabi-Yau theorem that eachKahler class contains a unique Ricci flat Kahler metric. It then follows from(A.3.1)-(A.3.2) that such a metric g also have: sg = 0 and WK = 0, henceis flat.

A.4. The four-dimensional casessfour

One specific feature of the curvature of a Kahler manifold of (real) di-mension 4 is that the term r0, · in the decomposition (A.1.7) is also ofKahler type, i.e. acts trivially on ΛJ,−M . This is because, ΛJ,−M is in-cluded in Λ+M , so that r0, ψ is antiselfdual for any ψ in ΛJ,−M ; on theother hand, r0, ψ certainly anti-commutes with J , so that r0, ψ is self-dual; it follows that r0, ψ = 0 for each ψ in ΛJ,−M .

On the other hand, since R acts trivially on ΛJ,−M the selfdual Weyltensor W+ in the Singer-Thorpe decomposition (A.1.8) is reduced to

W+W+ (A.4.1) W+ =

s6 0 00 − s

12 00 0 − s

12

with respect to the decomposition Λ+M = Rω ⊕ ΛJ,−M or else

(A.4.2) W+ = − s

12Π+ +

s

8ω ⊗ ω.

In particular,

W+normW+norm (A.4.3) |W+|2 =s2

24.

In general, the selfdual Weyl tensor W+ is not of Kahler type, neither thescalar part s

12IΛ2M of R in (A.1.7), but the sum s12IΛ2M +W+ = s

12(Π1,1 +


ω⊗ω) certainly is and actually coincides with the scalar part of R in (A.2.6).It follows that the latter can be written

R =s

12(Π1,1 + ω ⊗ ω)

+1

2(ω ⊗ ρ0 + ρ0 ⊗ ω)

+W−.

(A.4.4)

In particular, we have that

(A.4.5) WK = W−.

The Apte formulae then become

c21[M ] =

1

4π2

∫M

(s2

8− |ρ0|2) vg,

c2[M ] =1

8π2

∫M

(s2

12− |ρ0|2 + |W−|2) vg.

apte4apte4 (A.4.6)

We infer

toptop (A.4.7) (c21 − 2c2)[M ] =

1

4π2(

∫M

s2

24− |W−|2)vg.

It turns out that for any compact complex surface (M,J) the two Chernnumbers c2

1[M ] and c2[M ] only depends on the underlying oriented 4-manifold.More precisely, if χ and τ denote the Euler characteristic and the signatureof M , we have that

chitauchitau (A.4.8) χ = c2[M ], τ =1

3(c2

1 − 2c2)[M ].

Remark A.4.1. For any compact (oriented, connected) 4-dimensionalriemannian manifold (M, g), the Gauss-Bonnet formula gives the followinggeneral expression for the Euler characteristic:

GBGB (A.4.9) χ =1

8π2

∫M

(s2

24− |r0|2

2+ |W |2) vg,

where r0 denotes the trace-free part of the Ricci tensor r, whereas, by theHirzebruch signature formula, the signature is expressed as

toptautoptau (A.4.10) τ =1

12π2

∫M

(|W+|2 − |W−|2) vg.

In the Kahler case, these expressions can be readily deduced from (A.4.6),(A.4.7), (A.4.8) and (A.4.3) (beware however that according to our general

convention — cf. Section 1.3 — we have that |ρ0|2 = |r0|22 ) and the square

norms |W |2, |W+|2, |W−|2 which appear in (A.4.9)-(A.4.10) and in otherparts of these notes may differ by a factor 1

4 from other conventions ocurringin the literature).

APPENDIX B

The space of almost complex structurescompatible with a symplectic form

appcomplexB.1. The Cayley transformation

sscayleyIn this section, E ∼= R2m denotes a real vector space of even dimension

n = 2m and we fix a complex structure J0 of E.For any complex structure J of E, we denote by E1,0

J , resp. E0,1J , the

subspace of elements of type (1, 0), resp. (0, 1), of EC := E⊗C with respectto J .

Then, J is said to be commensurable to J0 if the projection of E0,1J on

E0,1J0

along E1,0J0

is onto, hence an isomorphism (in particular J0 is always

commensurable to itself but never to its conjugate −J0).

By definition, J is commensurable to J0 if and only if E0,1J ∩E

1,0J0

= 0,if and only if E0,1

J is the graph of a C-linear homomorphism, µ, from E0,1J0

to E1,0J0

, i.e.

comcom (B.1.1) E0,1J = Z + µ(Z) | Z ∈ E0,1

J0.

The homomorphism µ is the restriction to E0,1J of a unique real C-linear

endomorphism of EC , still denoted by µ as well as its restriction to E; thelatter is then a R-linear endomorphism of E which anticommutes with J0:µ J0 = −J0 µ.

Conversely, any such µ determines a complex structure J of E commen-surable to J0 if and only if the space E0,1

J defined by (B.1.1) satisfies the

condition E0,1J ∩E

0,1J = 0. This condition is satisfied if and only if 1−µ2 is

one to one, or, equivalently, if and only if 1+µ and 1−µ are both invertible(as endomorphisms of E).

The type decomposition of any element X of E relatively to J can bewritten as follows:

(B.1.2) X = Z + µ(Z) + Z + µ(Z),

where Z is a well-defined element of E1,0J0

; then, X1 = Z+Z is a well-definedelement of E, with

(B.1.3) X = X1 + µ(X1),

and

(B.1.4) JX = J0X1 − J0µ(X1).

We infer that

cayley1cayley1 (B.1.5) J = J0(1− µ)(1 + µ)−1 = (1 + µ) J0 (1 + µ)−1.

299

300 B. THE SPACE OF ALMOST COMPLEX STRUCTURES

It follows that J + J0 = 2J0(1 + µ)−1 is invertible and that

cayley2cayley2 (B.1.6) µ = (J + J0)−1(J0 − J) = (J − J0)(J + J0)−1.

Conversely, for any complex structure J of E such that J + J0 is invert-ible, the endomorphism µ of E defined by (B.1.6) is J0-antilineaire (directconsequence of the easy identity J0(J + J0)−1 = (J + J0)−1J); moreover,1 + µ = 2J(J + J0)−1 = 2(J + J0)−1J0 and 1 − µ = 2J0(J + J0)−1 =2(J + J0)−1J are both invertible.

The above discussion can be summarized as follows:

propcayley Proposition B.1.1. (i) A complex structure J of E is commensurablewith J0 if and only if J + J0 is invertible.

(ii) The Cayley correspondence J ↔ µ given by (B.1.5)-(B.1.6) thenestablishes a natural identification of the space of J0-commensurable complexstructures of E with the space of J0-antilinear endomorphisms of E such that(1± µ) are invertible.

In the sequel, µ will be referred to as the Cayley transform of J .The differential of the map µ→ J is easily checked to be given by

diffcayleydiffcayley (B.1.7) µ→ J = [µ(1 + µ)−1, J ].

We infer

(B.1.8) J0µ→ JJ.

Indeed, we have that [µ(1 + µ)−1, J ] = µ(1 + µ)−1J − Jµ(1 + µ)−1, whereas[J0µ(1 + µ)−1, J ] = −[µJ0(1 + µ)−1, J ] = −[µ(1 + µ)−1J, J ] = µ(1 + µ)−1 +Jµ(1 + µ)−1J = J [µ(1 + µ)−1, J ].

It follows that the space of complex structures of E which are commen-surable with J0, when equipped with the natural almost complex structure Jdefined by J(J) = JJ , is naturally identified with on open subset of the vec-tor space of J0-antilinear endomorphisms of E, viewed as a complex vectorspace by setting iµ = J0 µ = −µ J0.

B.2. Symplectic complex structuressssymplectic

Let ω0 be a symplectic form on E, i.e. an element of Λ2E∗ such thatωm0 6= 0. A complex structure J on E is said to be ω0-compatible if ω0(J ·, J ·) =ω0 — so that g := ω0(·, J ·) is symmetric — and if g0 is positive definite.The set of ω0-compatible complex structures of E will be denoted by C(ω0).

We now fix J0 in C(ω0) and we denote by g0 the (positive definite) innerproduct ω0(·, J0·).

prop1 Proposition B.2.1. (i) Any J in C(ω0) is commensurable to J0.(ii) A complex structure of E commensurable to J0 belongs to C(ω0) if

and only if its Cayley transform µ is symmetric and 1±µ are positive definitewith respect to g0.

Proof. (i) By definition, g := ω0(·, J ·) is positive definite: we then haveg = g0(·, S·), where S is symmetric and positive definite with respect to g0

and g, and J = J0S. We thus get J + J0 = J0(1 + S), which is evidentlyinversible. By Proposition B.1.1, J is then commensurable to J0.

B.2. SYMPLECTIC COMPLEX STRUCTURES 301

(ii) From the latter and (B.1.6), we get

muSmuS (B.2.1) µ = (1− S)(1 + S)−1, S = (1− µ)(1 + µ)−1.

The conclusion follows easily.

We denote by Sp(ω0) the symplectic group relative to ω0, defined as thegroup of linear transformations γ of E which preserve ω0, i.e. such thatω0(γ·, γ·) = ω0. By considering J0 and the corresponding inner product g0

as in the previous section, the elements of Sp(ω0) are also characterized byγ∗J0γ = J0, where γ∗ stands for the adjoint of γ with respect to g0.

The Lie algebra, sp(ω0), of Sp(ω0) is the Lie algebra of linear transforma-tions A of E such that ω0(a·, ·)+ω0(·, a·) = 0. Equivalently, a∗J0 +J0a = 0.This means that a linear transformation a of E belongs to sp(ω0) if andonly if J0a is symmetric with respect to g0 (in particular, the dimension ofsp(ω0), hence of Sp(ω0), is equal to m(2m+1)). It follows that Sp(ω0) splitsinto the direct sum of the following two subspaces: (i) the space, u(J0), ofthose a in End(E) which are g0-anti-symmetric and commute to J0, hencethe Lie algebra of the unitary group U(J0), which is the commutator of J0

in Sp(ω0); (ii) the space, m(J0), of those a in End(E) which are symmetricand anti-commute to J0. The pair (u(J0),m(J0)) is a symmetric pair in theLie algebra sp(ω0), in the sense that

(B.2.2) [u(J0),m(J0)] ⊂ m(J0), [m(J0),m(J0)] ⊂ u(J0).

The image of m(J0) by the exponential map from sp(ω0) to Sp(ω0) isa closed submanifold of Sp(ω0) — namely the space of those γ in Sp(ω0)which are symmetric and satisfy J0γ = γ−1J0 — and the map a → expais a diffeomorphism from m(J0) to its image. The product in Sp(ω0) theninduces a diffeomorphism from U(J0) × expm(J0) to Sp(ω0), which makesthe latter diffeomorphic to U(J0)×m(J0). In particular, U(J0) is a maximal(connected) compact subgoup of Sp(ω0).

The quotient Sp(ω0)/U(J0) is a hermitian symmetric spaces of non-compact type, hence a Hadamard space. At the origin x0, the tangentspace of Sp(ω0)/U(J0) is naturally identified with m(J0) and each point xof Sp(ω0)/U(J0) is therefore of the form x = exp(a) · x0, for a unique a inm(J0). The curve exp(a) ·x0 is the unique geodesic in Sp(ω0)/U(J0) joiningJ0 to J .

The group Sp(ω0) acts on C(ω0), by

actionaction (B.2.3) γ · J = γJγ−1

for any γ in Sp(ω0) and any J in C(ω0). We then have

Proposition B.2.2. The action of Sp(ω0) on Cω0 given by (B.2.3) istransitive, hence identifies C(ω0) with the symmetric space Sp(ω0)/U(J0),for any chosen base-point J0 in C(ω0),

Proof. The isotropy group of J0 for the action (B.2.3) is clearly the uni-tary group U(J0). It then only remains to check that the action is transitive.Any element J of C(ω0) satisfies ω0(J ·, J ·) = ω0, hence can be viewed as anelement of the group Sp(ω0). The operator S = J−1

0 J defined in the preced-ing section hence belongs to Sp(ω0) and we clearly have that SJ0 = J0S

−1.Since S is positive definite with respect to g, it admits a (unique) positive

302 B. THE SPACE OF ALMOST COMPLEX STRUCTURES

root, say S12 , and we again have that S

12J0 = J0S

− 12 ; indeed, J0S

− 12J0 is

also a positive root of S, hence is equal to S12 . It follows that S

12 also be-

longs to Sp(ω0) and J = S−12J0S

12 . We conclude that C(ω0) coincides with

the orbit of J0 under the action of Sp(ω0).

Via the above identification, the space C(ω0) admits a homogeneousKahler structure (g,J, κ) which is actually independent of the chosen basepoint J0 and is described as follows. At any point J of C(ω0), the tangent

space TJC(ω0) is naturally identified with the space of endomorphisms A = Jof E which anticommute with J and are symmetric with respect to the metricgω0(·, J ·) and we then have

(B.2.4) JA = JA, g(A1, A2) =1

2tr(A1A2), κ(A1, A2) =

1

2tr(JA1A2).

rema Remark B.2.1. Any J in C(ω0) is related to the chosen base-point J0

by J = exp(a) J0 exp(−a) for a uniquely defined element a in TJC(ω0). Wethen have that S = J−1

0 J = exp(−2a) and, by (B.2.1), a is related to theCayley transform µ of J with respect to J0 by µ = tanh a.

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