geometric quantization in complex and harmonic analysisgm/geoquant.pdfconsider smooth manifolds over...

Geometric Quantization in Complex and Harmonic

Analysis

Harald Upmeier

December 17, 2018

These are informal notes, subject to continuous changes and corrections

1

Contents

0 Overview 5

0.1 d+-Quantization, d ≥ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

0.2 0+-Quantization, Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 5

0.3 1+-Quantization, Conformal Field Theory . . . . . . . . . . . . . . . . . . . . . . . 6

0.4 2+-Quantization,Topological Quantum Field Theory . . . . . . . . . . . . . . . . 6

0.5 3 ≤ d ≤ 8, Higher gauge theory and special holonomy . . . . . . . . . . . . . . . . 6

1 Manifolds, Connexions and Curvature 7

1.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

• Jordan manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

• Restricted Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

• Loop groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

• Conformal blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.1.1 Covered manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10


1.1.2 Homogeneous manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12


1.2 Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14





1.3 0-Geometry: Hermitian Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19





1.4 1-Geometry: Connexions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21




2

1.5 2-Geometry: Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28



2 Classical Phase Spaces 32

2.1 Symplectic Manifolds and Kahler Manifolds . . . . . . . . . . . . . . . . . . . . . 32



• Loop groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34





• Loop groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39


2.2 Hamiltonian vector fields, Poisson bracket . . . . . . . . . . . . . . . . . . . . . . 40

2.3 Moment Map and Classical Reduction . . . . . . . . . . . . . . . . . . . . . . . . . 40




2.3.1 Homogeneous Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4 Quantum line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43


• Loop groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46


3 Quantum State Spaces 49

3.1 Reproducing kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49


3.2 Compact Lie Groups and Borel-Weil-Bott Theorem . . . . . . . . . . . . . . . . 52


3.2.1 Borel Subgroups and full Flag Manifolds . . . . . . . . . . . . . . . . . . . 53

3.2.2 0-Cohomology: Borel-Weil theorem . . . . . . . . . . . . . . . . . . . . . . 54

3.2.3 Parabolic subgroups and flag manifolds . . . . . . . . . . . . . . . . . . . . 56

3.2.4 q-Cohomology: Bott’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3 Compact Kahler Manifolds and Kodaira Embedding Theorem . . . . . . . . . 63

3.3.1 Chern Classes, Divisors and Positivity . . . . . . . . . . . . . . . . . . . . . 63

3.3.2 Blow-up process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3.3 Proof of the Kodaira embedding theorem . . . . . . . . . . . . . . . . . . . 69

3

Chapter 0

Overview

0.1 d+-Quantization, d ≥ 0

border: d-dimensional manifold S, closed (compact) but possibly disconnected (many-particle system)

bordism: d+ 1 manifold Σ, connected but non-closed, with boundary ∂Σ = S

border symplectic manifold M

border complex manifold: family of Kahler manifolds Mτ

border complex quantization: family of Hilbert spaces H2(Mτ )

border symplex quantization: projectively flat connexion on bundle of Hilbert spaces H2(Mτ )

classical bordism: flow of symplectomorphisms

quantum bordism: flow of unitary operators

0.2 0+-Quantization, Quantum Mechanics

border: point S = S0 or finite number of points

bordism: interval [0, t], 1-manifold with boundary

Example 0.2.1. Q configuration space

border symplectic manifold T ∗Q

border complex manifold: family of Kahler manifolds Mτ


border symplex quantization: projectively flat connexion on bundle of Hilbert spaces L2(Q)

classical bordism: geodesic flow

quantum bordism: time evolution etH

Example 0.2.2. G compact Lie group, T maximal torus

border symplectic orbit G/T

border complex orbit: family of Kahler manifolds GC/GCτ

border complex quantization: family of highest weight Hilbert spaces GC/GCτ

border symplex quantization: projectively flat connexion on bundle of Hilbert spaces

quantum bordism: no time evolution H = 0

4

0.3 1+-Quantization, Conformal Field Theory

border: circle S = S1, or disjoint union of circles=compact 1-manifold without boundary

bordism: cylinder [0, 1]× S1 or connected Riemann surface Σ with boundary

Example 0.3.1. G compact Lie group

border symplectic quotient

C∞(S1, G)/G

border complex quotient: family of Kahler manifolds

O(D, GC)

border complex quantization: positive energy representations of loop group (G. Segal)


Example 0.3.2. 1+1 gravity, G = SL2(R) non-compact

Example 0.3.3. Restricted Grassmannian, 2d QCD (Rajeev-Turgut)

0.4 2+-Quantization,Topological Quantum Field Theory

border: non-connected compact oriented surface S without boundary

bordism: connected non-compact 3-manifold Σ with boundary ∂Σ = S

Example 0.4.1. Chern-Simons theory: G=compact Lie group

border symplectic quotient (compact)

H1(S,G) = Hom(π1(S), G)

border complex quotient: family of Kahler manifolds

H1(Sτ , GC)



Example 0.4.2. 2+1 gravity=Chern-Simons theory for non-compact Lie group SL2(R) (Verlinde)

0.5 3 ≤ d ≤ 8, Higher gauge theory and special holonomy

• gauge theory in 4 dimensions, SU(2)-holonomy

• Calabi-Yau manifolds in 6 dimensions, SU(3)-holonomy

• G2-manifolds in 7-dimensions

• Spin(7)-manifolds in 8 dimensions

Since spacetime is supposed to have dimension ≤ 11 (M -theory) or ≤ 12 (F -theory), Kaluza-Klein

compactification to 4-dimensional Minkowski space yields ’border’ manifolds of dimension ≤ 8.

5

Chapter 1

Manifolds, Connexions and

Curvature

1.1 Manifolds

Consider smooth manifolds over R and complex manifolds over C. We use the term K-manifold for

K = R,C. If not specified otherwise, maps, functions, sections etc. will be smooth for K = R or

holomorphic for K = C.

• Jordan manifolds

Jordan manifolds are symmetric manifolds of arbitrary rank, associated with Jordan algebras and

Jordan triples. The basic example is projective space (rank 1)

Ps = E ⊂ K1+s : dimE = 1.

Let Z be a K-vector space, endowed with a ternary composition Z × Z × Z → Z, denoted by

(x, y, z) 7→ x; y; z,

which is bilinear symmetric in (x, z) and anti-linear in the inner variable. Define

D(x, y)z := x; y; z.

Then Z is called a Jordan triple if the Jordan triple identity[D(x, y), D(u, v)

]= D(x; y;u, v)−D(u, v;x; y)

holds. Z is called hermitian (over K) if the sesqui-linear form

(x, y) 7→ tr D(x, y)

is non-degenerate and hermitian

tr D(x, y) = tr D(y, x).

A hermitian Jordan triple is called 0hermitian, if the trace form (??) is positive definite. If there

are q negative eigenvalues, then Z is called qhermitian. We will mostly be concerned with complex0hermitian Jordan triples.

6

The basic example is Z = Kr×s with the ternary composition

u; v;w := uv∗w + wv∗u

which makes sense for rectangular matrices. More generally, the full classification of irreducible

complex 0hermitian Jordan triples is

• matrix triple Z = Cr×s, x; y; z = xy∗z+zy∗x, rank = r ≤ s, a = 2 (complex case), b = s−r

• r = 1, Z = C1×s = Cs, x; y; z = (x|y)z + (z|y)x

• symmetric matrices a = 1 (real case)

• anti-symmetric matrices a = 4 (quaternion case)

• spin factor Z = Ca+2, x; y; z = (x · y) z + (z · y) x+ (x · z)y, r = 2, b = 0

• exceptional Jordan triples of dimension 16 (r = 2) and 27 (r = 3), a = 8 (octonion case)

For (u, v) ∈ Z2 := Z × Z the endomorphism

Bu,vz := z − u; v; z+1

4u; v; z; v;u

of Z is called the Bergman operator. For matrices it becomes

Bu,vz = (Ir − uv∗)z(Is − v∗u)

which again makes sense for rectangular matrices.

A pair (x, y) ∈ Z2 is called quasi-invertible if Bx,y is invertible. In this case the element

xy := B−1x,y(x− x; y;x)

in Z is called the quasi-inverse. For rectangular matrices the quasi-inverse is given by

xy := (Ir − xy∗)−1x = x(Is − y∗x)−1

which is again a rectangular matrix.

By [?, ], we have the addition formulas

Bx,y+z = Bx,y Bxy,z

and

xy+z = (xy)z.

This implies that

[x, a] = [y, b] ⇔ (x, a− b) quasi-invertible and y = xa−b

defines an equivalence relation on Z2. Informally, [x, a] = [xa−b, b]. The compact quotient manifold

Z = Z2/R = [m, a] : z, a ∈ Z

is a compact symmetric space called the conformal hull of Z. Its non-compact dual is the connected

0-component

Z := m ∈ Z : Bz,z invertible0,

which is a bounded symmetric domain in its circular and convex Harish-Chandra realization.

7

Example 1.1.1. For the matrix triple Z = Kr×s, Z can be identified with the Grassmannian

Gr(Kr+s) = E ⊂ Kr+s : dimE = r.

The embedding σ0 : Z ⊂ Z is given by mapping m ∈ Kr×s to its graph

σ0m := (ξ, ξz) : ξ ∈ K1×r ⊂ K1×r ×K1×s = K1×(r+s).

of m ∈ Kr×s. Via this embedding, we have

Z = m ∈ Z : Ir − zz∗ > 0 = m ∈ Z : Is − z∗z > 0.

For r = 1, Z becomes projective space Ps and Z is the unit ball Bs.

A basic theorem of M. Koecher characterizes hermitian symmetric spaces in terms of Jordan triples:

Theorem 1.1.2. In the complex setting, for every +hermitian Jordan triple Z the conformal hull Z is

a compact hermitian symmetric space, and every such space arises this way. Similarly, every hermitian

bounded symmetric domain can be realized as the spectral unit ball Z of a hermitian Jordan triple Z.

Thus there is a 1-1 correspondence between 0hermitian Jordan triples and 0hermitian symmetric

spaces of compact/non-compact type. Via this correspondence the two exceptional symmetric spaces

can be treated on an equal footing with the classical types. For real Jordan triples and symmetric

spaces, the above 1-1 correspondence is ’almost’ true (some exceptional symmetric spaces are missing).

*Peirce manifolds

*Jordan-Kepler manifolds

*Jordan-Schubert varieties

• Restricted Grassmannian

We now describe an infinite-dimensional example.

Example 1.1.3. Let A be an associative unital Banach algebra. The set

S := s ∈ A : s2 = 1

of all symmetries in A is a Banach manifold, with tangent space

TsS = s ∈ A : ss+ ss = 0.

The set

P := p ∈ A : p2 = p

of all idempotents in A is a manifold, with tangent space

TpP = p ∈ A : pp+ pp = p.

If A is a ∗-algebra, one obtains real manifolds by restricting to self-adjoint symmetries or projec-

tions, resp.

Lemma 1.1.4. There is a 1-1 correspondence between (self-adjoint) symmetries s ∈ S and idempotents

p ∈ P given by p = s+12 and s = 2p− 1, respectively.

Proof. We have (s+ 1

2

)2

=1

4(s2 + 2s+ 1) =

1

4(1 + 2s+ 1) =

s+ 1

2and

(2p− 1)2 = 4p+ 1− 4p = 1.

8

Identifying a subspace E with its orthogonal projection pE or the corresponding symmetry sE =

2pE − 1, the complex Grassmannian becomes a connected component of the manifold of self-adjoint

projections (resp. symmetries) for the block-matrix algebra

A = C(r+s)×(r+s) =

(Cr×r Cr×s

Cs×r Cs×s

).

This is more precisely the symplectic realization of the complex Grassmannian.

*The infinite-dimensional restricted Grassmannian Gres arises by taking symmetries s in A =

L(H), for a complex Hilbert space H, such that s−1 is of trace class. In the approach by Rajeev-Turgut,

it plays a basic role in 2-dimensional QCD.

• Loop groups

Let G be a compact connected 1-connected simple Lie group. Let

(ξ|η) := −tr(adξadη)

be the negative Killing form. Let S := S1 be the circle and

C∞∗ (S, G) = m : S→ G : m(1) = e

be the based loop group. It has the tangent space

TmC∞∗ (S, G) = C∞∗ (S, g) = u : S→ g : u(1) = 0.

• Conformal blocks

Let S be a compact oriented surface. Let G be a compact Lie group with Lie algebra g. Then the set

Ω1(S,G)

of all connexions A on the trivial G-bundle S × G is an affine space of infinite dimension. It has the

tangent space

TA(Ω1(S,G)) = Ω1(S, g)

at any A ∈ Ω1(S, g).

In the following, most manifolds will be constructed as quotient manifolds under an equivalence

relation. Let N be a (not necessarily connected) manifold and R ⊂ N × N be a closed submanifold

which defines an equivalence relation on N. Then M := N/R is a manifold if the *Godement properties

[?, ] hold: The projections R→M must be submersions. For u ∈ N let [u] denote the equivalence class

in the quotient manifold M = N/R.

1.1.1 Covered manifolds

A covered manifold is a K-manifold M endowed with an open covering by local charts σa : Ua →M,

where Ua is a domain in a vector space L ≡ Kn. Then the open sets Va := σa(Ua) ⊂ M cover M.

Denote by

σa := (σa)−1 : Va → Ua ⊂ L

the inverse of σa. The charts are related by transition maps

σab = σb σa

9

satisfying σa = σb σab , σab σa = σb and

σac = σbc σab .

We define two (closely related) equivalence relations for a covered manifold. First, consider the

disjoint union

U :=⋃Ua × a

endowed with the equivalence relation

(x, a) ≈ (y, b)⇔ x ∈ Ua, y ∈ Ub, σax = σby.

Equivalently, y = σba(x). In the following, we often write argument variables, such as x, y, as a subscript,

in order to save brackets. Now consider the disjoint union

V :=⋃Va × a

endowed with the equivalence relation

(m, a) ∼ (m, b)⇔ m ∈ Va ∩ Vb.

Then M = U/ ≈= V/ ∼ .


Example 1.1.5. Consider the projective space M = Ps. For 0 ≤ i ≤ s let Ui = L = Cs and define the

charts

σi : Cs → Ps, σi(z0, . . . , 1i, . . . , zs) := [z0, . . . , 1i, . . . , zs].

Conversely, put

Vi := [ζ] ∈ Ps : ζi 6= 0.

Then

σi : Vi → Cs, σi[ζ] =(ζ0

ζi, :, 1

i, :,ζs

ζi

).

The transition maps (for i < j) are given by

σij(z0, . . . , 1

i, . . . , zj , . . . , zs) =

(z0

zj, . . . ,

1i

zj, . . . , 1

j, . . . ,

zs

zj

).

In this way Ps = U/ ≈ becomes a covered manifold. In the special case s = 1 (Riemann sphere) we

obtain

σ0(z1) := [1, z1], σ0[ζ] :=ζ1

ζ0

σ1(z0) := [m0, 1], σ1[ζ] :=ζ0

ζ1

σ01(z1) =

1

z1, σ1

0(z0) =1

z0.

*finite charts for Grassmannian

For the conformal hull Z of a hermitian Jordan triple Z, instead of a finite covering we have a

’continuous’ covering by local charts

σa : Z → Z, z 7→ σaz := [z, a]

10

for any a ∈ Z. Thus U = Z × Z =: Z2 in this case, so that

Z = Z2/ ≈ .

In the special case a = 0 we write z0 = z and obtain the affine embedding

σ0 : Z ⊂ Z.

If (z, a) is quasi-invertible, then σaz = [z, a] = [za, 0] = (za)0 = za. In view of the addition formula (??)

the transition map between two local charts σa and σb is given by

σab (z) = za−b

on the open set z ∈ Z : (z, a− b) quasi-invertible.

1.1.2 Homogeneous manifolds

Another basic type of quotient manifolds are the homogeneous manifolds. Let G be a Lie group with

a closed subgroup H ⊂ G. Then the equivalence relation R := (g, gh) : g ∈ G, h ∈ H on G is

invariant under left G-translations and hence

M = G/H = G/R

becomes a quotient manifold with a left G-action.


projective space

Ps = SU(1, s)/U(s)

Grassmannian: Let Z = Cr×s, endowed with the operator norm ‖z‖ = sup spec(zz∗)1/2. Then the

matrix unit ball

Z = m ∈ Cr×s : ‖z‖ < 1 = m ∈ Cr×s : I − zz∗ > 0

is a symmetric domain under the pseudo-unitary group

G = U(r, s) =

(a b

c d

)∈ GL(r + s) :

(a b

c d

)(1 0

0 −1

)(a∗ c∗

b∗ d∗

)=

(1 0

0 −1

),

which acts on Z via Moebius transformations(a b

c d

)(m) = (az + b)(cz + d)−1.

Its maximal compact subgroup is

K =

(a 0

0 d

): a ∈ U(r), d ∈ U(s)

.

with the linear action m 7→ azd∗. For the compact dual we have

Gr(Cr+s) = U(r + s)/U(r)× U(s)

For a general hermitian Jordan triple, let K = Aut(Z) denote the compact linear Lie group of all

Jordan triple automorphisms of Z. The structure group K ⊂ GL(Z) is generated by all invertible

11

Bergman operators Ba,b, where (a, b) ∈ Z2 is quasi-invertible. It acts via linear transformations on Z.

On the other hand, the non-linear transformations of Z are the translations

taz := z + a

and the quasi-inverse maps

t∗az := z−a

for a ∈ Z. The conformal group G of Z is an algebraic Lie group generated by these three types of

transformations. It acts transitively on Z, giving a conformal realization

Z = G/G0

as a flag manifold. Here the parabolic subgroup

G0 := g ∈ G : g(0) = 0

is generated by K and the quasi-inverse maps (??). Putting B∗a,b = Bb,a, one can show that K carries

an involution such that

K = k ∈ K : k∗ = k−1.

This can be extended to an involution of G mapping ta to t∗a. Then

G = g ∈ G : g∗ = g−1

is a compact subgroup of G which still acts transitively on Z and satisfies

G ∩ G0 = K.

This yields a metric realization

Z = G/K

of Z as a compact hermitian symmetric space. Let

s0z := −z

denote the symmetry at the origin 0 ∈ Z. Then

G := g ∈ G : g−1 = s0g∗s0

is a non-compact subgroup of G which acts transitively on the spectral unit ball Z and also satisfies

G ∩ G0 = K.

This gives a metric realization

Z = G/K

of Z as a non-compact hermitian symmetric space. In summary, we have a diagram of Lie groups

G

G

??

G

__

K

__ ??

For K = C the structure group K is a complexification of K, and the conformal group G is a

complexification of G and of G. Moreover, G is the full biholomorphic automorphism group of Z, and

G is the full biholomorphic automorphism group of Z. (These remarks hold more precisely for the

connected components of the identity.)

In terms of the classification of complex hermitian Jordan triples we have

12

• K = U(r)× U(s) : z 7→ uzv, u ∈ U(r), v ∈ U(s), Z = Cr×s

• K = U(s), Z = C1×s = Cs

• symmetric matrices a = 1 (real case)

• anti-symmetric matrices a = 4 (quaternion case)

• K = T · SO(a+ 2) spin factorZ = Ca+2

• K =?, Z = C16exc and K = T · E6, Z = C27

exc.

1.2 Bundles

For any fibre bundle B over a manifold M, let Γ(B) denote the set of all sections (smooth/holomorphic)

Φ : M → B, satisfying π Φ = IM .. For the trivial bundle B = M × F with fibre F we write

Γ(M × F ) = Γ(M,F ).

Thus Γ(M,F ) = C∞(M,F ) for K = R and Γ(M,F ) = O(M,F ) for K = C. Denote by TM the tangent

bundle if K = R and the holomorphic tangent bundle if K = C. Thus in the complex case we have the

complexified tangent space

TCmM := TmM ⊕ TmM

and the real tangent space TRmM is the real subspace

TRmM = v + v : v ∈ TmM.

The complex structure Jm : TRmM → TR

mM is given by

Jm(v + v) = iv + iv

for all v ∈ TmM.

Example 1.2.1. For a domainM ⊂ C, with coordinate z = x+iy, we have the holomorphic/antiholomorphic

tangent vectors∂

∂z=

1

2

( ∂∂x− i ∂

∂y

),∂

∂z=

1

2

( ∂∂x

+ i∂

∂y

)satisfying

∂

∂x=

∂

∂z+

∂

∂z, i

∂

∂y=

∂

∂z− ∂

∂z.

The complex structure is

J∂

∂x= J

( ∂∂z

+∂

∂z

)= i

∂

∂z− i ∂

∂z= − ∂

∂y

J∂

∂y= −iJ

( ∂∂z− ∂

∂z

)= −i

(i∂

∂z+ i

∂

∂z

)=

∂

∂z+

∂

∂z=

∂

∂x.

Let P be a principal fibre bundle with Lie structure group H, called an H-bundle in the following,

over M = P/H. Any H-module (E, π) (i.e., a finite dimensional vector space E endowed with a

representation π of H), gives rise to an associated vector bundle

Pπ×HE = [p, φ] = [ph, h−πφ] : p ∈ P, h ∈ H, φ ∈ E.

13

Let [p] ∈M denote the equivalence class of p ∈ P.Writing Φ[p] = [p, Φp], for the so-called homogeneous

lift Φ of a section Φ, one obtains an isomorphism

Γ(Pπ×HE) ≡ f ∈ Γ(P,E) : fph = h−π fp ∀p ∈ P, h ∈ H.

In the following, principal bundles and their associated vector bundles will be defined in terms of

cocycles.

Proposition 1.2.2. Let M = N/ ∼ be a quotient manifold for an equivalence relation R ⊂ N. Let

β : R → H be a smooth map into a Lie group H, denoted by (u, v) 7→ βvu, which has the cocycle

property

βvu βwv = βwu

for all triples u ∼ v ∼ w in N. Then

Nβ×∼H := [u, h] = [v, (βvu)−1h] : (u, v) ∈ R, h ∈ H

becomes an H-bundle over M = N/R, with projection N ×β∼H →M, [u, h] 7→ [u].

As a consequence any H-module E gives rise to an induced vector bundle

Nβ,π×∼E := (N

β×∼H)

π×HE = [u, φ] = [v, (βvu)−πφ] : (u, v) ∈ R, φ ∈ E

over M. Writing Φ[u] = [u, Φu] one obtains an isomorphism

Γ(Nβ,π×∼E) ≡ f ∈ Γ(N,E) : fv = (βvu)−π fu ∀ (u, v) ∈ R.

We often omit the reference to π if the context is clear.


For a covered manifold M consider maps βab : Va ∩ Vb → H satisfying the cocycle property

βab (m) βbc(m) = βac (m)

for all m ∈ Va ∩ Vb ∩ Vc. Then

βb,ma,m := βba(m)

defines a cocycle β : R→ H in the sense of (??). Hence

Vβ×∼H = [m,h]a = [m,βba(m)h]b : m ∈ Va ∩ Vb, h ∈ H

becomes an H-bundle over the quotient manifold M = V/ ∼ . Any H-module E gives rise to an

induced vector bundle

Vβ×∼E := (V

β×∼H)×

HE = [m,φ]a = [m,βba(m)φ]b : m ∈ Va ∩ Vb, φ ∈ E

over M. Writing Φ[m] = [m,Φa(m)]a one obtains an isomorphism

Γ(Vβ×∼E) ≡ (Φa) ∈

∏a

Γ(Va, E) : Φa(m) = βab (m) Φb(m) ∀ m ∈ Va ∩ Vb.

By local triviality, every principal bundle and every vector bundle can be realized this way.

14

The tangent bundle arises as follows. For a covering family of charts σa ofM and a map f : Va → E

we write∂f

∂σa(m) := (f σa)′(x) ∈ Hom(L,E)

if m = σax ∈ Va. Applying this notation to E = L and f = σb, we obtain

∂σb∂σa

(m) = (σb σa)′(x) = (σab )′(x) ∈ L(L) endomorphisms

for m = σax ∈ Va ∩ Vb. Now let m = σax = σby. Since y = σab (x), the chain rule yields

∂σc∂σa

(m) = (σac )′(x) = (σbc)′(y)(σab )′(x) =

∂σc∂σb

(m)∂σb∂σa

(m)

for all x, with y := σab (x). We sometimes write this relation in the opposite order

t · dxσac = (t · dxσab ) · dyσbc

for all t ∈ L. It follows that

σba(m) :=∂σa∂σb

(m)

defines a GL(L)-valued cocycle on R. Hence we obtain a GL(L)-bundle

Vσ×∼GL(L) = [m,h]a = [m,h

∂σb∂σa

(m)]b : m ∈ Va ∩ Vb

over M, called the bein bundle. Via the defining representation of GL(L), we obtain the associated

vector bundle

Vσ×∼L = [m, t]a = [m,

∂σb∂σa

(m)t]b : m ∈ Va ∩ Vb, t ∈ L

which is isomorphic to the tangent bundle TM by identifying [m, t]a with (Txσa)t for m = σax. In fact,

we have

(Txσa)t = Tx(σb σab )t = (Tyσ

b)(σab )′xt = (Tyσb)∂σb∂σa

(m)t

for m = σax = σby. The corresponding sections (vector fields) are

Γ(Vσ,ι×∼L) ≡ (T a) ∈

∏a

Γ(Va, L) : T bm =∂σb∂σa

(m)T am ∀m ∈ Va ∩ Vb.

Similarly, the cotangent bundle T ∗M is isomorphic to the cocycle bundle

Vσ×∼L∗ = [m,ϑ]a = [m,ϑ ∂σa

∂σb(m)]b : m ∈ Va ∩ Vb, ϑ ∈ L∗

by identifying [m,ϑ]a with ϑ (Tmσa) when m = σax. In fact, we have

ϑ (Tmσa) = ϑ Tm(σba σb) = ϑ (σba)′(x)(Tmσb) = ϑ ∂σa∂σb

(m)(Tmσb)

for m = σax = σby. The corresponding sections (1-forms) are

Γ(Vσ,ι×∼L∗) ≡ (Θa) ∈

∏a

Γ(Va, L∗) : Θb

m = Θam

∂σa∂σb

(m) ∀m ∈ Va ∩ Vb.

These bundles can also be described in the setting M = U/ ≈ . The formulas are

Uσ×∼GL(L) = [x, h]a = [y, h(σab )′x]b : x ∈ Ua, y ∈ Ub, σax = σby

15

Uσ×∼L = [x, t]a = [y, (σab )′xt]b : x ∈ Ua, y ∈ Ub, σax = σby, t ∈ L

Γ(Uσ×∼L) ≡ (T a) ∈

∏a

Γ(Ua, L) : T by = (σab )′xTax ∀x ∈ Ua, y ∈ Ub, σax = σby,

Uσ×∼L∗ = [x, ϑ]a = [y, ϑ (σba)′y]b : x ∈ Ua, y ∈ Ub, σax = σby, ϑ ∈ L∗

Γ(Uσ×∼L∗) ≡ (Θa) ∈

∏a

Γ(Ua, L∗) : Θb

y = Θax (σba)′y x ∈ Ua, y ∈ Ub, σax = σby.

In case M carries an H-structure, for a closed subgroup H ⊂ GL(L), the transition maps σab can

be chosen such that ∂σb∂σa

(m) = (σab )′x ∈ H, and we obtain H-bundles instead.


Example 1.2.3. For the projective space M = Ps taking derivatives of (??), we obtain

ek · (∂mzi

zj) =

∂

∂zk∂m

zi

zj=δik z

j − zi δjk(zj)2

for the standard base ek of Cs. For any other u = uk ek ∈ Cs we obtain

u · (∂mzi

zj) = uk ek · (∂m

zi

zj) = uk

δik zj − zi δjk(zj)2

=ui zj − zi uj

(zj)2

Proposition 1.2.4. For a hermitian Jordan triple Z the conformal hull carries a K-structure. More

precisely, the map β : Z2 → K defined by

βw,bz,a := Bz,a−b

is a cocycle, and the induced K-bundle Z2×β∼ K over Z is the bein (tangent frame) bundle.

Proof. The cocycle property follows from the addition formula (??). The well-known identity

∂zt∗a = B−1

z,−a

implies that the transition map σab = t∗b−a has the derivative

∂zσab = ∂zt

∗b−a = B−1

z,a−b.

This gives the bein bundle.

As a consequence, any K-module E yields an induced vector bundle

Z2β×∼E := (Z2

β×∼K)

π×K

E = [z, φ]a = [za−b, B−πz,a−bφ]b : (z, a− b) quasi-invertible

over Z = Z2/ ≈ . Writing Φ[z,a] = [z,Φaz ]a the sections Φ are described by

Γ(Z2β×∼E) ≡ (Φa) ∈ ΠaΓ(Z,E) : Φbza−b = B−πz,a−bΦ

az.

Since Z ⊂ Z is a dense open subset via the embedding z 7→ z0 = [z, 0], a section Φ is uniquely

determined by its trivialization Φ := Φ0. Thus the mapping Φ 7→ Φ identifies Γ(Z2×β∼E) with a vector

space of maps from Z to E. For the defining representation K ⊂ GL(Z) we obtain the tangent bundle

Z2β×∼Z := (Z2

β×∼K)×

K

Z = [z, t]a = [za−b, B−1z,a−bt]b : (z, a− b) quasi-invertible, t ∈ Z ≡ T Z

16

and the cotangent bundle

Z2β×∼Z∗ = (Z

β×∼K)×

K

Z∗ = [z, ϑ]a = [za−b, ϑ Bz,a−b]b : (z, a− b) quasi-invertible ≡ T ∗Z.

Example 1.2.5. Riemann sphere


For a Lie group G with a closed subgroup H ⊂ G, we may regard

G = G×HH

as an H-bundle over M := G/H. The homogeneous vector bundle associated to an H-module E of

H is given by

Gπ×HE := [g, φ] = [gh, h−πφ] : g ∈ G, h ∈ H, φ ∈ E.

It is G-equivariant under the action

gg′H [g′, φ] := [gg′, φ].


The derivative ∂0q of q ∈ G0 belongs to K, and

∂0 : G0 → K, q 7→ ∂0q.

is a homomorphism.

Proposition 1.2.6. The mapping

[z, h]a 7→ [t∗−atz, h]

induces an isomorphism

Z2β×∼K ≡ G

∂0

×G0

K

of K-bundles over Z.

Proof. The transformation g := t∗−atz ∈ G has the derivative

∂0g = (∂zt∗−a)(∂0tz) = ∂zt

∗−a = B−1

z,a.

Now let [z, φ]a = [w,B−πz,a−bφ]b. Then t∗−atz(0) = t∗−a(z) = za = wb = t∗−b(w) = t∗−btw(0). Hence there

exists q ∈ G0 such that t∗−btw = t∗−atzq. Then

B−1w,b = ∂0(t∗−btw) = ∂0(t∗−atzq) = ∂0(t∗−atz) ∂0q = B−1

z,a ∂0q.

Therefore ∂0q = Bz,a B−1w,b = Bz,a−b by the addition formula (??). This implies

[t∗−atz, h] = [t∗−btwq−1, h] = [t∗−btw, (∂0q)

−1h] = [t∗−btw, B−1z,a−bh].

Hence the assignment (??) is a well-defined map Z2×β∼ K → G×∂0

G0K, which is a bijection.

Thus for any G0-module E the mapping [z, φ]a 7→ [t∗−atz, φ] induces a vector bundle isomorphism

Z2β×∼E ≡ G

∂0

×G0

E.

As a consequence, the vector bundle Z2×β∼E carries a G-action. This is not obvious in the coordinate

chart picture.

17

1.3 0-Geometry: Hermitian Metrics

A 0-geometry on a vector bundle is a hermitian metric. We can also allow pseudo-metrics of

indefinite signature and speak generally of metric vector spaces and vector bundles. The positive

definite case will be called 0metric (0 negative eigenvalues).

Let P be an H-bundle over M = P/H. Then any metric H-module E defines a metric

([p, φ]|[p, η] := (ξ|η)

on the associated vector bundle P ×πH E. This is well-defined since

([ph, h−πφ]|[ph, h−πφ]) = (h−πξ|h−πξ) = (ξ|η).


Consider an H-valued cocycle βba and a metric H-module E.

Lemma 1.3.1. Let E be a metric vector space, with inner product (ξ|η). A family of smooth maps

ha : Va → H×(E) (self-adjoint invertible), satisfying the compatibility condition

ham = βba(m)∗ hbm βba(m)

for all m ∈ Va ∩ Vb defines a metric on V ×β∼E via

([m,φ]a|[m, η]a)m := (ξ|hamη).

For E = C, a family of smooth functions ha : Va → R>, satisfying the compatibility condition

ham = |βba(m)|2 hbm

for all m ∈ Va ∩ Vb defines a 0metric on the line bundle V ×β∼C via

([m,φ]a|[m, η]a)m := (ξ|hamη).

Proof. The identification [m,φ]a = [m,βba(m)φ]b yields

([m,βba(m)φ]b|[m,βba(m)η]b) = (βba(m)ξ|hbmβba(m)η)

= (ξ|βba(m)∗hbmβba(m)η) = (ξ|hamη) = ([m,φ]a|[m, η]a).

Proposition 1.3.2. Let E be a 0metric vector space. Let (χa) be a partition of unity subordinate to V.Then the family

ha :=∑c

βc∗a χc βca, ham =

∑c

βca(m)∗ χc(m) βca(m)

defines a 0metric on V ×β∼E. For E?C, the family

ha :=∑c

|βca|2 χc, ham =∑c

|βca(m)|2 χc(m)

defines a 0metric on the line bundle V ×β∼C.

18

Proof. Since the sum (??) is locally finite and the χc(m) add up to 1, (??) defines a smooth map from

Va to the positive definite matrices. For m ∈ Va ∩ Vb the cocycle property (??) implies

βba(m)∗ hbm βba(m) = βba(m)∗∑c

βcb(m)∗ χc(m) βcb(m) βba(m)

=∑c

(βcb(m)βba(m))∗ χc(m) βcb(m)βba(m) =∑c

βca(m)∗ χc(m) βca(m) = ham

Proposition 1.3.3. Let (ha) be a 0metric on V ×β∼E. Then

κab (m) := (ham)1/2 βab (m) (hbm)−1/2

defines a unitary cocycle, i.e. κag(m) ∈ U(E) for all m ∈ Va ∩ Vb.

Proof. The cocycle property follows from

γab γbc = (ha)1/2 βab (hb)−1/2 (hb)1/2 βbc (hc)−1/2 = (ha)1/2 βab β

bc (hc)−1/2 = (ha)1/2 βac (hc)−1/2 = γac .

Moreover, we have

κa∗b κab = ((ha)1/2 βab (hb)−1/2)∗ (ha)1/2 βab (hb)−1/2 = (hb)−1/2 βa∗b ha βab (hb)−1/2 = (hb)−1/2 hb (hb)−1/2 = I.

As a consequence we may form the 0metric vector bundle

Vγ×∼E = 〈m, ξ〉a = 〈m, γab (m)ξ〉b : m ∈ Va ∩ Vb.

It carries the 0metric

(〈m, ξ〉a|〈m, η〉a) = (ξ|η).

since the condition (??) is trivially satisfied by ham = IE .

For the tangent bundle, a family of smooth maps ha : Va/Ua → H×(L), satisfying the compatibility

condition

ham =∂σa∂σb

(m)∗ hbm∂σa∂σb

(m), hax = (σba)′(y)∗ hby (σba)′(y)

on Va ∩ Vb/Ua ∩ Ub defines a tangent metric on V/U ×σ∼ L ≡ TM via the assignment

([m,u]a|[m, v]a)m := (u|hamv), ([x, u]a|[x, v]a) := (u|haxv),

Similar for the cotangent bundle. In the positive case the family

ham :=∑c

∂σa∂σc

(m)∗ χc(m)∂σa∂σc

(m)

induces a tangent 0metric on M. The associated unitary cocycle is

κab (m) := (ham)1/2 ∂σb∂σa

(m) (hbm)−1/2

19


Example 1.3.4. For K = C consider the holomorphic tangent bundle TP on the Riemann sphere P1,

endowed with the tangent metric

h0z = (1 + zz)−2.

The coordinate change w := 1z yields

h1w(

∂

∂w,∂

∂w) = (1 + ww)−2.

*The metric is invariant under SU(2).


If H ⊂ G is a closed subgroup and E is a metric H-module, then G×H E becomes a G-equivariant

metric vector bundle with respect to the fibre metric

([g, φ]|[g, ψ])gH := (φ|ψ).

For line bundles, with φ, ψ ∈ C, the 0metric is

([g, φ]|[g, ψ])gH := φψ.


Any unitary K-representation (E, π) has a holomorphic extension to K. Then the mapping [z, φ]a 7→[t∗−atz, φ] induces an isomorphism

Z2κ×∼E ≡ G

π×KE

of hermitian holomorphic vector bundles. As a consequence, the restricted G-action on the vector

bundle Z2×β∼E is isometric.

1.4 1-Geometry: Connexions

The Lie algebra Γ(TM) of vector fields on M is endowed with the commutator

[X,Y ]m = Xm · dmY − Ym · dmX.

The infinitesimal action of vector fields on maps Φ : G→ E is given by

(dXΦ)g := Xg · TgΦ.

Proposition 1.4.1.

dX(dY Φ)− dY (dXΦ) = d[X,Y ]Φ.

For a real manifold M let

Ωr(M,R) = Γ(T rM)

denote the space of all smooth real r-forms over M. Thus

Ω0(M) = Γ(M ×R) = Γ(M,R)

consists of all smooth functions on M. The exterior derivative

d : Ωr(M)→ Ωr+1(M)

20

is defined by the Palais formula: Given vector fields ξ0, . . . , ξp then

(dω)(X0, . . . , Xp) =

p∑i=0

(−1)i Xiδ ω(X0, . . . , X

i, . . . , Xp)

+∑i<j

(−1)j−iω([Xi, Xj ], X0, . . . , Xi, . . . , X

j, . . . , Xp).

This definition differs from [?, Proposition 3.11] by a factor of 1p+1 , but makes sense in any characteristic.

For a 2-form ω we obtain

(X,Y, Z)dωm = X · (Y, Z)ω − Y · (X,Z)ω + Z · (X,Y )ω − ([X,Y ], Z)ω − ([Y,Z], X)ω + ([X,Z], Y )ω

If B is a smooth vector bundle over a real manifold M, one can still define differential forms Ωr(B), but

the exterior differential d makes sense only if B = M ×E is trivial. In this case we write Ωr(M ×E) =

Ωr(M,E).

For a complex manifold M, the complexified tangent space splits into the holomorphic and anti-

holomorphic tangent space. The complexified smooth differential forms have a splitting

Ωr(M,C) =∑p+q=r

Ωp,q(M)

into (p, q)-forms. Accordingly, the differential

d : Ωr(M,C)→ Ωr+1(M,C)

splits as d = ∂ + ∂, with

∂ : Ωp,q(M)→ Ωp+1,q(M), ∂ : Ωp,q(M)→ Ωp,q+1(M).

If B is a holomorphic vector bundle over a complex manifold M, the anti-linear part ∂ of the exterior

differential is still well-defined.

In general, for a G-bundle P over M = P/G let P ×adG g denote the adjoint g-bundle of P. The space

Ω1(P ) of all G-connexions on P over M is an affine space, with tangent space

TAΩ1(P ) = Ω1(Pad×Gg)

at any A ∈ Ω1(P ). We write Ω1(M,G) for the space of connexions on the trivial G-bundle M ×G, with

tangent spaces Ω1(M, g).

Let P ×πGE be an associated vector bundle. Let m ∈ M and u ∈ TmM. Choose p ∈ P with

m = [p] = π(p). For any connexion A ∈ Γ1(P ) the horizontal subspace TAp P yields an isomorphism

Tpπ : TAp P → TmM.

Hence there exists a unique horizontal tangent vector uA ∈ TAp P such that Tp(π)uA = u. Given a

section Φ, apply uA to the smooth function Φ : P → E we obtain uA · dpΦ ∈ E. Then

u · dAmΦ = [p, uA · dpΦ]

is independent of the choice of p [?, Section III.1, Lemma on p. 115], and we obtain the covariant

differential as a 1-form dAΦ. The map

dA : Ω0(Pπ×GE)→ Ω1(P

π×GE), Φ 7→ dAΦ

21

satisfies the Leibniz rule

dA(fΦ) = df ∧ Φ + f · dAΦ

for all sections Φ ∈ Ω0(P ×πGE) and functions f ∈ Ω0(M,K). On the other hand, given a vector field

X ∈ Γ(TM) we define the covariant derivative dAX acting on sections. The two notions are related

by

X · dAΦ = dAX · Φ.

The value at a given point m ∈M is denoted by

(X · dAΦ)m = (dAX · Φ)m = Xm · (dAmΦ)

Thus there is a canonical mapping

Ω1(P )× Ω0(Pπ×GE)→ Ω1(P

π×GE), (A,Φ) 7→ dAΦ.

If P ×H E →M is a holomorphic vector bundle one can also consider the anti-linear part

∂A

: Ω0(P ×HE)→ Ω0,1(P ×

HE).

Proposition 1.4.2. For any tangent metric g there is a unique Levi-Civita connexion ðg which

satisfies

dXg(Y,Z) = g(dðgX Y,Z) + g(Y, dðgX Z)

and is torsion-free, i.e.,

dðgX Y − dðgY X = [X,Y ].

It is given by

2g(dðgX Y,Z) = dXg(Y, Z) + dY g(Z,X)− dZg(Y,X) + g(Z, [X,Y ])− g(Y, [X,Z])− g([Y, Z], X)

Proof. Combining the two properties yields

dXg(Y,Z)+dY g(Z,X)−dZg(Y,X) = g(dðgX Y, Z)+g(Y, dðgX Z)+g(dðgY Z,X)+g(Z, dðgY X)−g(dðgZ Y,X)−g(Y, dðgZ X)

= g(dðgX Y+dðgY X,Z)+g(Y, dðgX Z−dðgZ X)+g(dðgY Z−dðgZ Y,X) = g(2dðgX Y+[Y,X], Z)+g(Y, [X,Z])+g([Y, Z], X)

The definition of the Levi-Civita connexion ðg is analogous to the exterior derivative

dω(X,Y, Z) = dXω(Y,Z)− dY ω(X,Z) + dZω(X,Y )− ω([X,Y ], Z) + ω([X,Z], Y )− ω([Y,Z], X)

= dXω(Y,Z) + dY ω(Z,X)− dZω(Y,X) + ω(Z, [X,Y ])− ω(Y, [X,Z])− ω([Y,Z], X)

of a 2-form ω.

Now let P = N ×β∼H be a cocycle H-bundle on M = N/R. Then the adjoint bundle is

(Nβ×∼H)

ad×Gh = N

β,ad×R

h.

Hence the affine space Ω1(N ×β∼H) of all H-connexions on N ×β∼H has the tangent space

TA(Ω1(Nβ×∼H)) = Ω1(N

β,ad×R

h

at any A ∈ Ω1(N ×β∼H).

22


For covered manifolds, connexions are constructed as follows. A connexion A on V ×β∼E is given by

the covariant differential

dA : Ω0(Vβ×∼E)→ Ω1(V

β×∼E).

Given v ∈ TmM and a section Φ ∈ Ω0(V ×β∼E) we have local representatives (v ·dAmΦ)a ∈ E for m ∈ Va.

Proposition 1.4.3. A family m 7→ Aam of gl(E)-valued 1-forms on Va such that

Aa = βab

(dβba +Ab βba

), Aam = βab (m)

(dmβ

ba +Abm βba(m)

)for m ∈ Va ∩ Vb, as an identity of linear functionals TmM → gl(E), defines a (global) connexion A on

V ×β∼E with covariant derivative

(v · dAΦ)am = v · dmΦa + (v ·Aam)Φam.

Here v · dmΦa ∈ E and v ·Aam ∈ L(E). For E = C, a family of 1-forms Aam on Va, satisfying

Aam −Abm =dmβ

ba

βba(m)= dm log βba

for all m ∈ Va ∩ Vb, yields a global connexion A on V ×β∼C with covariant derivative (??).

Proof. In order to define a global connexion, we need to check the compatibility relation

(v · dAmΦ)a = βab (m)(v · dAmΦ)b

for m ∈ Va ∩ Vb and v ∈ TmM. The condition (??) becomes

v ·Aam = βba(m)(v · dmβab + (v ·Abm) βab (m)

)with v · dmβab ∈ L(E). Since v · dmΦb = v · dm(βbaΦa) = (v · dmβba)Φa(m) + βba(m)(v · dmΦa) by the

product rule, we have

βab (m)(v · dmΦb) = βab (m)(v · dmβba) Φa(m) + v · dmΦa.

Hence (??) implies

(v · dAΦ)a(m) = v · dmΦa + (v ·Aam)Φa(m) = v · dmΦa + βab (m)(v · dmβba + (v ·Abm) βba(m)

)Φa(m)

= v·dmΦa+βab (m)(v·dmβba)Φa(m)+βab (m)(v·Abm) Φb(m) = βab (m)(v·dmΦb+(v·Abm)Φb(m)

)= βab (m)(v·dAΦ)b(m).

For E = C, we have

βab (m) Abm βba(m) = Abm βab (m) βba(m) = Abm.

Thus (??) simplifies to (??).

The space Ω1(V ×β∼GL(E)) of all connexions on V ×β∼E is an affine space, with tangent space

TA(Ω1(Vβ×∼GL(E)) = Ω1(V

β×∼gl(E)).

In fact, let (Aa1) and (Aa2) be two connexions on V ×β∼E. Then

Aa := Aa1 −Aa2

is a smooth mapping Va ∩ Vb → gl(E) such that

Aa = βab Λb βba

on Va ∩ Vb. Thus (Aa) defines a global gl(E)-valued 1-form on M.

23

Proposition 1.4.4. Let (χa) be a partition of unity subordinate to (Va). Then the family

Aa =∑c

χc βca(dβac )

defines a global connexion on V ×β∼E.

Proof. On Va ∩ Vb ∩ Vc we have

βca(dβac ) = −(dβca)βac = −(d(βbaβcb))β

ac = −(dβba)βcbβ

ac − βba(dβcb)β

ac

= −(dβba)βab − βba(dβcb)βbcβ

ab = βba(dβab ) + βbaβ

cb(dβ

bc)β

ab = βba

(dβab + βcb(dβ

bc)β

ab

).

it follows that

Aa =∑c

χcβca(dβac ) =

∑c

χcβba

(dβab + βcb(dβ

bc)β

ab

)= βba

(dβab +

∑c

χcβcb(dβ

bc)β

ab

)= βba(dβab +Abβab )

For the tangent bundle, a family m 7→ Aam of gl(L)-valued 1-forms on Va such that

Aa =∂σb∂σa

(d∂σa∂σb

+ Ab ∂σa∂σb

)on Va ∩ Vb, defines a global tangent connexion A on M = V/R, with covariant derivative

(v · dAX)am = v · dmXa + (v ·Aam)Xa

m.

Here v · dmXa ∈ L and v ·Aam ∈ L(L).

If M is a complex manifold, we consider holomorphic vector bundles over M defined by holomorphic

cocycles βab .

Theorem 1.4.5. Let M be a complex manifold, with a metric on (ha) on V ×β∼E. Then the family

(ðh)am := (ham)−1 ∂mha

of (1, 0)-forms induces a (unique) connexion ðh on V ×β∼E which satisfies

dX(ξ|η) = (dðhX ξ|η) + (ξ|dðhX η)

for all real vector fields X ∈ Γ1(MR), and the (’torsion-free’) condition

∂ðh

Φ = 0

for all holomorphic sections Φ ∈ Γ(V ×β∼E). For E = C, given a 0metric (ha) on the line bundle

V ×β∼C, the family

(ðh)am :=∂mh

a

ham= ∂m log ha

of (1, 0)-forms induces the Chern connexion ðh on V ×β∼C.

Proof. Since we take the C-linear Wirtinger derivative ∂mha, it follows that (ðh)a is a (1, 0)-form with

values in gl(E). Since βba is holomorphic in m we have ∂mβb∗a = 0 and ∂mβ

ba = dmβ

ba. Applying the

product rule to ham = βba(m)∗ hbm βba(m) we obtain

∂mha = βba(m)∗

((∂mh

b)βba(m) + hbm (∂mβba))

= βba(m)∗(

(∂mhb)βba(m) + hbm (dmβ

ba)).

24

It follows that

(ðh)am = (ham)−1 ∂mha =

(βab (m)(hbm)−1βab (m)∗

)βba(m)∗

((∂mh

b)βba(m) + hbm (dmβba))

= βab (m)(

(hbm)−1(∂mhb)βba(m) + dmβ

ba

)= βab (m)

((ðh)bm βba(m) + dmβ

ba

).

Thus (??) is satisfied. For the second assertion, let Φ = (Φa) be a holomorphic section. Then the Φa

are holomorphic and hence ∂mΦa = 0. It follows that

(v · dðhm Φ)a = v · dmΦa + (ham)−1(v · ∂mha) Φa(m) = v · ∂mΦa + (ham)−1(v · ∂mha) Φa(m)

is C-linear in v. Therefore the anti-linear part (∂ðhm Φ)a vanishes for all a and hence ∂

ðhΦ = 0.

For a tangent metric (ha) the family

(ðh)am :=∂mha

ham

of (1, 0)-forms induces the Chern connexion ðh on the tangent bundle V ×β∼ L ≡ TM. A complex

manifold M endowed with a tangent 0metric is called a hermitian manifold. For a cocycle description,

endow L with an inner product (ξ|η).


Example 1.4.6. For the holomorphic tangent bundle of P1, with metric (??), the general formula (??)

yields the connexion 1-form

(ðh)0z =

∂h0z

h0z

= (1 + zz)2 ∂

∂z(1 + zz)−2dz = −2(1 + zz)2 (1 + zz)−3 z dz =

−2z

1 + zzdz

of type (1, 0).


We first construct some vector fields on M = G/H. Consider the left translation action

gLg′ := gg′

of G on itself. For γ ∈ g define a vector field γL ∈ Γ(TG) by

γLg := (TegL)γ = γ · (TegL) ∈ TgG

Lemma 1.4.7. The vector field γL on G is left-invariant, i.e. for each g ∈ G the left translation gL

on G satisfies

gL∗ γL = γL.

Proof. This follows from

(gL∗ γL)gg′ = (Tg′g

L)γLg′ = (Tg′gL)(Teg

′L)γ = Te(gL g′L)γ = Te((gg

′)L)γ = γLgg′ .

25

Consider the left translation action g 7→ gλ of G on G/H given by

gλ(g′H) := gg′H.

Then the canonical projection π : G→ G/H satisfies

π gL = gλ π

for all g ∈ G. For γ ∈ g define a vector field γλ ∈ Γ(G/H) by

γλgH := (Tegλ)(Teπ)γ

Lemma 1.4.8. The vector field γλ on G/H is left-invariant, i.e. for each g ∈ G the left translation gλ

on G/H satisfies

gλ∗γλ = γλ.

Proof. This follows from

(gλ∗γλ)gg′H = (Tg′Hg

λ)γλg′H = (Tg′Hgλ)(THg

′λ)(Teπ)γ = TH(gλg′λ)(Teπ)γ = TH((gg′)λ)(Teπ)γ = γλgg′ .

Lemma 1.4.9. For all γ ∈ g we have

π∗γL = γλ,

i.e.,

γλgH = (Tgπ)γLg .

Proof.

γλgH := (Tegλ)(Teπγ) = (Tgπ)(Teg

Lγ)

Lemma 1.4.10. The left-invariant vector field γ satisfies

(dγL f)g = ∂0t fg exp(tγ)

Proof.

∂0t fg exp(tγ) = ∂0

t (f gL)(exp(tγ)) = de(f gL)γ = (dgf)(TegL)γ = γLg · dgf = (dγL f)g.

Let M = G/H. Then we have a commuting diagram

TgHM THMTHg˜oo

TgG

Tgπ

OO

TeGTeLgoo

Teπ

OO

TΘg G

OO

mTeLg

oo

OO

Lemma 1.4.11. For η ∈ h we have

dηL Φ = −ηπ Φ

26

Proof. It follows from (??) and Lemma (??) that

(dηL Φ)g = ∂0t Φg exp(tη) = ∂0

t exp(tη)−π Φg = −ηπ Φg.

Consider a vector space splitting

g = h⊕m

which is AdH -invariant. Thus [h, h] ⊂ h and [h,m] ⊂ m, but not necessarily [m,m] ⊂ h. For γ ∈ g

we write γh and γm for the projections. Given an H-module (E, π) we consider the corresponding

infinitesimal action

ηπ := ∂0t exp(tη)π

for all η ∈ h.

Proposition 1.4.12. The left-invariant connexion A associated with a splitting (??) has the covariant

derivative

(dAγλ Φ)∼ = dγLΦ + γπh Φ, (γλ · dAΦ)∼g = γLg · dgΦ + γπh Φg

for all γ ∈ g.

Proof. For M = G/H every tangent vector in TgHM can be written as

γλgH = (γ · Teπ) · (THgλ)

for a uniquely determined γ ∈ m. Then γLg = γ · (TegL) belongs to (TegL)m = TAg G, since the connexion

is left-invariant, and the projection is

(Tgπ)γLg = (Tgπ)(TegL)γ = (THg

λ)(Teπ)γ = γλgH .

Therefore γLg = (γλgH)A is the horizontal lift of γλ. Now (??) implies

(dAγλ Φ)gH = γλgH · dAgHΦ = [g, (γλgH)A · dgΦ] = [g, γLg · dgΦ] = [g, (dγLΦ)g].

Equivalently, we have (dAγλ Φ)∼ = dγLΦ for all γ ∈ m. This implies

(dAγλ Φ)∼ = dγLm Φ = dγL Φ− dγLh Φ

for all γ ∈ g, since, by (??), we have γλ = 0 on M for all γ ∈ h = Ker Teπ and hence both sides of (??)

vanish. Applying (??) to η := γh, the assertion follows.

1.5 2-Geometry: Curvature

For every G-connexion A ∈ Ω1(P ) the covariant derivative (??) has a canonical extension

dA : Ωj(Pπ×GE)→ Ωj+1(P

π×GE)

for j ≥ 0, satisfying a graded Leibniz rule

dA(ϑ ∧ Φ) = dϑ ∧ Φ + (−1)iϑ ∧ dAΦ

for all Φ ∈ Ωj(P ×πGE) and ϑ ∈ Ωi(M,K). Thus there is a canonical mapping

Ω1(P )× Ωj(Pπ×GE)→ Ωj+1(P

π×GE), (A,Φ) 7→ dAΦ.

If P ×H E →M is a holomorphic vector bundle one can also consider the anti-linear part

∂A

: Ωp,q(P ×HE)→ Ωp,q+1(P ×

HE).

27

Proposition 1.5.1. The square

dAdA : Ω0(P ×HE)→ Ω2(P ×

HE)

can be written as

dA(dAΦ) = (dAA) ∧ Φ.

for the curvature 2-form dAA ∈ Ω2(P ×adH h) More generally, the square

dAdA : Ωp(P ×HE)→ Ωp+2(P ×

HE)

is given by

dAdA(ϑ⊗ Φ) = (dAA) ∧ (ϑ⊗ Φ)

Proof. Using ⊗ also for multiplication by functions, we have

dA(dA(f⊗Φ)) = dA(df⊗Φ+f⊗dAΦ) = d(df)⊗Φ−df ∧dAΦ+df ∧dAΦ+f⊗ (dAdAΦ) = f⊗ (dAdA)Φ

since ddf = 0. Thus dAdA commutes with multiplication by functions f and is therefore a multiplication

by a 2-form with values in the bundle P ×adH h.

For a matrix group the curvature is given by

dAA = dA+ [A ∧A].

Thus the curvature depends in a non-linear, quadratic manner on A. For abelian groups. we have

dAA = dA.

For a holomorphic vector bundle with metric h we have the Chern connexion ðh, with covariant

derivative dðh and curvature dðhðh.


For a covered manifold M this looks as follows. For any vector field X ∈ Γ(TM) we put

(X · dAΦ)am := (Xm · dAΦ)a.

Then the family (X ·dAΦ)a of smooth maps Va → E is a localized section. The curvature of A is defined

by

dAX(dAY Φ)− dAY (dAXΦ)− dA[X,Y ]Φ = dAA(X,Y ) · Φ

Proposition 1.5.2. The curvature of (Aa) is given by the family

(u, v) · (dAA)am := v · (u · dmAa)− u · (v · dmAa) + [u ·Aam, v ·Aam].

of gl(E)-valued 2-forms. Here [S, T ] = ST − TS is the commutator in gl(E). For E = C the curvature

of (Aa) simplifies to

(u, v) · (dAA)am := v · (u · dmAa)− u · (v · dmAa) = (dAa)m(u, v).

Proof. Let X,Y be smooth vector fields on M. Then

(Ym · dAmΦ)a = Ym · dmΦa + (Ym ·Aam)Φam

and hence

(Xm · dAm(Y · dAΦ))a = Xm · dm(Ym · dAmΦ)a + (Xm ·Aam)(Ym · dAmΦ)a

28

= Xm · dm(Ym · dmΦa + (Ym ·Aam)Φam

)+ (Xm ·Aam)

(Ym · dmΦa + (Ym ·Aam)Φam

)= (Xm · dmY ) · dmΦa + (Xm, Ym)d2

mΦa + ((Xm · dmY ) ·Aam)Φam + (Ym · (Xm · dmAa))Φam

+(Xm ·Aam)(Ym ·Aam)Φam + (Ym ·Aam)(Xm · dmΦa) + (Xm ·Aam)(Ym · dmΦa).

For the commutator we obtain, using symmetry of the second derivative d2mΦa and the symmetry of

the last two summands,

(Xm · dAm(Y · dAΦ))a − (Ym · dAm(X · dAΦ))a

= (Xm · dmY − Ym · dmX) · dmΦa + ((Xm · dmY − Ym · dmX) ·Aam)Φam

+(Ym · (Xm · dmAa)−Xm · (Ym · dmAa))Φam +(

(Xm ·Aam)(Ym ·Aam)− (Ym ·Aam)(Xm ·Aam))

Φam

= [X,Y ]m · dmΦa + ([X,Y ]m ·Aam)Φam

+(Ym · (Xm · dmAa)−Xm · (Ym · dmAa))Φam + [Xm ·Aam, Ym ·Aam]Φam.

For a line bundle, the commutator part [u ·Aam, v ·Aam] vanishes.

Proposition 1.5.3. For a holomorphic metric vector bundle V ×β∼E the Chern connexion ((ðh)a)

satisfies

∂(ðh)a = (ðh)a ∧ (ðh)a

and

∂(ðh)a = (dðhðh)a.

In other words, the exterior differential dA = ∂A+∂A

has the (2, 0)-part (ðh)a∧(ðh)a and the (1, 1)-part

is given by the curvature (dðhðh)a.

Proof. Consider first the wedge product. Since

(Y · (ðh)a)m = (ham)−1(Ym ∂mha)

we have

(ðh)am ∧ (ðh)am = [Xm · (ðh)am, Ym · (ðh)am] = [(ham)−1(Xm ∂mha), (ham)−1(Ym ∂mh

a)]

= (ham)−1(Xm ∂mha)(ham)−1(Ym ∂mh

a)− (ham)−1(Ym ∂mha)(ham)−1(Xm ∂mh

a).

and hence

ham((ðh)am ∧ (ðh)am) = (Xm ∂mha)(ham)−1(Ym ∂mh

a)− (Ym ∂mha)(ham)−1(Xm ∂mh

a).

Therefore (ðh)a ∧ (ðh)a is a differential form of type (2, 0) since both X and Y involve holomorphic

Wirtinger derivatives. Consider now the exterior differential

(X,Y )dΘ = dX(Y · (ðh))− dY (X · (ðh))− [X,Y ] · (ðh)

for vector fields X,Y. The product and quotient rules imply

dX(Y · (ðh)a)m = (ham)−1(dX · (Ym ∂mha))− (ham)−1(dXh

a)(ham)−1(Ym ∂mha).

Therefore

ham dX(Y ·(ðh)a)m = dX ·(Ym ∂mha)−(dXh

a)(ham)−1(Ym ∂mha) = (XmdmY )(∂mh

a)+Ym(Xmdm∂mha)−(dXh

a)(ham)−1(Ym ∂m ha).

It follows that

ham ((X,Y )dΘ) = (XmdmY − YmdmX) ∂mha + Ym(Xmdm∂mh

a)−Xm(Ymdm∂mha)

29

−(dXha)(ham)−1(Ym ∂mh

a) + (dY ha)(ham)−1(Xm ∂mh

a)− [X,Y ]m ∂mha

= Ym(Xmdm∂mha)−Xm(Ymdm∂mh

a)− (Xmdmha)(ham)−1(Ym ∂mh

a) + (Ymdmha)(ham)−1(Xm ∂mh

a),

since the first and last terms cancel. Subtracting (??) we obtain the curvature

ham ((X,Y )Ω) = ham ((X,Y )dΘ− (ðh)am ∧ (ðh)am)

= Ym(Xmdm∂mha)−Xm(Ymdm∂mh

a)− (Xm∂mha)(ham)−1(Ym ∂mh

a) + (Ym∂mha)(ham)−1(Xm ∂mh

a).

Finally, the second holomorphic derivatives Ym(Xm ∂m∂mha) = Xm(Ym ∂m∂mh

a) vanish by Schwarz’

theorem. Therefore

ham ((X,Y )Ωa)

= Ym(Xm∂m∂mha)−Xm(Ym∂m∂mh

a)− (Xm∂mha)(ham)−1(Ym ∂mh

a) + (Ym∂mha)(ham)−1(Xm ∂mh

a).

It follows that Ω is a differential form of type (1, 1), involving only mixed derivatives. In summary,

d(ðh) has the (1, 1)-part dðhðh and the (2, 0)-part (ðh) ∧ (ðh). Since (ðh) is of type (1, 0), d(ðh)a has

no (0, 2)-part, and the assertion follows.

Proposition 1.5.4. For hermitian holomorphic line bundles the curvature (1, 1, )-form (dðhðh)a is

closed.

Proof. The curvature is given by

(dðhðh)a = ∂(ðh)a = ∂∂ log ha.

Since ∂2

= ∂2 = ∂∂ + ∂∂ = 0, it follows that

d(∂∂ log ha) = (∂ + ∂)(∂∂ log ha) = ∂∂(∂ log ha) + ∂∂(∂ log ha) = −∂∂∂(∂ log ha) = 0


Consider the invariant connexions on homogeneous vector bundles over G/H given by a splitting (??)

of the Lie algebra g.

Proposition 1.5.5. For γ, δ ∈ m the curvature is given by the ’multiplication operator’

(dAA(γ˜, δ˜) Φ)∼ = −[γ, δ]πh Φ.

Proof. For γ, δ ∈ m we have vanishing h-projection. Hence (??) implies

(dAγ˜ (dAδ˜ Φ)− dAδ˜ (dAγ˜ Φ)− dA[γ˜,δ˜] Φ)∼

= dγ dAδ˜ Φ− dγ dAγ˜ Φ− d[γ,δ]Φ− [γ, η]πhΦ

= dγ dδ Φ− dγ dγ Φ− d[γ,δ]Φ− [γ, η]πhΦ = −[γ, η]πhΦ.

Here we used that π∗γ = γ˜ and π∗δ = δ˜ implies π∗[γ, δ] = [γ˜, δ˜], so that [γ, δ] is a horizontal lift of

[γ˜, δ˜].

30

Chapter 2

Classical Phase Spaces

2.1 Symplectic Manifolds and Kahler Manifolds

A 2-form ω ∈ Ω2(M) on a smooth manifold M is called symplectic if dω = 0 and ω is non-degenerate,

i.e. for each m ∈M the canonical map

ωm : TmM → T ∗mM,

arising as a special case of (??), is a linear isomorphism. Alternatively (in the finite-dimensional case),

ωm(u, v) = 0 for all v ∈ TmM implies u = 0. A symplectic manifold is a manifold M endowed with

a symplectic 2-form ω. Then dimM = 2n is even. The Liouville measure is defined by the 2n-form

1

n!ωn.

Proposition 2.1.1. Let Q be a real manifold (configuration space). Then the cotangent bundle

M = T ∗Q

is a symplectic manifold (phase space), with symplectic form

ωx,ξ(x, ξ, y, η) = xη − yξ

for all x, y ∈ TxQ and ξ, η ∈ T ∗xQ

Proof. Let π : T ∗Q → Q denote the canonical projection. Then Tx,ξπ : Tx,ξ(T∗Q) → TxQ. Define a

global 1-form ϑ ∈ Ω1(T ∗Q) by

ϑx,ξv := ξ((Tx,ξπ)v).

for all v ∈ Tx,ξ(T ∗Q). Thus we apply ξ ∈ T ∗xQ to (Tx,ξπ)v ∈ TxQ. Then

ω := dϑ

is closed, since d2 = 0, and non-degenerate.

The symplectic manifold T ∗Q is given in its real polarization. We will work instead with complex

polarizations. This is crucial for harmonic analysis but also quantum field theory.

Lemma 2.1.2. Let (M,J,h) be a hermitian complex manifold. Then

ωm(u+ u, v + v) :=hm(u, v)− hm(v, u)

2i

31

is a (not necessarily closed) non-degenerate 2-form ω ∈ Ω2(M,R), satisfying

ωm(JmX, JmY ) = ωm(X,Y )

for all X,Y ∈ TRmM.

Proof. Since J(u+ u) = iu+ iu we have

ωm(Jm(u+ u), Jm(v + v)) = ωm(iu+ iu, iv + iv) =hm(iu, iv)− hm(iv, iu)

2i

=hm(u, v)− hm(v, u)

2= ωm(u+ u, v + v)

Define a Riemannian metric g on M by

gm(X,Y ) := ωm(JmX,Y ).

Then

gm(u+u, v+v) = ωm(Jm(u+u), v+v) = ωm(iu+iu), v+v) =hm(iu, v) + hm(v, iu)

2i=

hm(u, v) + hm(v, u)

2.

Then

gm(u+ u, u+ u) = hm(u, u).

If h is a 0metric (positive definite), it follows that gm(X,X) > 0 for all 0 6= X ∈ TRmM. The hermitian

metric h can be recovered from ω and g via

hm(u, v) = gm(u+ u, v + v) + iωm(u+ u, v + v).

Thus on a symplectic manifold (M,ω) the formula (??) yields a 1-1 correspondence between almost

complex structures J and Riemannian (pseudo)-metrics g. An almost complex structure J on (M,ω) is

called compatible if (??) is a positive-definite Riemannian metric. By Proposition ?? every symplectic

manifold has a compatible almost complex structure, which however may not be integrable. This leads

to the important

Definition 2.1.3. The following equivalent definitions define a 0Kahler manifold:

• A symplectic manifold (M,ω) with a compatible almost complex structure J which is integrable

(vanishing Nijenhuis tensor) and hence, by the Newlander-Nirenberg theorem, is a complex struc-

ture.

• A 0hermitian manifold M such that the resulting 2-form ω is closed

• A 0-hermitian manifold such that the tangent Chern connexion ðh and the Levi-Civita connexion

ðg coincide (after proper identification)

Example 2.1.4. For Q = Rn we take M = Cn with it standard complex structure J. The hermitian

metric

hz(∂

∂z,∂

∂z) = 1

introduced in (??) leads to

ωz(∂

∂y,∂

∂x) = ωz(i

( ∂∂z− ∂

∂z

),∂

∂z+

∂

∂z) =

1

2i(hz

(i∂

∂z,∂

∂z

)− hz

( ∂∂z, i∂

∂z

)) =

1

2i(2i) = 1.

32

It follows that h induces the symplectic form

ω = dpj ∧ dqj ,

when we identify q = x and p = y. In differential form language we have

ω = dy ∧ dx =dz − dz

2i∧ dz ∧ dz

2=

1

4i(dz ∧ dz − dz ∧ dz) =

1

2idz ∧ dz.

Thus hz corresponds to the (1, 1)-form dz ∧ dz.


The Bergman metric hm(u, v) = tr D(B−1z,zu, v) is positive definite and we obtain a symplectic form

ωm(u+ u, v + v) :=hm(u, v)− hm(v, u)

2i

which is closed, as will be shown later. Hence the Jordan manifolds

Z ⊂ Z ⊂ Z

are Kahler manifolds.


Proposition 2.1.5. On the space S of symmetries the imaginary symplectic form

ω = trs ds ds = sji dskj ∧ dski

is closed.

• Loop groups

For the (parallelizable) loop space Γ(S, G) the tangent space Γ(S, g) has a class of hermitian Sobolev

type metrics

(u|v)k =

∫S

ds ((∆ku)s|vs).

For k = 0 this gives the basic L2-metric

(u|v)0 =

∫S

ds (us|vs).

For k = 1/2 one obtains a Kahler metric

(u|v)1/2 =

∫S

ds ((|D|u)s|vs)

with Kahler form

ωe(u, v) =1

2π

∫S

ds (u′s|vs).

There is also a 1-metric

(u|v)1 =

∫S

ds ((∆u)s|vs) =

∫S

ds (u′s|v′s)

33


Proposition 2.1.6. Let S be a compact oriented surface. Let G be a compact Lie group with Lie

algebra g. Then the affine space Ω1(S,G) of all connexions A on the trivial G-bundle S ×G carries the

symplectic form

ωA(A1, A2) =

∫S

tr[A1 ∧ A2]

where A ∈ TA(Ω1(S,G)) = Ω1(S, g).

Proof. We write

Λ = λi ⊗ γi

for scalar 1-forms λi ∈ Ω1(S) and a basis γi ∈ g. Choose a U -invariant inner product tr[γ, γ′] on g, for

example the negative Killing form in the semi-simple case. Then the scalar 2-form

tr[Λ ∧ Λ′] := λi ∧ λ′j tr[γi, γj ] ∈ Ω2(S)

is independent of the choice of basis γi and can be integrated over S. for U = SUn(C) we use −trγγ′

Is this of complex type?


Let G be a Lie group with a (right) action

M ×G→M, (m, g) 7→ mg˜on a manifold M. The corresponding infinitesimal action

M × g→ TM, (m,u) 7→ u˜mof the Lie algebra g is defined by

u˜m = ∂0t (m · exp(tu)).

Here exp : g→ G is the exponential map. For any m ∈M the stabilizer subgroup

Gm := g ∈ G : m · g = m

has the Lie algebra

gm := u ∈ g : u˜m = 0.

The quotient manifold

Gm := Gm\G

has the tangent space

Tm(Gm) = u˜m : u ∈ g = gm\g

Lemma 2.1.7.

u˜m(Tmg˜) = Ad−1g u

˜ m·g

Proof.

u˜m(Tmg˜) = ∂0t (m · exp(tu))(Tmg˜) = ∂0

t

((m · exp(tu)) · g

)= ∂0

t

(m · (exp(tu) g)

)= ∂0

t

(m · g · (g−1 exp(tu) g)

)= ∂0

t

(m · g · exp(t Ad−1

g u))

= Ad−1g u

˜ m·g

34

Put Intg(g′) := gg′g−1. Then

Adg := Te(Intg)

defines an action of G on the Lie algebra g = TeG. The co-adjoint action g∗ ×G→ g∗ on the linear

dual space M := g∗ is defined by

(m Adg)u := m(Adgu).

for all m ∈ g∗ and u ∈ g. Put

aduv = [u, v].

Lemma 2.1.8. For any m ∈ g∗ the stabilizer subgroup

Gm := g ∈ G : m Adg = m

has the Lie algebra

gm := u ∈ g : m adu = 0.

Proof. Let u ∈ g and let gt ∈ G be a smooth curve with g0 = e and ∂0t gt = u. Then

∂0t (m (Adgt)|v) = ∂0

tm((Adgt)v)) = m(∂0t (Adgt)v) = m[u, v] = (m adu)v

Since v ∈ g is arbitrary, it follows that

∂0tm Adgt = m adu.

Regarding m Adgt as a curve in the orbit Gm = Gm\G it follows that

Tm(Gm) = m adu : u ∈ g.

For u ∈ g let

u˜m := u+ gm ∈ TmGm

denote the equivalence class. For each ξ ∈ g∗ we have the action

(ξ (Adg)|v) := (ξ|(Adg)v)

for all v ∈ g. Now we define

ωξ(ξ (adu)|ξ (adv)) := ξ[u, v]

Theorem 2.1.9. For m ∈ g∗ define a bilinear form ωm on TmGm by

ωm(u˜m, v˜m) := m[u, v]

for all u, v ∈ g. This is well-defined and yields a G-invariant symplectic form on the coadjoint orbit Gm.

Proof. Suppose u, u′ ∈ g and v, v′ ∈ g satisfy u− u′ ∈ gm and v − v′ ∈ gm. Then m ad(u− u′) = 0 =

m ad(v − v′) and

m[u, v]−m[u′, v′] = m[u− u′, v] +m[u′, v − v′] = (m adu−u′)v − (m adv−v′)u′ = 0.

This shows that m[u, v] depends only on the equivalence class u˜m, v˜m. Hence (??) is well-defined.

The tangent space TmGm consists of all linear functionals u˜m = m adu, for u ∈ g. Suppose that

u˜m ∈ TmGm belongs to the radical of ωm. Then

(m ad)u(v) = m(aduv) = m[u, v] = ωm(u˜m, v˜m) = 0

35

for all v ∈ g. Thus m adu = 0 as a tangent vector to Gm. Therefore ω is non-degenerate.

To show that ω is G-invariant, we apply (??) and obtain

(u˜m, v˜m)(g˜∗ω)m = (u˜m(Tmg˜), v˜m(Tmg˜))ωm·g = (Ad−1g u

˜ m·g, Ad−1

g v

˜ m·g)ωm·g

= (m · g)[Ad−1g u,Ad−1

g v] = m(Adg[Ad−1g u,Ad−1

g v]) = m[u, v] = (u˜m, v˜m)(g˜∗ω)m.

Thus we have

g˜∗ω = ω

for all g ∈ G.Every u ∈ g induces a vector field u˜ on Gm by

m · exp(tu˜) = m Adexp(tu).

For fixed v ∈ g consider the smooth function

fvm := m|v.

Then

(u˜∂fv)m = ∂0t f

vm·exp(tu˜) = ∂0

t fvm Adexp(tu)

= (∂0t fm Adexp(tu)

)v = m(∂0tAdexp(tu)v) = m[u, v].

Since

ωm(v˜m, w˜m) = f [v,w]m

it follows that (u˜∂ω(v˜, w˜))m = m|[u[v, w]] and the Jacobi identity implies

u˜∂ω(v˜, w˜) + v˜∂ω(w˜ , u˜) + w˜∂ω(u˜, v˜) = 0.

On the other hand, we have

[u˜, v˜] = [u, v]˜

and hence

ωm([u˜, v˜]m, w˜m) = ωm([u, v]˜m

, w˜m) = m[[u, v]w].

Using the Jacobi identity again, we obtain

ωm([u˜, v˜]m, w˜m) + ωm([v˜, w˜ ]m, u˜m) + ωm([w˜ , u˜]m, v˜m) = m([[u, v]w] + [[v, w]u] + [[w, u]v]) = 0.

In summary, dω(u˜, v˜, w˜) = 0. Thus dω = 0.

For u0, u1, u2 ∈ g we consider the right invariant vector fields

u˜g := u · Te(Rg)

acting on G and also on Gξ = Gξ\G. Consider the function

f(m) := m|[u1, u2]

on the orbit. Then

f(o · g0t ) = (o · g0

t )|[u1, u2] = o|g0t · [u1, u2]

and therefore

(u˜0 · f)(o) = ∂0t f(o · g0

t ) = o|∂0t g

0t · [u1, u2] = o|[u0[u1, u2]].

Thus the first three terms sum up to zero by the Jacobi identity. For the second type we have

ωm([u˜, v˜]m, w˜m) = ωm([u, v]˜m

, w˜m) := m|[[u, v], w].

Thus the last three terms sum up to zero by the Jacobi identity.

36


*projective space

*Grassmannian

Proposition 2.1.10. Let G be a real semi-simple Lie group of hermitian type, with maximal compact

subgroup K. Then the ’symmetric domain’ Z = G/K is a coadjoint orbit, whose (Kostant-Kirillov)-

symplectic structure agrees with(??). Moreover, the compact dual space (conformal hull)

Z = GC/KC · Z

is a compact Kahler manifold, and ?? we have

(Z, ω) = (GC/GC

+ , Imh)

Proof. Define m : g→ iR by

m(γ) = (iz∂

∂z|γ′0)

Proposition 2.1.11. Let G be a simply-connected compact Lie group, with maximal torus T . Then the

full flag manifold

G/T = T = Gm

is a coadjoint orbit for the linear functional m : g→ iR given by

mY := ρY0.

Here Y 7→ Y0 is the projection onto t.

Proof. With respect to the root decomposition

g = t⊕∑β∈∆

gβ

we write elements in g as

Y = Y0 +∑β∈Λ

Yβ .

Define a linear form m : g→ C by

mY := ρY0.

Let X = X0 +∑α>0

(Xα −X∗α) ∈ g satisfy m adX = 0. Let β > 0 and Y ∈ g−β be arbitrary. Then

[X,Y ] = [X0, Y ] +∑α>0

[Xα −X∗α, Y ] = −(βX0)Y +∑α>0

([Xα, Y ]− [X∗α, Y ])

has the t-projection

[X,Y ]0 = [Xβ , Y ] = c ·Hβ .

Since β > 0 we have ρHβ > 0. Therefore 0 = (m adX)Y = ρ[X,Y ]0 = c · ρHβ implies c = 0. Hence

[Xβ , Y ] = 0 for all Y ∈ g−β , showing that Xβ = 0 for β > 0. Thus X = X0 ∈ t.

complex structure on coadjoint orbits

Lemma 2.1.12. For each w ∈WGw> · TC ·Gw< ⊂ GC

is open.

37

Moreover

G/T = GC/GC+

is a compact Kahler manifold, and we have

(G/T, ω) = (GC/GC+ , Imh)

*Peirce manifolds as coadjoint orbits


• Loop groups

Let G be a simply-connected and simply laced (ADE) Lie group. Put S := S1 and let

L = C∞(S, G)

with Lie algebra

Λ := C∞(S, g).

Then the (smooth) dual is

Λ+ := C∞(S, g∗)

under the pairing

(m|γ) :=

∫S

ds msγs

for all m ∈ Λ,m ∈ g∗. The coadjoint action is

Its orbit of 0 is the loop space

Ω(S) = m ∈ C∞(S, g∗) : m0 = m2π

It carries the symplectic form

ω(ξ, η) :=1

2π

2π∫0

ds(ξ′(s), η(s))


Proposition 2.1.13. For the affine symplectic space Ω1(S,G) of G-connexions on a compact oriented

surface S we consider the group

Ω0(S,G)

acting by gauge transformations

g ·A := gAg−1 + g−1dg.

This action preserves the symplectic structure (??).

Thus Ω1(S,G) becomes a Ω0(S,G)-equivariant symplectic manifold. The Lie algebra of Ω0(S,G) is

identified with Ω0(S, g) under the pointwise Lie bracket. Define a pairing Ω0(S, g)⊗ Ω2(S, g)→ R by

(γ,Θ) 7→∫Σ

tr[γ ·Θ].

Here we write

Θ = ϑ⊗ η

38

for some 2-form ϑ ∈ Ω2(S,R) and η ∈ g. Then the g-valued 2-form

[Θ · γ] = ϑ [η, γ]

gives rise to a scalar 2-form

tr[Θ · γ] = ϑ tr[η, γ]

which can be integrated over S. Via this pairing we identify Ω2(S, g) with a subspace of the dual space

Ω0(S, g)∗. The full continuous dual should be of distribution type.

2.2 Hamiltonian vector fields, Poisson bracket

C∞(M,R)

Hamiltonian vector fields: For any function f ∈ C∞(M,R) define a vector field f on M by

ωm(fm, Ym) := dm(f)Ym

for all Y ∈ Γ(M,T ∗M). Then the Poisson bracket is defined by

˜f1, f2 = [f1, f2]

We say that f1, f2 are in involution if ˜f1, f2 = 0. Completely integrable classical systems f1, . . . , fnin pairwise involution. classical dynamics: Geodesic flow on T ∗Q multi-flow: A-action, G = KAN

Iwasawa decomposition

Prequantization f 7→ f + i∇f quantum dynamics: ei∆ on L2(Q), quantization of geodesic flow multi-

dynamics: Berezin transform for real Jordan manifolds

2.3 Moment Map and Classical Reduction

Coadjoint orbits, moment map and symplectic quotient

T ∗Q×Diff(X)→ T ∗Q, σ 7→ T ∗σ

is a symplectic action with moment map

µ : T ∗X → Γ1(X)+

µx,ξv = ξvx

for all v ∈ Γ1(X)

A symplectic manifold (M,ω) endowed with a smooth (right) action M×G→M, g 7→ Rg preserving

ω

R∗gω = ω

is called a G-equivariant symplectic manifold. Let M be a symplectic manifold endowed with a

symplectic G-action. The associated infinitesimal action of the Lie algebra g defines a tangent vector

γm := ∂0t (gt ·m) ∈ TmM

for every m ∈M. Here gt ∈ G is a smooth curve with g0 = I and ∂0t gt = γ.

39

Definition 2.3.1. A smooth map

µ : M → g∗

is called a moment map if for each v ∈ g the smooth function µv : M → R, defined by

µv(m) := µ(m)v

for the standard pairing g∗ × g→ R, has the differential

dm(µv) = ωmv˜mfor all m ∈M. Here ωmv˜m ∈ T 1

mM since v˜m ∈ TmM and ωm ∈ T 2mM. Thus

t · dmµv = ωm(t, v˜m)

for all u ∈ TmM.

Formally, we have dµ = ω. This ’explains’ that a moment map is unique up to a constant if M is

connected

Theorem 2.3.2. Let µ : M → g∗ be a Hamiltonian G-action. Suppose that G acts freely and properly

on µ−1(0). Then

M//G := µ−1(0)/G

is a smooth manifold of dimension dimM − 2 dimG, which carries a unique symplectic form ω0 ∈Ω2(M//G) satisfying

p∗ω=ι∗ω,

where p : µ−1(0)→M is the inclusion map and p : µ−1(0)→M//G is the canonical projection.

Proposition 2.3.3. Suppose that µ is a moment map for the symplectic G-action on M. Then, for

each γ ∈ g the smooth function µγ has the Hamiltonian vector field γM acting on M. (One says that

the G-action is hamiltonian)

***Symplectic quotient


Example 2.3.4. The torus action Cn × Tn → Cn is a hamiltonian action. We have the Lie algebra

t = iRn has the dual space

t+ = nR

and the moment map µ : Cn → nR has the form

µ(m) = (|z1|2, . . . , |zn|2)


Example 2.3.5.

Grres(H) = GLres(H)/Bres

has the symplectic structure

ω =i

4trΦ dΦ dΦ

where Φ2 = 1. Hence

Φ dΦ = −dΦ F

40

and therefore

dω = dΦ3 = trΦ2(dΦ3) = −trΓ(dΦ)3Φ = −tr(dΦ)2Φ2 = −dω

so ω is closed.

The moment map is

µ(Φ)u = −tr(Φu)

where u ∈ g(H) satisfies u∗ = εuε. This follows from the computation

2trΦ[u,Φ]dΦ = −dtruΦ.

Example 2.3.6. Let Q be a Riemannian manifold. then T ∗Q is a symplectic quotient. In particular,

for Q = Rn, it follows that T ∗Rn is a symplectic quotient.

Remark 2.3.7. Every classical physical system is a symplectic quotient.


Theorem 2.3.8. The action of Γ0(S,G) on Γ1(S,G) by gauge transformations has a moment map

Γ1(S,G)→ Γ2(S, g) ⊂ Γ0(S, g)∗

given by the curvature

µA = ðΘ = dΘ + [Θ ∧Θ] ∈ Γ2(S, g)

Proof. We have to show that for each γ ∈ Γ0(S, g) the smooth function

(µγ)(A) := (µ(A)|γ) = (ðΘ|γ) =

∫Σ

tr[γ · ðΘ]

of the argument A ∈ Γ1(S,G) has the differential

dA(µγ)A = ωA(γA, A) =

∫S

tr[γA ∧ A]

where A and the value γA of the vector field at A belong to TA(Γ1(S,G)) = Γ1(S, g). For the left hand

side consider a curve At with A0 = A and ∂0tAt = A. Since

∂0t (dAtAt) = ∂0

t (dAt + [At ∧At]) = dA+ [A ∧A] = ðA

it follows that

dA(µγ)A = ∂0t

∫S

tr[γ · ðΘ] =

∫S

tr[γ · ∂0t ðΘ] =

∫S

tr[γ · ðA].

For the right hand side, consider a curve gt ∈ Γ0(S,G) with g0 = I and ∂0t gt = γ. Differentiating

gt ·A = gtAg−1t − g−1

t dgt

at t = 0 we obtain, using Schwarz rule to exchange the diffentiation in t and on S,

γA = ∂0t (gt ·A) = γA−Aγ − dγ = −ðγ ∈ Ω1(S, g) ≡ TA(Ω1(S,G)).

Since ð is a graded derivation after applying the trace, it follows that

−∫S

tr[γA · A] = −∫S

tr[ðγ ∧ A] =

∫S

tr[γ ∧ ðA].

41

Corollary 2.3.9. µ−1(0) = A : dAA = 0 consists of all flat C-connexions on S, and the symplectic

quotient Ω1(S,G)//Ω0(S,G) agrees with the non-abelian 1-cohomology

H1c (S,G) = µ−1(0)/Ω0(S,G)

which can be identified with Hom(π1(S), C)/C. This is a compact symplectic orbifold.

Note that in the non-abelian case higher order cohomology cannot be defined directly (higher cate-

gories).

Theorem 2.3.10. (Narasimhan-Seshadri) The symplectic quotient

H1(S,C) := Ω1(S,C)//Γ0(S,C) = Ω1flat(S,C)/C = Hom(π1(S), C)/C

is the space of all flat C-connexions on S, modulo conjugation by C. It is an orbifold with smooth part

consisting of all irreducible connexions.

Theorem 2.3.11. Fix a complex structure τ on S. Then the complex-analytic quotient H1(Sτ , CC)

consists of all semi-stable holomorphic vector bundles over Sτ . It is an complex-analytic space, with

regular part consisting of all stable vector bundles.

2.3.1 Homogeneous Manifolds

For a coadjoint orbit Gm we have

Proposition 2.3.12. For any m ∈ g∗, the inclusion ι : Gm → g∗ is a moment map for the co-adjoint

action.

Proof. We have to show that for each v ∈ g the mapping ιvm := ιmv = mv has the differential

u˜m(dmιv) = ωm(u˜m, v˜m).

This follows from

u˜m(dmιv) = (∂0

t m · exp(tu))(dmιv) = ∂0

t ιvm·exp(tu) = ∂0

t (m · exp(tu))v

= ∂0tm(Adexp(tu)v) = ∂0

tm((exp(t adu)v)) = m(aduv) = m[u, v] = ωm(u˜m, v˜m)

2.4 Quantum line bundles

We call a symplectic form ω integral, if

1

2πi

∫S

ω ∈ Z

for all 2-cycles S ⊂M. This means that ω2πi ∈ H

2(M,Z).

Theorem 2.4.1. Let ω2πi ∈ H

2(M,Z) be an integral symplectic form. Then there exists a complex pre-

quantum line bundle endowed with a hermitian metric h and a metric connection A with curvature

dAA = ω (called the first Chern class). Conversely, the integrality condition is also necessary for the

existence of a prequantum line bundle.

42

Proof. Choose a Leray open cover Va of M, meaning that all finite intersections are contractible. Since

dω = 0, the Poincare Lemma implies that for each a there exists a potential Aa ∈ Ω1(Va, iR) such that

dAa = ω|Va .

Then d(Aa − Ab)|Va∩Vb = ω − ω = 0. Applying the Poincare Lemma again there exist functions

àb ∈ Ω0(Va ∩ Vb, iR) such that

(Aa −Ab)|Va∩Vb = dàb .

Put

κab := exp(àb ) ∈ T

Since ω ∈ H1(M, 2πiZ) is integral, it follows that

(àb + `bc + `ca)|Va∩Vb∩Vc ∈ 2πiZ.

Therefore the cocycle property

(κab κbc κ

ca)|Va∩Vb∩Vc = exp(àb + `bc + `ca) = 1

holds. Hence we obtain a T-bundle V ×κ∼T and the associated line bundle

Vκ×∼

C = (Vκ×∼

T)×T

C = 〈m,φ〉a = 〈m,βab (m)φ〉b : m ∈ Va ∩ Vb, φ ∈ C

for the standard T-representation C. By corollary 3.2.11 it carries the hermitian metric

(〈m,φ〉a|〈m,ψ〉a) = φψ.

Since

κab (dκba) = dà − d`b = Aa −Ab,

the family Aa defines a connexion A. By (??) the curvature of (Aa) is given by

(dAA)am(u, v) = v · (u · dmAa)− u · (v ·Ab) = (dAa)m(u, v) = ωm(u, v).

Hence dAA = ω.

For a hermitian holomorphic line bundle, the curvature

ωa = ∂(ha)−1∂ha

defines an integer cohomology class

c1(L) :=1

2πiω ∈ H2(M,Z)

called the (first) Chern class. This is a conformal invariant: If the hermitian metric (ha) is changed by

a conformal factor ha

:= ef ha, where f ∈ Γ(M,R), then Aa

:= (ha)−1∂h

a= Aa + ∂f and therefore

ωa = ∂Aa

= ωa + ∂∂f.

Since ∂∂f = d∂−∂2 f, it follows that 12πi ω = 1

2πiω in H2(M,Z).

On a Kahler manifold M a quantum line bundle is a holomorpic hermitian line bundle whose

Chern connexion has curvature ω. We may also consider the scale of all k-th powers, with the inverse1k being interpreted as Planck’s constant.

43

Lemma 2.4.2. If a hermitian holomorphic line bundle (LL, h) on a Kahler manifold satisfies

hm(u, v) = ∂v∂u log hm

then ωm is the curvature of the Chern connexion ðh. Thus (L, h,ðh) becomes a (pre)-quantum line

bundle.

On a complex manifold M a smooth function ` : M → R is called plurisubharmonic (in short,

plush) if the Levi form

(∂i∂j`(m)) = (∂2

∂zi`∂zj(m))

is positive (semi-definite). If (??) is strictly positive, the ` is called strictly plurisubharmonic. In this

case the (1, 1)-form

∂∂` =∑i,j

∂i∂j`

on M is a strictly positive (imaginary) symplectic form on M. Consider the hermitian metric hm :=

exp `(m) on the holomorphic line bundle.


Example 2.4.3. For the holomorphic tangent bundle on P1 Proposition ?? and (??) yield the curvature

(1, 1)-form

∂A0 = −2∂

∂z

z

(1 + zz)2dz ∧ dz = −2

1 · (1 + zz)− zz(1 + zz)2

dz ∧ dz = −2dz ∧ dz

(1 + zz)2.

Lemma 2.4.4. ∫S2

dz ∧ dz(1 + zz)2

= 2πi.

Proof. We have

dz ∧ dz = (dx− i dy) ∧ (dx+ i dy) = 2idx ∧ dy.

Using polar coordinates z = r eis and putting u := r2 we obtain

1

2πi

∫S2

dz ∧ dz(1 + zz)2

=1

π

∫R2

dxdy

(1 + x2 + y2)2

=1

π

2π∫0

ds

∞∫0

r dr

(1 + r2)2= 2

∞∫0

r dr

(1 + r2)2=

∞∫0

du

1 + u2=−1

1 + u|∞0 = 1.

As a consequence we have ∫S2

ω = −4πi.

Therefore the symplectic form ω of the holomorphic tangent bundle is integral, but this bundle is not

the ’minimal’ line bundle associated with an integral symplectic form.

44

• Loop groups

The corresponding prequantum line bundle is the central extension viewed as a circle bundle over Ω(S).

Passing to Jordan manifolds, the quasi-determinant ∆z,w of an irreducible metric Jordan triple

Z satisfies

detBz,w = ∆pz,w,

where p is a numerical invariant called the genus. For matrices Z = Kr×s we have

∆z,w = det(Ir − zw∗) = det(Is − w∗z).

We also have the addition formula

∆z,u ∆zu,v = ∆z,u+v,

which is not trivial since a p-th root is involved. It follows that the map δ : R→ C∗ defined by

δw,bz,a := ∆z,a−b

is a cocycle with values in C∗. This cocycle and its integer power δn induces a line bundle

Z2δ,n×∼

C := (Z2δ×∼

C∗)n×C∗

C := [m,φ]a = [ma−b,∆−nz,a−bφ]b : φ ∈ C, ∆z,a−b 6= 0

over Z. A holomorphic section Φ ∈ O(Z2×δ,n∼ C) has the local trivializations

Φ[m,a] = [m, a,Φa(m)]

for a ∈ Z, where Φa : Z → C are holomorphic functions satisfying the compatibility condition

Φb(za−b) = ∆−nz,a−b Φa(m)

whenever a, b ∈ Z satisfy ∆z,a−b 6= 0. Since Z ⊂ Z is a dense open subset via the embedding z 7→ z0 =

[z, 0], a section Φ ∈ O(Z2×δ,n∼ C) is uniquely determined by its trivialization Φ := Φ0. Thus via the

mapping Φ 7→ Φ we may identify O(Z2×αn∼ C) with a vector space of entire functions on Z. Later, this

will be determined explicitly.

Proposition 2.4.5. For an irreducible 0metric Jordan triple Z, the G-invariant Bergman metric on

Z ⊂ Z is given by

(u|v)z = tr D(B−1z,zu, v) = ∂u∂v log ∆−pz,z.

It follows that Z ×δ,−p C, endowed with the 0metric

([z, u]|[z, v]) := ∆−pz,z (u|v)

is the (pre)-quantum bundle for the 0Kahler manifold Z. Similarly, the G-invariant Bergman metric on

Z ⊃ Z is given by

(u|v)z = tr D(B−1z,−zu, v) = ∂u∂v log ∆p

z,−z.

It follows that Z2 ×δ,p C, endowed with the 0metric

([z, u]|[z, v]) := ∆pz,−z (u|v)

is the (pre)-quantum bundle for the 0Kahler manifold Z.

Proof. We carry out the proof for the matrix case Z = Cr×s, where ∆z,w = det(Ir−zw∗) and p = r+s.

We have

det a det z = det(az) = (det La)(z)

45

and hence

det adet′eu = (det La)′eu = det′a(au).

It follows that

det′av = det a det′e(a−1v) = det a tr(a−1v).

Therefore

∂v log det(a) = tr(a−1v).

In the non-compact setting we obtain

∂v log det(1− zz∗) = tr(1− zz∗)−1∂v(1− zz∗) = −tr(1− zz∗)−1zv∗.

Therefore

−∂u∂v log det(1−zz∗) = tr ∂u((1−zz∗)−1zv∗) = tr(−(1−zz∗)−1∂u(1−zz∗)(1−zz∗)−1zv∗+(1−zz∗)−1uv∗

)= tr

((1− zz∗)−1uz∗(1− zz∗)−1zv∗ + (1− zz∗)−1uv∗

)= tr (1− zz∗)−1

(uz∗(1− zz∗)−1zv∗ + uv∗

)= tr (1− zz∗)−1

(uz∗z(1− z∗z)−1v∗ + u(1− z∗z)(1− z∗z)−1v∗

)= tr (1− zz∗)−1u(1− z∗z)−1v∗ = tr (B−1

z,zu)v∗ =1

ptr D(B−1

z,zu, v)

It follows that

tr D(B−1z,zu, v) = −p∂u∂v log det(1− zz∗) = dlu∂v log det(1− zz∗)−p.

In the compact setting we have ∆z,−w = det(Ir + zw∗) and obtain

∂v log det(1 + zz∗) = tr((1 + zz∗)−1(∂v(1 + zz∗))) = tr((1 + zz∗)−1zv∗).

Therefore

∂u∂v log det(1+zz∗) = tr ∂u((1+zz∗)−1zv∗) = tr(−(1+zz∗)−1∂u(1+zz∗)(1+zz∗)−1zv∗+(1+zz∗)−1uv∗

)= tr

((1 + zz∗)−1uz∗(1 + zz∗)−1zv∗ + (1 + zz∗)−1uv∗

)= tr (1 + zz∗)−1

(− uz∗(1 + zz∗)−1zv∗ + uv∗

)= tr (1 + zz∗)−1

(− uz∗z(1 + z∗z)−1v∗ + u(1 + z∗z)(1 + z∗z)−1v∗

)= tr (1 + zz∗)−1uz∗z(1 + z∗z)−1v∗ = tr (B−1

z,−zu)v∗ =1

ptr D(B−1

z,−zu, v)

It follows that

tr D(B−1z,−zu, v) = p∂u∂v log det(1 + zz∗) = ∂u∂v log det(1 + zz∗)p.

For Z = C the calculation simplifies to

∂u∂v log(1− zz) = ∂u−zv

1− zz=−uv(1− zv) + zv(−uz)

(1− zz)2= − uv

(1− zz)2= −hz(u|v)

46


Determinant line bundle Fix a complex structure Sτ . Then Ω1(S,G) acquires a complex structure J

and the covariant derivative dA of A has a (0, 1)-part ∂A.

Lemma 2.4.6. (Ω1(S,G), J) can be identified with the space

H1(Sτ , GC)

of all holomorphic GC-bundles over Sτ .

Holomorphic Quillen determinant line bundle over H1(Sτ , GC) with connexion whose curvature is

the Kahler form.

LA = detH1(Sτ , EA)⊗ detH0(Sτ , EA)

metric defined by regularized determinants of Laplacians.

47

Chapter 3

Quantum State Spaces

3.1 Reproducing kernels

On the other hand, we obtain an anti-holomorphic map

K : M → P(O(M ×β C))

by

w 7→ Kw ∈ O(M ×β C)


Example 3.1.1. Consider the projective space Pd. For 0 ≤ i ≤ d put

Vi := [ζ] ∈ Pd : ζi 6= 0

where [ζ] := Cζ for 0 6= µ ∈ Cd+1. Define βij : Vi ∩Pdj → G by

βij [ζ] :=ζi

ζj.

Note that the fraction depends only on [ζ]. Since the cocycle identity is satisfied, we obtain a C×-bundle

Vβ×∼

C× = [[ζ], h]i = [[ζ], hβji [ζ]]j = [[ζ], h]i = [[ζ],ζi

ζjh]j

over Pd = V/R. For each m ∈ N, let Cm[ζ] be the space of all m-homogeneous polynomials ψ(ζ) in

ζ = (ζ0, . . . , ζd) ∈ Cd+1. For ψ ∈ Cm[ζ], define a holomorphic function ψi : Vi → C by

ψi([ζ]) :=1

(ζi)mψ(ζ).

This depends only on [ζ] since ψ is m-homogeneous. For [ζ] ∈ Vi ∩ Vj we have

ψi([ζ]) :=(ζj)m

(ζi)mψj(ζ)

by definition. Hence the finite family (ψi) defines a holomorphic section of

Vβ,×∼mC = (V

β×∼

C×)m×C×

C.

48

Thus we obtain a linear map

Cm[ζ]→ O(Pd ×β Cm), ψ 7→ (αi)

which is a GLd+1(C)-equivariant isomorphism. After ’symmetry breaking,’ Cd+1, is isomorphic to the

space of all polynomials of degree ≤ d on Cd. For 0 ≤ a ≤ d we define a polynomial ψa in d variables

by

ψa(z0, . . . , za, . . . , zd) := ψ(z0, . . . , 1a, . . . , zd)

If ζa 6= 0 then

ψ(ζ) =1

(ζa)mψa(

ζ0

ζa, . . . ,

ζa

ζa,ζd

ζa)

It follows that

ψa = ψb σab .

Conversely, let ψa, 0 ≤ a ≤ d be polynomials in d variables of degree ≤ m such that (??) holds. Then

there is a unique section ψ ∈ O(U ×σ,m∼ C) satisfying (??). It follows that O(U ×σ,m∼ C) can be identified

with the space of all m-homogeneous polynomials in ζ = (ζ0, . . . , ζd). This space is irreducible under

the natural action of SLd+1(C). For d = 2 we obtain the tangent bundle and

O(Uσ,2×∼

C) = O1(Pd).

Example 3.1.2. The tautological bundle T over the Grassmannian M = Gr(Kr+s) has the fibre

U over U ∈M. Consider the dual bundle T ∗ and the line bundle ∧rT ∗, whose fibre over U consists of

all alternating r-multilinear maps from U to K. For any index chain 1 ≤ i1 < i2 < . . . < ir ≤ r + s of

length r there is a section σi1,...,ir of ∧rT ∗, defined by

U 7→ σi1,...,irU (v1 ∧ . . . ∧ vr) := det(vj |βik)rj,k=1

for all v1, . . . , vr ∈ U. For another w ∈ Cr×s we put vj := (βj , βjz) ∈ U and β′k := βk for 1 ≤ k ≤ r,

and β′′k := βkw. Then

(vj |(β′k, β′′k ) = (βj |βk) + (βjz|βkw) = (βj(1 + zw∗)|βk)

showing that the trivialization σz,w := σwG(m) = det(1 + zw∗). Comparing with (??) we see that

Z ×R

C ≡ ∧rT ∗

and hence Z ×nR C is the n-th power of ∧rT ∗. The action of G on H2π(Z,E) is given by

(U−1g Φ)(ζ) := (∂ζg)−π Φ(gζ)

for all Φ ∈ H2π(Z, E). Here we use the fact that ∂ζg ∈ K.

Example 3.1.3. In the rank 1 case Z = C1×d the homogeneous line bundle Z2×αnσ C over Z = Pd

has holomorphic sections Kw with affine trivialization

Kw(m) = (1 + (z|w))n,

where w ∈ C is arbitrary and (z|w) denotes the inner product. Thus O(Z2×αnσ C) ≡ Hn(Z,C) consists

of all polynomials in z of degree ≤ n, or equivalently, of all n-homogeneous polynomials in d+1 variables,

under the natural action of G = SU(d + 1). For d = 1 this space is also described by entire functions

f0, f∞ on C satisfying the compatibility condition

f∞(−1

z) = z−n f0(m)

for all m ∈ C∗.

49

More explicitly, for any w ∈ Z there exists a global holomorphic section Kwξ ∈ O(Z2×δnR C) with

local trivializations

m 7→ Kaz,w = Dnz,a−w,

since the relation (??) implies

Dnz,a−bKbza−b,w = Dn

z,a−bDnza−b,b−w = Dn

z,a−w = Kaz,w.

Proposition 3.1.4. There is a natural G-equivariant isomorphism

O(Z2δ,n×∼

C) ≡ Pn(Z) :=∑m≤n

Pm(Z).

Proof. Using the Faraut-Koranyi formula we obtain

Kz,w = K0z,w = Dn

z,−w =∑m≤n

(−n)mKm(z,−w) =∑m≤n

(−1)|m|(−n)mKm(z, w).

As a special case of (??) the action of G on H2n(Z,C) is given by

(U−1g Φ)(ζ) := det(∂ζg)−n/p Φ(gζ)

for all Φ ∈ H2n(Z,C). Since

detBz,w = Dpz,w,

where p is the genus of Z, the cocycles (??) and (??) are related by

detβw,bz,a = (δw,bz,a )p.

On the level of principal bundles this implies

Z2δp

×σ

C∗ = Z2det β×σ

C∗ = Gdet ∂0

×G0

C∗

for the p-th power cocycle δp. As a special case consider the determinant character δk := detZ k of K.

Then (??) implies

Gδ×K

C = Z2αp

×∼

C.

In this sense, the line bundle Z2×α∼C is more fundamental.

Proposition 3.1.5. Let (E, π) be a holomorphic representation of K. Then, for any w ∈ Z there exists

a global holomorphic section Kwξ ∈ O(Z2×π∼E) with local trivializations

m 7→ Kaz,wξ = Bπz,a−wξ.

In particular, we have

Kz,wξ = K0z,wξ = Bπz,−wξ.

Proof. This follows from (??) which implies

Bπz,a−bKza−b,wξ = Bπz,a−bBπza−b,b−wξ = Bπz,a−wξ = Kaz,wξ.

50

3.2 Compact Lie Groups and Borel-Weil-Bott Theorem

Theorem 3.2.1. For a metric Jordan triple Z, the associated Jordan manifolds Z ⊂ Z ⊂ Z

Z = G/K = G/Gm,

are coadjoint orbits, for the linear functional m : g→ iR defined by

mX = tr∂0X

where ∂0X = X ′(0) ∈ k ⊂ gl(Z).

Proof. The complexified Lie algebra g = g⊗C consists of all vector fields

X = a+ `+ b∗

where a ∈ Z is the constant vector field,

b∗(z) =1

2z; b; z

is a quadratic vector field given by the Jordan triple product, and ` = `(z) ∈ k is a linear vector field.

Define an invariant inner product on g by

〈a+ `+ b∗|α+ λ+ β∗〉 := tr(D(a, β) + `λ+D(α, b))

for all a, b, α, β ∈ Z and `, λ ∈ k. Via the inner product we may identify g and its dual g∗. We have to

find an element m ∈ g∗ ≈ g such that

k = gm = X = a+ `+ b∗ : m adX = 0.

Since Z is circular, we have

I := z∂

∂z∈ k.

This element generates the center of k and corresponds to the identity on Z. Now define

mX = m(a+ `+ b∗) = 〈I|X〉 = tr`.

The commutator

[X,Y ] := dXY − dYXyields [a, α] = 0 and [b∗, β∗] = 0. Moreover [`, α] = `α as a constant vector field. Moreover,

[`, β∗] =1

2[`z, z;β; z] = `z;β, z − 1

2`z;β; z = −1

2z; `∗β; z = −(`∗β)∗

as a quadratic vector field. Finally,

[a, b∗] = [a,1

2z; b; z] = a; b; z = D(a, b)z = D(a, b)

viewed as a linear vector field. Therefore

adX(α+λ+ β∗) = [a+ `+ b∗, α+λ+ β∗] = ([`, α]− [λ, a]) + ([a, β∗] + [`, λ]− [α, b∗]) + ([`, β∗]− [λ, b∗])

= (`α− λa) + (D(a, β) + [`, λ]−D(α, b∗)) + ((λ∗b)∗ − (`∗β)∗).

Therefore

(m adX)(α+ λ+ β∗) = tr(D(a, β) + [`, λ]−D(α, b)) = tr(D(a, β)−D(α, b))

since [`, λ] is a commutator in k. For X ∈ g we need b = εa, β = εα where e = ±. Thus

(m adX)(α+ λ+ εβ∗) = εtr(D(a, α)−D(α, a)).

By polarization, it follows that m adX = 0 if and only if xtrD(a, α) = 0 for all α ∈ Z. Since Z is

non-degenerate, this means a = 0 and therefore X ∈ k.

51

Consider a compact complex projective manifold M, with structure sheaf O. Let Oq denote the sheaf

of germs of holomorphic sections of the q-th exterior power T qM. Then On belongs to the canonical

bundle of n-forms.

For any holomorphic vector bundle V over M, let O ⊗ V denote the sheaf of germs of holomorphic

sections of V, and let Op ⊗ V denote the sheaf of germs of holomorphic p-form sections of V. Since M

is compact, the sheaf cohomology groups Hq(M,Op⊗ V ) are finite-dimensional complex vector spaces.

The Serre duality theorem states that Hq(M,O ⊗ V ) is in duality with Hn−q(M,On ⊗ V ∗), where

V ∗ is the dual vector bundle of V. Thus

Hq(M,O ⊗ V )∗ = Hn−q(M,On ⊗ V ∗).


Example 3.2.2. The group G = SL1+n(C) contains the parabolic subgroup

G− = p =

(p0

0 b

0 d

)

fixing the line C

(1

0

). Then G/G− = Pn. For integers m, consider the homogeneous line bundle

O(m) = Gm×G−

C = [g, φ] = [gp, (p00)mφ]

associated with the character p 7→ (p00)m of G−. Its holomorphic sections

H0(Gm×G−

C) =

Cm[ζ0, . . . , ζn] m ≥ 0

0 m < 0.

Then G×n+1

G−C = ∧n(TM) and

G−n−1×G−

C = ∧n(T ∗M)

is the canonical bundle. By a theorem of Serre, we have

Hq(Gm×G−

C) = 0

for 0 < q < n. For the n-th cohomology, we apply Serre duality:

Hn(Gm×G−

C)∗ = H0((G−n−1×G−

C)⊗ (Gm×G−

C)∗) = H0((G−n−1×G−

C)⊗ (G−m×G−

C)) = H0(G−n−m−1×G−

C).

Thus

Hn(Gm×G−

C) =

0 n+m ≥ 0

C−n−m−1[ζ]∗ n+m < 0,

where Ck[ζ]∗ carries the contragredient representation.

3.2.1 Borel Subgroups and full Flag Manifolds

A semisimple complex Lie algebra g with maximal torus t ⊂ g has a root decomposition

g = t⊕∑α∈∆

gα.

52

For every root α ∈ ∆ there exists a unique ’coroot’ Hα ∈ [gα, g−α] satisfying αHα = 2. The weight

lattice

T∗

:= λ ∈ t∗

: λHα ∈ Z ∀ α ∈ ∆

is a free abelian group of rank dim t, containing the roots. The elements of T∗

correspond to characters

of the group T under taking ’logarithms,’ whence the notation. The Weyl group W acts on ∆ and on

T∗.

Fix a subset ∆+ ⊂ ∆ of positive roots. There exists a unique element w0 ∈ W satisfying w0∆+ =

−∆+. Define

T∗+ := λ ∈ T

∗: λHα ≥ 0 ∀ α ∈ ∆+.

The half-sum ρ of positive roots belongs to T∗+, since ρHσ = 1 for each simple (positive) root σ. Define

g> :=∑α∈∆+

gα, g< :=∑α∈∆+

g−α.

Then we have the ’Gauss decomposition’

g = g< ⊕ t⊕ g>.

On the Lie group level this implies that

G< · T ·G> ⊂ G

is a dense open subset. Here G is assumed to be simply-connected, containing a maximal complex torus

T . Let g 7→ g∗ be an involution such that

g = γ ∈ g : γ∗ = −γ

is the Lie algebra of a compact form G of G.

The Theorem of the highest weight is the following:

Theorem 3.2.3. For every λ ∈ T∗+ there is a finite-dimensional irreducible g-module, denoted by Gλ,

with highest weight λ, and every finite-dimensional irreducible g-module is isomorphic to Gλ for a unique

λ ∈ T∗+.

Thus any choice of positive roots yields a bijection

T∗+ → G

∗,

where the right-hand side denotes the (discrete) set of all finite-dimensional irreducible g-modules.

3.2.2 0-Cohomology: Borel-Weil theorem

Lemma 3.2.4. For every GC-module E there is a G-equivariant mapping

E → O(GC,C), (ξ, η) 7→ ξ∗η

defined by

(ξ∗η)g := (ξ|gπη)

for all ξ, η ∈ E. We have

ρg(ξ∗η) = ξ∗(gπη)

Proof.

(ρg(ξ∗η))g1= (ξ∗η))g1g = (ξ|(g1g)πη) = (ξ|gπ1 gπη) = (ξ∗(gπη))g1

53

Assume that

bπ+ξ = χ(b+)ξ

is a highest weight vector. Then bπ

−ξ

(ξ∗η)b−g = (ξ|(b−g)πη) = (ξ|bπ− gπη) = (bπ∗− ξ|gπη)

= (bπ

−ξ|gπη) = χ(b−)(ξ|gπη) = χ(b−)(ξ|gπη) = χ(b−)(ξ∗η)g.

In particular,

(ξ∗η)tg = χ(t)(ξ∗η)g = t−χ(ξ∗η)g.

This shows that

ξ∗η ∈ O(GCχ×B−

C).

Consider the simply-connected complex Lie group G with Lie algebra g.

Theorem 3.2.5. Borel-Weil Theorem: Let λ ∈ T∗

and consider the induced line bundle G×λG−

C,

with trivial action of G<. Then

H0(Gλ×G−

C) =

Gλ λ ∈ T

∗+

0 λ /∈ T∗+

Here Gλ is ’the’ irreducible G-module with highest weight λ.

Proof. The holomorphic sections H0(G×λG−

C) are identified with the subspace

f ∈ O(G,C) : fgb = b−λ fg ∀ b ∈ G−.

Assume first that H0(G×λG−

C) 6= 0. Then there exists a (non-zero) highest weight vector f0 ∈H0(G×λ

G−C) satisfying

an f0 = aχf0, f0ag = a−χ f0

g

for all a ∈ G+, where χ is a character of G

+. For c ∈ G

>, t ∈ T , d ∈ G< we have (ct)χ = tχ and

(td)λ = tλ. Hence (??) and (??) imply

f0ctd = (ct)−χ f0

d = t−χ f0e = (td)−λf0

c = t−λf0e .

If f0e = 0 then f0 vanishes on the dense open subset G

>T G< ⊂ G. Hence f0 = 0 by continuity,

a contradiction. Thus f0e 6= 0 and (??) shows λ = χ. Since χ is dominant, λ ∈ T

∗+ and the second

assertion follows.

Conversely let λ ∈ T∗+ be a dominant weight such that (??) holds. Let Gλ be an irreducible G-

module with highest weight λ. Consider the involution g 7→ g∗ on G such that the compact real form

G acts unitarily. Let v0 ∈ Gλ be a non-zero highest weight vector (unique up to a scalar multiple). For

any v ∈ Gλ define a holomorphic function v on G by

vg := (v0|g−λv).

For all b = ct ∈ G− we have b∗ ∈ G+

and hence

b−λ∗v0 = b∗−λv0 = t−λv0.

It follows that

vgb = (v0|(gb)−λv) = (v0|b−λ(g−λv)) = (b∗−λv0|g−λv) = (b∗−λv0|g−λv) = t−λ(v0|g−λv) = t−λvg.

54

Therefore, via the identification (??), we have v ∈ H0(G×λG−

C). The computation

(g · v)g′ = vg−1g′ = (v0|(g−1g′)−λv) = (v0|(g′)−λ(gλv)) = gvg′

for g, g′ ∈ G shows

g · v = gv.

It follows that the C-linear mapping

Gλ → H0(Gλ×G−

C), v 7→ v

is G-equivariant. We have to show that it is an isomorphism. Suppose that v = 0 for some v ∈ Gλ.Then

(g∗v0|v) = (v0|gλv) = vg−1 = 0

for all g ∈ G. By irreducibility, the orbit Gλv0 is total in Gλ. It follows that v = 0. Thus (??) is also

injective, and the range of (??) is a G-submodule of H0(G×λG−

C). For c ∈ G>, t ∈ T , d ∈ G< we

have c−λv0 = v0 = d−λ∗v0 and t−λv0 = t−λv0. It follows that

v0ctd = (v0|(ctd)−λv0) = (v0|d−λt−λc−λv0) = (v0|d−λt−λv0) = (v0|d−λt−λv0)

= t−λ(v0|d−λv0) = t−λ(d−λ∗v0|v0) = t−λ(v0|v0).

On the other hand, let f0 ∈ H0(G×λG−

C) be any highest weight vector. Then c−λf0 = f0 and hence

f0ctd = (c−λf0)td = f0

td = t−λf0e .

Since G>T G< is dense in G, a continuity argument shows

f0 =f0e

(v0|v0)v0.

Thus all highest weight vectors in H0(G×λG−

C) are proportional. Since distinct irreducible summands

would contain distinct highest weight vectors, H0(G×λG−

C) is irreducible. Therefore (??) defines a

G-equivariant isomorphism.

3.2.3 Parabolic subgroups and flag manifolds

We now pass from a maximal torus to an arbitrary torus. Let Π ⊂ ∆ be a set of simple (positive) roots.

Let Φ ⊂ Π be any subset, including the empty set Φ = ∅. Then

Φt := H ∈ t : αH = 0 ∀ α ∈ Φ

is a subtorus whose centralizer

c := X ∈ g : [X, Φt] = 0

is a reductive Lie algebra, with Levi decomposition

c = Φt⊕ gΦ

Its semi-simple commutator ideal gΦ has itself a Gauss decomposition

gΦ = gΦ< ⊕ t

Φ ⊕ gΦ>

55

where

tΦ

:=< Hα : α ∈ ∆ ∩ Z · Φ >⊂ t

and

gΦ> =

∑α∈∆+∩Z·Φ

gα, gΦ< =

∑α∈∆+∩Z·Φ

g−α.

On the other hand, define

Φg> :=∑

α∈∆+\Z·Φ

gα,Φg< :=

∑α∈∆+\Z·Φ

g−α.

Then the parabolic subalgebra is

Φg− = Φt⊕ gΦ ⊕ Φg< = Φt⊕ gΦ< ⊕ t

Φ ⊕ gΦ> ⊕ Φg< = t⊕ g< ⊕ gΦ

> = g− ⊕ gΦ>

since t = tΦ ⊕ Φt and gΦ

< ⊕ Φg< = g<.

Thus in the non-empty case Φ 6= ∅ the reductive torus centralizer Φt⊕ gΦ plays the role of the torus

t and Φg< is the unipotent radical. Compared to the line bundles in the case Φ = ∅, we now have vector

bundles since the semi-simple part gΦ has higher dimensional irreducible highest weight representations.

In the special case Φ = ∅ we have ∅t = 0, ∅t = t, ∅g = 0, since t is maximal. Therefore

∅g− = t⊕∑α∈∆+

g−α = t⊕ g< = g−

is a Borel subalgebra. In the opposite case Φ = Π we have Πt = t, Πt = 0 and hence Πg = Πg− = g is

the full Lie algebra.

3.2.4 q-Cohomology: Bott’s Theorem

For passing from 0-cohomology to q-cohomology, in case λ is not dominant, we use reflections by simple

roots. Let Φ = σ, where σ ∈ Π is a simple root. Then there is a splitting

t = σt⊕ tσ

where tσ

:= C ·Hσ and σt := H ∈ t : σH = 0. The torus centralizer

c := X ∈ g : [X, σt] = 0 = σt⊕ gσ

is a reductive Lie algebra, and the Gauss decomposition of its semi-simple commutator ideal gσ simplifies

to

gσ = g−σ ⊕C ·Hσ ⊕ gσ ≡ sl2(C),

since gσ< = g−σ and gσ> = gσ. On the other hand, define

σg> :=∑

α∈∆+\σ

gα,σg< :=

∑α∈∆+\σ

g−α.

The parabolic subalgebra is

σg− = σt⊕ gσ ⊕ σg< = t⊕ gσ ⊕∑α∈∆

g−α = gσ ⊕ g−.

Thus we have added one positive root space gσ to the Borel subalgebra g−. Since dim gα = 1 we have

σG−/G− = P1.

56

Take Eσ ∈ gσ, Fσg−σ with [Eσ, Fσ] = Hσ. Since G is supposed to be simply-connected, σG− has a

Levi decompositionσG− = σT G

σ σG<

where the semi-simple part Gσ≡ SL2(C) has the Lie algebra

gσ =< Eσ, Fσ, Hσ >

and the complex torus σT ⊂ T has the Lie algebra

σt := H ∈ t : σH = 0.

Let λ ∈ T ∗ satisfy m := λHσ ≥ 0. Let

σgm :=< vm, vm−2, . . . , v2−m, v−m >

be the m+ 1-dimensional ’spin’ representation of σg ≡ sl2(C). Then

Hσvk = k vk, Eσvk ∈ Cvk+2, Fσvk ∈ Cvk−2

for all k, putting vk = 0 if |k| > m. Since G is assumed to be simply-connected, one can show that

σG ≡ SL2(C) and hence the infinitesimal action on σgm can be integrated to an action π of σG denoted

by σGm. The highest weight vector vm satisfies p(Hσ)vm = p(m)vm for all polynomials p and hence

exp(zHσ)πvm = ezmvm

for all z ∈ C. Now suppose that t ∈ Tσ ∩ σG. Then t = exp(zHσ) for some z ∈ C. This implies

tλ · vm = exp(zHσ)λ · vm = ezλHσ · vm = ezmvm = exp(zHσ)πvm = tπvm.

Since σG centralizes σT , it follows that

tπ(sπvm) = (ts)πvm = (st)πvm = sπtπvm = sπ(tλvm) = tλ · (sπvm)

for all s ∈ σG. Since the set σGπvm is total in σGm, it follows that

tπv = tλv

for all v ∈ σGm. Thus the two representations agree on σG ∩ σT and therefore induce an irreducible

representation of σGσT which extends trivially to a representation of σG−. We denote this module by

σG−m.

Lemma 3.2.6. Let λ ∈ T ∗ satisfy m := λHσ ≥ 0. Then there is an exact sequence of G−-modules

0→M → σG−λ → G

−λ → 0

such that M = 0 m = 0

M = G−sσλ

m = 1

0→ G−sσλ→M → σG

−λ−σ → 0 m ≥ 2

Proof. Define a G−-submodule

M :=< vm−2, . . . , v2−m, v−m >

Since

Gσ

λ/M =< vm >

57

with tvm = Φ(t)vm and ϑ · vm = Φvm = λvm, we obtain an exact sequence

0→M → Gσ

λ → Cλ → 0.

If m = λHσ = 0 then M = 0 by definition. If m = λHσ = 1 then

M =< v−1 >= Cλ−σ = Csσλ

since in general vk has weight λ − m−k2 σ. If m = λHσ ≥ 2 then < v−m > is a G−-submodule of M

isomorphic to Cλ−mσ = Csσλ. The quotient module is

M/ < v−m >= Gσ

λ−σ.

This yields the exact sequence

0→ Csσλ →M → Gσ

λ−σ.

Lemma 3.2.7. Let π : G → GL(E) be a holomorphic representation and consider the restricted

representation π : H → GL(E). Then the map

G×HE → G/H × E, [g, v] 7→ (gH, gπv)

is an isomorphism.

Proof. The calculation

[g, v] = [gh, h−πv] 7→ (ghH, (gh)πh−πv) = (gH, gπv)

shows that the map (??) is well-defined. It is clearly surjective. To show injectivity, let (gH, gπv) =

(g1H, gπ1 v1). Then h := g−1g1 ∈ H and h−πv = g−π1 gπv = v1. Thus [g, v] = [gh, h−πv] = [g1, v1].

Proposition 3.2.8. Let V be a (holomorphic) σG−-module and let λ ∈ T ∗ satisfy λHσ = −1. Then

Hk(G ×G−

(G−λ ⊗ V )) = 0 ∀ k ≥ 0.

Proof. We haveσG− ×

G−

(G−λ ⊗ V )) = (σG− ×

G−

G−λ )⊗ (σG− ×

G−

V )

and the condition λHσ = −1 implies that

σG− ×G−

G−λ = O(−1)

as a line bundle over σG−/G− ≡ P1. By Proposition ?? it follows that

Hq(σG− ×G−

G−λ ) = Hq(P1,O(−1)) = 0

for q = 0, 1. On the other hand,σG− ×

G−

V = P1 × V

is a trivial vector bundle by Lemma 3.2.7, since V carries a representation of σG− ⊃ G−. Therefore

Hq(σG− ×G−

(G−λ ⊗ V )) = Hq((σG− ×

G−

G−λ )⊗ (σG− ×

G−

V )) = 0

58

for all q. The fibrationσG−/G− → G/G− → G/σG−

induces the Leray spectral sequence

Hp+q(G ×G−

(G−λ ⊗ V )) = Hp(G ×

σG−

Hq(σG− ×G−

(G−λ ⊗ V )).

The assertion follows.

For each simple root σ, the reflection sσ ∈W acts on t∗ by

sσ(λ) := λ− (λHσ)σ = λ− (λ|σ)σ.

These reflections generate the Weyl group WT (G). Define an affine action of W on t∗ by

w · λ := w(λ+ ρ)− ρ.

Lemma 3.2.9. Let σ ∈ Π be a simple root and λ ∈ T ∗ such that (λ+ ρ)Hσ ≥ 0. Then, as G-modules,

Hk(Gλ×G−

C) ≡ Hk+1(Gsσ·λ×G−

C) ∀ k ∈ Z.

Proof. Assume m = (λ+ ρ)Hα ≥ 2. Then Lemma 3.2.6 yields exact G−-module sequences

0→M → σG−λ+ρ → G

−λ+ρ → 0,

0→ G−sσ(λ+ρ) →M → σG

−λ+ρ−σ → 0

Tensoring with G−−ρ yields exact G−-module sequences

0→M ⊗ G−−ρ → σG

−λ+ρ ⊗ G

−−ρ → G

−λ → 0,

0→ G−sσ·λ →M ⊗ G

−−ρ → σG

−λ+ρ−σ ⊗ G

−−ρ → 0.

The corresponding sequences of holomorphic G-module sheaves are also exact. Since ρHσ = 1, Propo-

sition 3.2.8 yields

Hk(G ×G−

(σG−µ ⊗ G

−−ρ)) = 0

for µ = λ+ ρ and µ = λ+ ρ− σ. Therefore the corresponding exact cohomology sequence implies

Hk(G ×G−

G−λ ) ≡ Hk+1(G ×

G−

(M ⊗ G−−ρ)) ≡ Hk+1(G ×

G−

G−sσ·λ)

for all k ∈ Z.

Lemma 3.2.10. Let λ ∈ T∗

with λ+ ρ ∈ T∗+. Then, as G-modules, for all w ∈W

Hk(Gλ×G−

C) = Hk+|w|(Gw·λ×G−

C) ∀ k ∈ Z

Proof. The proof uses induction over ` ≥ 1. For ` = 1, we have w = sσ for some simple root σ and

Lemma 3.2.9 applies. Now let w = s0 · · · s` be a product of minimal length ` + 1, with sk = sαk for

simple roots αk. Suppose we have

sk−1 · · · s1α0 = αk

59

for some 1 ≤ k ≤ `. Then (sk−1 · · · s1)s0(s1 · · · sk) = sk and hence

w = s0 · · · sk−1sk+1 · · · s`

has length ≤ `, a contradiction. Thus (??) cannot happen for any k. Since

sσ(∆+ − σ) = ∆+ − σ

for any simple root σ, it follows that w′ := s1 · · · s` satisfies

w′−1α0 = s` · s1α0 ∈ ∆+.

Putting σ = α0 we have

(w′ · λ+ ρ)Hσ = w′(λ+ ρ)Hσ = (λ+ ρ)Hw′−1σ ≥ 0

since λ + ρ ∈ T∗+. Applying the induction hypothesis to w′, of length ≤ `, and Lemma 3.2.9 to w′ · λ

we obtain G-module isomorphisms

Hk(Gλ×G−

C) = Hk+`(Gw′·λ×G−

C) = Hk+`+1(Gsσ·w′·λ×G−

C) = Hk+`+1(Gw·λ×G−

C)

Corollary 3.2.11. Let λ+ ρ ∈ T∗+. Then

Hk(Gλ×G−

C) = 0 ∀ k > 0.

Proof. An element w ∈ W of maximal length satisfies w(∆+) = ∆−. This implies that ` = `(w) =

dimC G/G−. Applying Lemma 3.2.10 we obtain

Hk(Gλ×G−

C) = Hk+`(Gw·λ×G−

C) = 0

for k > 0, since k + ` > dimC G/G−.

A linear form µ ∈ T∗

is called regular, if µHα 6= 0 for all α ∈ ∆. Then there exists a unique

w = wµ ∈ W such that w(µ) ∈ T∗+.

Theorem 3.2.12. (Bott) Let ΦG−λ be irreducible with highest weight λ. If λ+ ρ is singular, then

Hk(G ×ΦG−

ΦG−λ ) = 0

for all k ≥ 0. If λ+ ρ is regular, let w ∈W be the unique element such that w(λ+ ρ) ∈ T∗+. Then

Hk(G ×ΦG−

ΦG−λ ) =

Gw·λ k = |w|0 k 6= |w|

.

Here Gw·λ is ’the’ irreducible G-module of highest weight w · λ.

Proof. We first consider line bundles over G−. (Φ = ∅.). Choose w ∈W with w ·λ+ρ = w(λ+ρ) ∈ T∗+.

Assume first that λ + ρ is singular. Then w · λ + ρ = w(λ + ρ) is also singular. Thus there exists a

60

simple root σ with (w · λ + ρ)Hσ = 0. Hence (w · λ)Hσ = −ρHσ = −1. Applying Lemma 3.2.10 and

Proposition 3.2.8 (for V = C) we obtain

Hk(Gλ×G−

C) = Hk+`(Gw·λ×G−

C) = 0.

Now let λ+ ρ be regular. Then w is unique. Since w · λ+ ρ ∈ T∗+, Lemma 3.2.10 implies

Hk+`(Gλ×G−

C) = Hk+`(Gw−1·w·λ×G−

C) = Hk(Gw·λ×G−

C)

for all k ∈ Z. For k < 0 this vanishes trivially. For k > 0 this vanishes by Corollary 3.2.11. For k = 0

we obtain

H`(Gλ×G−

C) = H0(Gw·λ×G−

C) = Gw·λ

by the Borel-Weil Theorem 3.2.5. Here we use that w · λ ∈ T∗+ since w · λ + ρ is regular, so that

(w·λ+ρ)Hσ ≥ 1 for all simple roots σ, and therefore (w·λ)Hσ = (w·λ+ρ)Hσ−ρHσ = (w·λ+ρ)Hσ−1 ≥0.

The final step in the proof is achieved by

Proposition 3.2.13. Let ΦG−λ be an irreducible holomorphic representation of ΦG− with highest weight

λ. Then, as G-modules,

Hk(G ×G−

G−λ ) = Hk(G ×

ΦG−

ΦG−λ ) ∀ k ≥ 0.

Proof. We first show that

H0(ΦG−λ×G−

C) = ΦG−λ

is an irreducible ΦG-module of highest weight λ. The parabolic subgroup ΦG− has a Levi decomposition

ΦG− = ΦT ΦG ΦG<

where ΦG is semi-simple and connected, ΦT ⊂ T is a complex torus and ΦG< is the unipotent radical

of ΦG−. We have

ΦG− = ΦT ΦG ΦG< = ΦT ΦG<ΦT ΦG

> ΦG< = T ΦG>G< = ΦG

>G−.

Since the unipotent radical always acts trivially we have to check the actions of ΦG and ΦT on

H0(ΦG−×λG− C). The semi-simple Lie group ΦG has the Borel subgroup

ΦG− = ΦT ΦG<.

It follows thatΦG−/G− = ΦG

>= ΦG/ΦG−.

Hence the inclusion map ι : ΦG → ΦG− induces a biholomorphic map ι : ΦG/ΦG− → ΦG−/G−satisfying

ι∗(ΦG− ×G−

G−λ ) = ΦG

λ′

×ΦG−

C,

where λ′ := λ|tΦ . This implies

H0(ΦG−λ×G−

C) = H0(ΦGλ′

×ΦG−

C)

61

as ΦG-modules. Applying the first part of the proof (Bott’s theorem for line bundles) to ΦG/ΦG− it

follows that

H0(ΦGλ′

×ΦG−

C) = ΦGλ′

is an irreducible ΦG-module of highest weight λ′. Let f0 ∈ ΦGλ′ be a highest weight vector. Since

H0(ΦG−×λG− C) is irreducible under ΦG, it is ’a fortiori’ irreducible under ΦG−. In order to find its

highest weight, recall that

H0(ΦG− ×G−

G−λ ) = f ∈ O(ΦG−) : f(pb) = b−Φ f(b) ∀ p ∈ ΦG−, b ∈ G−.

For t ∈ ΦT and p = scu, with s ∈ ΦG, c ∈ ΦT , u ∈ ΦG< we have t−1sc = sct−1 since ΦT ΦG is the

centralizer of ΦT in G. Let λ′′ := λΦt. Then

(t · f0)p = f0(t−1p) = f0(t−1scu) = f0(t−1sc) = f0(sct−1) = tλ′′f0(sc) = tλ

′′f0(scu) = tλ

′′f0(p).

Hence f0 is a highest weight vector for the weight λ = (λ′, λ′′) under the action of T = Φ T Φ T.

Therefore (??) holds. Under the inclusion map ι : ΦG−/G− → G/G− the pull-back is the homogeneous

line bundle

ι∗(Gλ×G−

C) = ΦG− ×G−

G−λ .

Applying the Leray spectral sequence

Hp+q(G ×G−

G−λ ) = Hp(G ×

ΦG−

Hq(ΦG− ×G−

G−λ ))

to the special case q = 0 and using (??) yields the assertion

Hk(G ×G−

G−λ ) = Hk(G ×

ΦG−

H0(ΦG− ×G−

G−λ )) = Hk(G ×

ΦG−

ΦG−λ ).

Let G/T be a compact flag manifold, where T is the centralizer of a torus. Consider the complexified

Lie algebra

gC,

with Cartan subalgebra tC and Weyl group W := N(T )/T. For every w ∈ W we obtain a Borel

subalgebra gw ⊂ gC such that

gw+ ∩ gw− = tC.

T ⊂ G

torus, centralizer CG(T ) G/CG(T ) flag domain

3.3 Compact Kahler Manifolds and Kodaira Embedding The-

orem

3.3.1 Chern Classes, Divisors and Positivity

Recall that for a positive definite hermitian metric∑i,j

hijdzidzj

62

with (hij) > 0 positive definite, the associated (1, 1)-form

−iω :=∑i,j

hijdzi ∧ dzj

is called a Kahler form if dω = 0.

Lemma 3.3.1. On a ball U ⊂ Cn a (1, 1)-form ω is closed if and only if −iω = ∂∂K for some smooth

real function K.

Proof.

Corollary 3.3.2. The (1, 1)-form ω associated with a 0metric h is a Kahler form if and only if for a

covering (Va) there exist smooth functions Ka : Va → R such that

−iω|Va = ∂∂Ka

for all a.

Proof. dω = 0 if and only if ω|Va is closed for all a.

A smooth function K : M → R is called 0plurisubharmonic if the hermitian Levi form∑i,j

∂2K

∂zi∂zjdzidzj

is 0positive. In this case

−iω = ∂∂K =∑i,j

∂2K

∂zi∂zjdzi ∧ dzj

is a Kahler form.

Proposition 3.3.3. Pn is a Kahler manifold.

Proof. For 0 ≤ a ≤ n let z′ = (z0, . . . , za, . . . , zn) with zj = ζj

ζa . Define Ka : Va → R by

Ka[ζ] = log(1 + (z′|z′)) = log(1 +∑j 6=a

|zj |2) = log(1 +∑j 6=a

| ζj

ζa|2) = log |ζ|2 − log |ζa|2.

Then on Va ∩ Vb we have

Ka[ζ]−Kb[ζ] = log | ζb

ζa|2 = log |σba(z)|2 = log σba + log σba

evaluated on Ua ∩ Ub. Hence ∂∂Ka = ∂∂Kb on Ua ∩ Ub and we obtain a global (1, 1)-form ω with

−iω = ∂∂Ka on Ua, satisfying −idω = (∂ + ∂)∂∂Ka = ∂∂∂Ka = −∂∂∂Ka = 0. For positivity, we

compute

∂Ka = ∂ log(1 + (z′|z′)) =(z′|dz′)

1 + (z′|z′)and hence

∂∂Ka =∂(z′|dz′)

1 + (z′|z′)+(∂

1

1 + (z′|z′)

)∧ (z′|dz′) =

1

1 + (z′|z′)∑j 6=a

dzj ∧ dzj − (dz′|z′) ∧ (z′|dz′)(1 + (z′|z′))2

=1

(1 + (z′|z′))2

∑i,j 6=a

(δji (1 + (z′|z′))− zizj) dzi ∧ dzj .

By Cauchy-Schwarz, the n× n matrix (δji (1 + (z′|z′))− zizj) (for indices 0 ≤ i, j ≤ n distinct from a)

is positive definite. Hence ω is a Kahler form.

63

Definition 3.3.4. The Chern class of a cocycle line bundle V ×β∼C is the integral 2-cocycle

c(Vβ×∼

C) ∼ 1

2πi(log βba + log βcb + log βac ) ∈ H2(M,Z).

Lemma 3.3.5. In terms of a metric ha satisfying ha|βba|2 = hb the Chern class is cohomologous to the

family of closed (1, 1)-forms

c(F ) ∼ 1

2πi∂∂ log ha.

Proof. Identifying the Cech and Dolbeault description, the closed (1, 1)-forms ∂∂ log ha correspond to

the Cech 2-cocycle log βba + log βcb + log βac .

Definition 3.3.6. A line bundle L on M is said to be 0positive on an open subset V ⊂M, if

ic(L) ∼∑i,j

hijdzi ∧ dzj ,

where (hij) is 0positive on V.

In general, let D ⊂M be a divisor (irreducible subvariety of codimension 1) in a compact complex

manifold M. For a coordinate cover (Va) there exist holomorphic functions fa : Va → C such that

D ∩ Va = m ∈ Va : fa(m) = 0. We may choose fa such that

βba(m) :=fa(m)

fb(m)

is holomorphic and nowhere zero on Va ∩ Vb. Then the cocycle (βba) ∈ H1(M,C×) defines a line bundle

[D] = Vβ×∼

C

which corresponds to the divisor D.

3.3.2 Blow-up process

Let L = Cn. For projective space P(L) = Pn−1 consider the open subsets

Vi := [ζ] ∈ P(L) : ζi 6= 0 ⊂ P(L)

for 1 ≤ i ≤ n, with coordinate charts

τ i : Cn\i → Vi, ζn\i 7→ [ζn\i; 1i].

The set

N := (z, [ζ]) ∈ L×P(L) : z ∈ [ζ] = (z, [ζ]) ∈ L×P(L) : ziζj = zjζi ∀ 1 ≤ i, j ≤ n

= (z, [ζ]) ∈ L×P(L) : rank

(z1, . . . , znζ1, . . . , ζn

)≤ 1.

is an n-dimensional submanifold of L×P(L). The canonical projection

π : N → P(L), (z, [ζ]) 7→ [ζ]

is a submersion. Consider the open covering

Ni := (z, [ζ]) ∈ N : ζi 6= 0 = π−1(Vi)

of N.

64

Lemma 3.3.7. We have coordinate charts

τ i : Cn → Ni, τ i(t′, ti) := (ti(t′, 1i), [t′, 1i]),

where t′ ∈ Cn\i.

Proof. Let t′′ ∈ Cn\i,j . The equality

τ i(t′′, ti, tj) = (ti(t′′, 1i, tj), [t′′, 1i, tj ]) = τ j(s′′, si, sj) = (sj(s′′, si, 1j), [s′′, si, 1j ])

for ti 6= 0 6= tj shows

τ ij(t′′, ti, tj) =

( 1

tjt′′,

1i

tj, titj

).

Note that

(s′′, si, 1j) =( 1

tjt′′,

1i

tj, 1j)

=1

tj(t′′, 1i, tj).

Proposition 3.3.8. Let M be a complex n-manifold and p ∈M. Choose a chart σ : U → V ⊂M such

that 0 ∈ U ⊂ L and p = σ0 ∈ V . Put

N := (z, [ζ]) ∈ N : z ∈ U.

Then the disjoint union

M := (M \ p) ∪ P(L)

becomes a manifold such that M \ p ⊂ M is an open subset and the bijective map

F : N → (V \ p) ∪ P(L) ⊂ M,

defined by

F (z, [ζ]) :=

σz ∈ V \ p ⊂M \ p z 6= 0

[ζ] ∈ P(L) z = 0,

is biholomorphic.

Proof. Put U i := τi(N i) and define charts ρi : U i → M by

ρi(t) = F (ti(t′, 1i), [t′, 1i]) =

σ(ti(t′, 1i)) ti 6= 0

[t′, 1i] ti = 0.

Then Wi := F (N i) = ρi(U i) ⊂ M are open subsets and, in view of (??) and (??), there is a commuting

diagram

N iF // Wi

U i

τ i

__

ρi

?? .

We also have the charts σa : Ua → Va covering W0 := M \ p. Thus

M = W0 ∪W1 ∪ . . . ∪Wn.

We show that the collection σa, ρi are local charts for M (the chart σ is not needed any more.) Since

ρi = (F ∪I) τ i the transition maps

ρji := ρi ρj = ((F ∪I) τ i)−1 ((F ∪I) τ j) = τi τ j = τ ji

65

are biholomorphic. Now let m ∈ Va ∩ MT

S = Va ∩ MS . On V ′i := τi(Ni \ T ), the diagram (??) simplifies

to

NTF // MS

Ni \ T

∪

OO

V ′iτ ioo

ρi

OO

Thus the identity σa(m) = ρi(w) = F (τ i(w)) implies

σa ρi(w) = z = (σa F−1 τ i)(w), ρi σa(m) = w = (ρi τi F )(m).

Since f is biholomorphic, the assertion follows.

The manifold

M = (M \ p) ∪ P(L) = U/ ∼

is called the blow-up of M at the point p.

Lemma 3.3.9. The collection of holomorphic functions

βi

0 : Wi ∩W0 → C∗, βi

0(F (z, [ζ])) := zi,

βj

i : Wi ∩Wj → C∗, βj

i (F (z, [ζ])) :=ζj

ζi

form a cocycle on M = (M \ p) ∪ P(L).

Proof. Note that βi

0(F (z, [ζ])) = zi is non-zero since Va ∩P(L) = ∅. On W0 ∩Wi ∩Wj we have

βi

0(m) βj

i (m) = βi(F−1(m)) βji (F−1(m)) = βj(F−1(m)) = β

j

0(m)

and

βi

0(m) β0

j (m) = βi(F−1(m))1

βj(F−1(m))= β

j

i (m).

Lemma 3.3.10. The line bundle U ×β∼C over M = U/ ∼ associated with the cocycle (??) corresponds

to the divisor P(L) ⊂ M. In formulas

[P(L)] = Uβ×∼

C

Proof. We have W0 ∩P(L) = ∅. If i > 0, then every point in Wi has the form

m = F (ti(t′, 1i), [t′, 1i]) =

σ(ti(t′, 1i)) ti 6= 0

[t′, 1i] ti = 0.

Thus the intersection Wi∩P(L) on the coordinate chart Wi correspond to ti = 0. Therefore, on Wi∩Wj ,

the cocycle associated with P(L) is given by ti

tj = βi

j(m).

Our next goal is to determine the Chern class of this line bundle in terms of a metric. Choose a

smooth function h : M → R+ satisfying

h(m) = 1

for m ∈M \ V , and

h(σz) = (z|z)

for all z ∈ U with (z|z) < ε.

66

Lemma 3.3.11. The smooth functions

h0

: W0 → R+, h0(m) := h(m),

hi

: Wi → R+, hi(ρi(t′, ti)) :=

h(σ(ti(t′, 1i)))

|ti|2

define a 0metric on the line bundle [P(L)] = U ×β∼C.

Proof. If 0 < |t| < ε then (??) implies

h(σ(ti(t′, 1i)))

|ti|2=‖ti(t′, 1i)‖2

|ti|2= ‖(t′, 1i)‖2.

Therefore (??) defines a smooth function on Wi. By Proposition ?? we need to verify the property

hi

m = |βj

i (m)|2 hj

m

for 0 ≤ i, j ≤ n andm ∈Wi∩Wj .Assume first i, j > 0. Letm = F (z, [ζ]) = ρi(t′′, ti, tj) = ρj(s′′, si, sj) ∈Wi∩Wj . Then z = ti(t′′, 1i, tj) = sj(s′′, si, 1j) and [ζ] = [t′′, 1i, tj ] = [s′′, si, 1j ], Therefore ti = sjsi, sj =

titj and ζ = ζi(t′′, 1i, tj) = ζj(s′′, si, 1j). This implies tiζj = sjsiζj = sjζi and hence

|βj

i (m)|2 hj(m) = |ζ

j

ζi|2 h(σm)

|sj |2=h(σm)

|ti|2= h

i(m).

On the other hand, if m ∈W0 ∩Wi, for i > 0, then m = σm = ρi(t′, ti) for z = ti(t′, 1i) ∈ U \ 0. Hence

zi = ti and

|βi

0(m)|2 hi(m) = |zi|2 h(σm)

|ti|2= h(σm) = h

0(m).

Corollary 3.3.12. The Chern class is given by the family of (1, 1)-forms

c([P(L)]) = c(Uβ×∼

C) ∼( 1

2πi∂∂ log h

`)n`=0

.

Lemma 3.3.13. Let π : M →M be the canonical projection, mapping P(L) to p. Then the (1, 1)-form

∂∂(h π + log h`) = π∗(∂∂h) + ∂∂ log h

`)

on W` is 0positive on a neighborhood of P(L) ⊂ M.

Proof. For fixed ` > 0 and (t|t) < ε the condition (??) implies

h`(ρ`(t′, t`)) = ‖(t′, 1`)‖2 = 1 + (t′|t′).

Putting (t′|dt′) :=∑j 6=`

tj dtj, (dt′|t′) :=

∑i 6=`

tidti, we have

∂ log(1 + (t′|t′)) =∂(t′|t′)

1 + (t′|t′)=

(t′|dt′)1 + (t′|t′)

and hence

∂∂ log(1 + (t′|t′)) =∂(t′|dt′)

1 + (t′|t′)− (dt′|t′)

1 + (t′|t′)∧ (t′|dt′)

1 + (t′|t′)

=1

1 + (t′|t′)∑i,j 6=`

δji dti ∧ dtj − (dt′|t′) ∧ (t′|dt′)

(1 + (t′|t′))2

67

=1

(1 + (t′|t′))2

∑i,j 6=`

dti ∧ dtj(δji (1 + (t′|t′))− ti tj

).

The matrix Aji := δji (1 + (t′|t′))− ti tj corresponds to the hermitian form

(ξ, η) 7→ (ξ′|η′)(1 + (t′|t′))− (ξ′|t′)(t′|η′) = (ξ′|η′) +(

(ξ′|η′)(t′|t′)− (ξ′|t′)(t′|η′)).

By Cauchy-Schwarz, this is semi-positive but vanishes on the hyperplane t` = 0. We need the extra

h-term for positivity: Near P(L) we have ‖t‖2 < ε and hence P ∗(∂∂h) = P ∗(∂∂(z|z)), with z = t`(t′, 1`)

and (z|z) = |t`|2(1 + (t′|t′)). Therefore

∂(z|z) = ∂(|t`|2(1 + (t′|t′))

)= (∂|t`|2)(1 + (t′|t′)) + |t`|2 ∂(t′|t′) = (t`dt

`)(1 + t′|t′) + |t`|2(t′|dt′)

and hence

∂∂(z|z) = ∂(t`dt`)(1 + t′|t′)− (t`dt

`) ∧ ∂(1 + t′|t′) + (∂|t`|2) ∧ (t′|dt′) + |t`|2 ∂(t′|dt′)

= dt` ∧ dt`(1 + t′|t′)− (t`dt`) ∧ (dt′|t′) + (t

`dt`) ∧ (t′|dt′) + |t`|2

∑j 6=`

dtj ∧ dtj .

By (??), the divisor P(L) corresponds to z = 0. On W` this is equivalent to t` = 0. If t` = 0 then

∂∂(z|z) = dt` ∧ dt`(1 + t′|t′).

By continuity, it follows that the sum

∂∂(h π + log h`) = ∂∂((z|z) + log(1 + (t′|t′)))

is positive definite on a neighborhood of P(L) in W`.

Proposition 3.3.14. The Chern class of the divisor P(L) ⊂ M is negative:

c[P(L)] < 0

near P(L).

Proof. By Lemma ?? P ∗(∂∂h) + ∂∂(log h) is strictly positive near P(L). Now

∂∂(h π) = ∂∂(π∗h) = d∂(π∗h) ∼ 0

is null-cohomologous in M. Hence c[−P(L)] is cohomologous to a positive line bundle.

3.3.3 Proof of the Kodaira embedding theorem

Proposition 3.3.15. The canonical line bundles K(M) and K(M) are related by

K(M) = π∗K(M) + (n− 1)[P(L)],

where π : M →M. is the canonical projection.

Proof. Relative to the coordinate charts σa : Ua → Va ⊂ M \ p and σ : U → V 3 p, the canonical line

bundle of M is given by the cocycle Jab = det ∂σb∂σa

on Va ∩ VbJa = det ∂σa∂σ on Va ∩ V

.

68

Passing to M, with coordinate charts σa : Ua → Va ⊂ W0 and ρi : Vi → Wi, the cocycle (??) is

supplemented by Jji := det ∂ρi∂ρj

on Wi ∩Wj

J ia := det ∂σa∂ρion M i ∩ Va

.

Now let s = ρij(t) = ρj ρi(t). Then σ(τ i(t)) = σ(τ j(s)) and hence

(ttt′′, ti, titj) = (sjs′′, sjsi, sj).

Therefore

sj = titj , si =ti

sj=

1

tj, s′′ =

ti

sjt′′ =

1

tjt′′

The partial derivatives ∂∂ti s

i = 0, ∂∂ti s

j = tj , ∂∂ti s

k = 0, ∂∂tj s

i = −1(tj)2 ,

∂∂tj s

j = ti, ∂∂tj s

k =

−tk(tj)2 ,

∂∂tk

si = 0, ∂∂tk

sj = 0, ∂∂tk

s` =δ`ktj yield the Jacobi matrix

∂ρi∂ρj

=

0 tj 0

−1/(tj)2 ti −t′′/(tj)2

0 0 I ′′/tj

with determinant

Jji = det∂ρi∂ρj

=1

(tj)n−2det

(0 tj

−1/(tj)2 ti

)=

1

(tj)n−1.

Finally, let σa(w) = ρi(t) = σ(m), where z = (tit′, ti). Since zi = ti, zk = titk we obtain ∂zi

∂ti = 1, ∂zk

∂ti =

tk, ∂zi

∂tk= 0, ∂z`

∂tk= tiδ`k. Therefore

∂z

∂t=

(1 t′

0 ti I ′

)has the determinant

det∂z

∂t= (ti)n−1.

It follows that

det∂w

∂t= det

∂w

∂zdet

∂z

∂t= (ti)n−1 det

∂w

∂z= (ti)n−1 det

∂σa∂σ

.

Proposition 3.3.16. Let E be a strictly positive line bundle on M. For distinct p, q ∈ M consider

M = (MP(L)p )

P(L)q , with canonical projection π : M → M. Then for k sufficiently large, the bundle

kπ∗E −KM − [Pp]− [Pq] on M is strictly positive.

Proof. By Proposition ?? we have

F := kπ∗E −KM − [Pp]− [Pq] = π∗(kE −KM )− n [Pp]− n [Pq].

It follows that

c(F ) = π∗c(kE −KM )− n c[Pp]− n c[Pq].

Since E > 0, there exists k so large that kE−KM > 0. Then π∗(kE−KM ) ≥ 0 on M and π∗(kE−KM ) >

0 on M \ (Pp ∪Pq), where π is biholomorphic. By Proposition 3.3.13, we have c[Pp] < 0 near Pp, and

similarly, c[Pq] < 0 near Pq. Therefore (??) is strictly positive on M.

Lemma 3.3.17. Let π : M = MPp → M where P ⊂ MP

p is a divisor isomorphic to P(L). For

a line bundle L over M let Op ⊗ L denote the sheaf of holomorphic sections M → L which vanish

at p. Let OP ⊗ π∗L denote the sheaf of holomorphic sections M → π∗L which vanish on P. Then

H1(M,OP ⊗ π∗L) = 0 implies H1(M,Op ⊗ L) = 0.

69

Proof. Let V = (Va) be an open covering of M such that L|Va is trivial. Then V a := π−1(Va) form an

open covering V of M. Let Φab ∈ Z1(V,Op⊗L) be a 1-cocycle. Thus Φab : Va∩Vb → L are holomorphic

sections vanishing on Vab ∩ p. Therefore Φab π : V a ∩ V b → π∗L are holomorphic sections vanishing

on V ab ∩ P. For any sheaf S, the canonical map

H1(V,S)→ H1(M,S)

is injective. Hence the assumption implies H1(V,OP ⊗π∗L) = 0. It follows that there exist holomorphic

sections ψa : V a → π∗L vanishing on V a ∩ P such that

Φab π = ψa − ψb.

Suppose first that p /∈ Va. Then V a ⊂ M \ P and π : V a → Va is biholomorphic. Therefore

Φa := ψa π−1 : Va → L

is a holomorphic section vanishing on Va ∩ p = ∅. Suppose now that p ∈ Va. Then the restriction

π : V a \ P → Va \ p is biholomorphic. Thus ψa π−1 : Va \ p → L is a holomorphic section. Since

L|Va is trivial, we may apply Hartogs’ extension theorem (for n > 2) to obtain a holomorphic section

Φa : Va → L satisfying

Φa|Va\p = ψa π−1.

Therefore Φa π = ψa on V a \ P. By continuity (or analytic continuation) it follows that

Φa π = ψa

on V a. This implies Φa(p) = ψa(P ) = 0. Thus we obtain a family (Φa) ∈ H0(V,Op ⊗ L) such that

Φab = Φa − Φb. Therefore (Φab) = 0 ∈ H1(V,Op ⊗ L). Since V is arbitrary and, in general,

Hq(M,S) = limVHq(V,S),

the assertion follows.

Theorem 3.3.18. (Kodaira Vanishing Theorem) Let L > 0. Then

Hq(X,Op ⊗ L) = 0 ∀ p+ q > n.

Proof. Since L > 0 the square (dA)2 of its Chern connexion is the exterior multiplication εω by a

positive (1, 1)-form ω. which is therefore a Kahler metric. Consider the operators

A := ∂A ∗∂A + ∗∂A ∂A, A

:= ∂A ∗∂

A+ ∗∂

A∂A

(Hodge-Laplacian). Thus

∧p+1,q

ιω

yy∧p,q−1 ∧p,q

∂Add

∗∂A

oo

ιωzz∧p−1,q−1

∂A

ee

and

∧p+1,q+1

ιω

yy∧p,q ∧p,q

εω

ee

[εω,ιω]oo

ιωyy∧p−1,q−1

εω

ee

.

70

For Kahler manifolds we have the Bochner-Kodaira-Nakano Identity

A −A = [εω, ιω].

By Dolbeault and Hodge theory we have

Hp(M,Ωq ⊗ L) = Hp,qA(M,L)

Now let u ∈ Hp,qA(M,L) = Hp(M,Ωq ⊗ L). Then∫

M

dm(Au|u)m =

∫M

dm(∂A ∗∂Au+ ∗∂A ∂Au|u)m =

∫M

dm(

(∗∂Au|∗∂Au) + (∂Au|∂Au)m

)≥ 0

and hence∫M

(Au|u) =

∫M

(Au+ [εω, ιω]u|u) =

∫M

(Au|u) +

∫M

([εω, ιω]u|u) ≥∫M

([εω, ιω]u|u).

If [εω, ιω] is positive definite on each fibre, then Au = 0 implies u = 0, i.e., Hp(M,Ωq ⊗ L) = 0. Since

(εω, ιω, (deg − n)I) is a so-called sl2-triple, we have

([εω], ιω]u|u) = (p+ q − n)‖u‖2

which is positive for p+ q > n.

Corollary 3.3.19. If F −KM > 0 then

Hq(M,O ⊗ F ) = 0 ∀ q > 0.

The Kodaira map is defined as follows: Let M be a Kahler manifold such that for all m ∈M there

exists Φ ∈ O(V∼ ×β C) with Φm 6= 0. Then O(V ×β∼ C) has finite dimension. We define a holomorphic

map

K : M → P(O(Vβ×∼

C)∗)

by the hyperplane

Kz := Φ ∈ O(Vβ×∼

C) : Φm = 0 = KerK∗m

as the kernel of the evaluation map.

The Kodaira embedding theorem (first half) is the following:

Theorem 3.3.20. Let E > 0 be a positive line bundle on a compact Kahler manifold M. Then, for k

large enough, the Kodaira map (??) for F = kE is injective.

Proof. For p 6= q in M consider the subsheaf Op,q ⊗ F ⊂ O ⊗ F of germs vanishing at p and q. Then

the so-called ’skyscraper sheaf’ S = O ⊗ F/Op,q ⊗ F has stalks Sp ≡ C ≡ Sq, whereas Sm = 0 for

m ∈M \ p, q. The exact sheaf sequence

0→ Op,q ⊗ F → O⊗ F → S → 0

induces an exact cohomology sequence

0→ H0(M,Op,q⊗F )→ H0(M,O⊗F )κ∗p,q−−→ H0(M,S) = C⊕C→ H1(M,Op,q⊗F )→ H1(M,O⊗F )→ H1(M,S)→ . . . ,

where κ∗p,q(Φ) = (Φap,Φbq), for p ∈ Va, q ∈ Vb, is the ’double’ evaluation map. In order to show that

the Kodaira map (??) is injective, it thus suffices to show that H1(M,Op,q ⊗ F ) = 0, since then κ∗p,q is

71

surjective for every pair p 6= q. Let π : M → M be the canonical projection, with P := π−1(p) = Pp

and Q := π−1(q) = Pq. By Lemma ?? it suffices to show that H1(M,OP∪Q ⊗ F ) = 0. Now F :=

π∗(kE) − [P ] − [Q] satisfies F − KM > 0 for k large enough, and hence, by corollary ??, we have

H1(M,O ⊗ F ) = 0. Since

O ⊗ F = O ⊗ (π∗(kE)− [P ]− [Q]) = OP∪Q ⊗ π∗(kE)

we finally obtain H1(M,OP∪Q ⊗ π∗(kE)) = 0 and hence H1(M,Op,q ⊗ (kE)) = 0. It follows that the

sheaf O ⊗ (kπ∗E − [Pp]− [Pq]) on M satisfies

H1(M,O ⊗ (kπ∗E − [Pp]− [Pq])) = 0.

72

geometric quantization in complex and harmonic analysisgm/geoquant.pdfconsider smooth manifolds over...

Documents