Almost Complex Structures on quaternion-Kahlermanifolds of positive type

Paul Gauduchon

Mathematics Days in Sofia, 7–11 July 2014

based on:P. Gauduchon–A. Moroianu–U. Semmelmsann, Almostcomplex structures on quaternion-Kahler manifolds and innersymmetric spaces, Invent. math. 184 (2012), 389–403.

Other reference:C. LeBrun–S. Salamon, Strong rigidity of positivequaternion-Kahler manifolds, Invent. math. 118 (1994), 109–132.

Quaternion-Kahler manifolds

A quaternion-Kahler — qK for short — manifold is aRiemannian manifold (M, g) of dimension n = 4k ≥ 8, whoseholonomy group is contained in Sp(k) Sp(1) ⊂ SO(4k).

Remark 1

1. Sp(k) Sp(1) =(Sp(k)× Sp(1)

)/±1

2. Sp(k) ⊂ U(2k): preserves the quaternionic structure j of C2k

(j anti-C-linear, j2 = −1), or, equivalently, the complexsymplectic 2-form Ω = 〈·, j ·〉

3. Sp(k) Sp(1) ⊂ SO(4k), via

(A, p) · u = A u p−1,

for any A in Sp(k), any p in Sp(1), any u in R4k = Hk .

Alternative, more concrete, definition

A quaternion-Kahler manifold is a (complete) orientedRiemannian manifold (M, g) of dimension n = 4k ≥ 8 equippedwith a rank 3 subbundle Q ⊂ A(M) of the bundle A(M) ofanti-symmetric endomorphims of the tangent bundle TM suchthat:

(i) Q is preserved by the Levi-Civita connection ∇ of g , and

(ii) Q is locally generated by positively oriented almost-complexstructures J1, J2, J3 such that J1J2J3 = −IdTM .

Classical examples1. Quaternionic projective space: M = HPk = Sp(k+1)

Sp(k)×Sp(1)2. Grassmannian of complex 2-planes in C2+k :

M = Gr2(2 + k ,C) = SU(k+2)

S(U(k)×U(2)

)3. Grassmannian of real oriented 4-planes in R4+k :

M = Gr4(4 + k ,R) = SO(k+4)SO(k)×SO(4)

In each caseTxM = HomK(x , x⊥),

for any x in M, with K = H,C,R respectively, and the definingbundle Q is then:

Qx = Φ ∈ Ax(M) of the form Φ(X ) = X φfor any X in TxM, where φ is either

1. a quaternionic skew-Hermitian endomorphism, or2. a trace-free, skew-hermitian endomorphism, or3. a self-dual skew-symmetric endomorphism

of x , i.e. an element of sp(x), su(x) or so(x), respectively.

Wolf spaces

The above classical examples, together with

G2

SO(4),

F4

Sp(3)sp(1),

E6

SU(6)Sp(1),

E7

Spin(12)Sp(1),

E8

E7Sp(1),

are so-called Wolf spaces, constructed, as well as their twistorspaces, by J. A. Wolf in 1965.

For any compact simple Lie group G , the corresponding twistorspace is realized as

1. the (co-)adjoint orbit in the Lie algebra g of the highestcoroot, J. A. Wolf 1965,

2. the projectivization P(Nmin) of the minimal (co-)adjoint orbitNmin of GC in gC , A. Beauville 1998.

Quaternion-Kahler manifolds of positive type

Theorem 1 (D. Alekseevsky 1968, S. Ishihara 1974, S.Salamon 1982, A. L. Besse 1987)

Quaternion-Kahler manifolds are Einstein.

Three types, according to the sign of the scalar curvature s:

1. s > 0 qK of positive type (compact)

2. s = 0 locally hyperkahler: Hol0(M, g) ⊂ Sp(k)

3. s < 0 qK of negative type

The talk is mainly devoted to qK manifolds of positive type, whichwe simply call positive qk manifolds.

The twistor fibration

To any qK manifold (M, g ,Q) is associated its twistor space,Z = Z (M), introduced and worked out independently by S.Salamon and L. Berard Bergery in 1982, defined by

Z (M) = S(Q) (sphere bundle ofQ)

= bundle of compatible complex structures onTM.

Canonical almost complex structure J on Z : obtained bycombining the natural complex structure on the fibers with thetautological complex structure on the horizontal distribution,n H∇,determined by the Levi-Civita connection, ∇, of g .

The canonical almost complex structure J is integrable, makingZ into a complex manifold of (complex) dimension 2k + 1.Moreover, Z admits a natural real structure, κ, with no fixedpoint, namely κ(J) = −J, for any J in Z .

Main properties of the twistor spaceWe henceforth assume that (M, g ,Q) is a positive qK manifold.

Then,

1. The pull-back of g on Z , together with an appropriate scalingof the round metric on the fibers of Z , determines aRiemannian metric, g , which is Kahler-Einstein, withpositive scalar curvature, making Z into a Fano manifold (=the anti-canonical line bundle K−1Z of (Z ,J ) is ample).

2. The horizontal distribution H∇ is a J -holomorphicsubbundle of the (holomorphic) tangent bundle TZ .

3. The vertical projection, θ : TZ → TV , regarded as 1-form onZ with values in the holomorphic line bundle

L = TZ/H∇, (1)

is a holomorphic contact form, meaning that θ ∧ (dθ)k is anon-vanishing holomorphic section of KZ ⊗ Lk+1.

Inverse construction

It follows that Z is a contact Fano manifold, with

K−1Z = Lk+1

(in particular, the index of Z , as a Fano manifold, is at least equalto k + 1 = dimCZ+1

2 .)

Conversely,

Theorem 2 (C. LeBrun 1995, A. Moroianu 1998 if M is spin)

Any contact Fano manifold, admitting a Kahler-Einstein metricand a suitable real structure, is the twistor space of a positive qKmanifold.

Four-dimensional qK manifolds

The above definitions for qK manifolds are void for 4-dimensionalmanifolds: indeed, Sp(1) Sp(1) = SO(4), whereas Q = A+M,bundle of self-dual elements of A(M), always satisfies requiredproperties for any oriented 4-dimensional riemannian manifold.

Definition 1A 4-dimensional oriented Riemannian manifold (M, g) is calledquaternion-Kahler if

1. g is Eintein, and

2. W + = 0 ⇔ the canonical almost complex structure of thetwistor space is integrable (Atiyah–Hitchin–Singer)

Positive 4-dimensional qK manifolds: HP1 = S4 and

Gr2(3,C) = CP2.

Facts and conjectures

At the moment, only known examples of qK manifolds of positivetype are Wolf spaces.Current conjecture (LeBrun–Salamon 1994): No other examplesexist. Established for k = 1 (Hitchin 1981, Friedrich– Kurke 1982)and for k = 2 (Y.-S. Poon–Salamon 1991, LeBrun–Salamon 1994).Open for k ≥ 3.

Known fact: For any k , finitely many 4k-dimensional qKmanifolds of positive type (LeBrun-Salamon 1994)

Theorem 3 (C. LeBrun–S. Salamon)

Let (M, g) be any 4k-dimensional qK manifold of positive type.Then, b2(M) ≤ 1, with equality iff M = Gr2(2 + k ,C).

Uses works on Fano manifolds, in the framework of the Moriprogram, in particular works of S. Mori, J. Kollar, Y. Miyaoka, , J.A. Wisniewski...

Almost complex structures on positive qK manifolds

Theorem 4 (D. Alekssevsky–S. Marchiafava–M. Pontecorvo1998)

Positive qK manifolds admit no compatible almost complexstructure. Equivalently, the twistor fibration of a positive qKmanifold has no global section.

Notice that the natural complex structure J of M = Gr2(2 + k,C)is defined by JX = X i , for X in TxM = HomC(x , x⊥), hence isnot a compatible complex structure.

The main goal of this talk is to give a proof of

Theorem 5 (P. Gauduchon–A. Moroianu–U. Semmelmann2011)

Let (M, g ,Q) be any positive qK manifold, of dimension 4k ≥ 8,different from Gr2(2 + k ,C). Then, M admits no almost complexstructure, even in the stable sense (= a complex structure onTM ⊕ R`, for some `).

Preliminary remarks

1. HP1 = S4 is stably complex, since TS4⊕R is trivial, but has noalmost complex structure: otherwise, χ+ τ would be divisible by 4,

but χ = 2 and τ = 0. Similarly, CP2 has no almost complexstructure compatible with the reversed orientation, as χ = 3 andτ = −1.

2. The non-existence of (stable) almost complex structures on Hk

was established by F. Hirzebruch (1953 for k 6= 2, 3, 1958 for anyk). Alternative argument by W. S. Massey in 1962.

3. The non-existence of (stable) almost complex structures onmost real Grasssmannians Gr4(4 + k ,R) was establihed by P.Sankaran in 1991 and Z.-Z. Tang in 1994 (only Gr4(8,R) andGr4(10,R) are not covered).

The Atiyah–Singer index theorem

The proof of Theorem 5 relies on the Atiyah–Singer theorem inthe following form: Let

1. M be any compact, oriented, spin, even-dimensionalRiemannian manifold,

2. Σ be its spinor bundle,

3. V be any auxiliary complex vector bundle over M,

4. DV : Σ+ ⊗ V → Σ− ⊗ V be the corresponding twisted Diracoperator,

then, the index of DV is given by

indV =(ch(V )A(TM)

)[M].

Still holds if M is not spin — so that Σ is only locally defined —and, similarly, V is only locally defined, provided that Σ⊗ V isglobally defined.

The canonical vector bundles H ,E

For any positive qK manifold (M, g ,Q) of dimension n = 4k, wehave

TCM = H ⊗ E ,

where:

1. TCM = TM ⊗ C: complexified tangent bundle,

2. H: rank 2 complex vector bundle induced by the standardrepresentation of Sp(1) = SU(2), and

3. E : rank 2k complex vector bundle induced by the standardrepresentation of Sp(k) ⊂ SU(2k).

Beware: E , H are only globally defined if the Sp(k)Sp(1)-structureof (M, g) lifts to a Sp(k)× Sp(1)-structure. This happens iffw2(Q) = 0, iff M = HPk ((S. Marchiafava–G. Romani 1976, S.Salamon 1982). For M 6= HPk , E ,H are only locally defined butany even tensor combination of them is globally defined.

Spinor bundles on qK manifolds

A 4k-dimensional positive qK manifold (M, g ,Q) is spin iff: k iseven, or M = HPk (S. Salamon 1974). In both cases, the spinorbundle is

Σ =⊕

r+s=k

R r ,s ,

withR r ,s = SymrH ⊗ Λs

0E

where:

1. SymrH: r -th symmetric tensor power of H, and

2. Λs0H: primitive part with respect to the (complex) symplectic

form of E of the s-th exterior power of E .

(Kramer–Semmelmann-Weingart 1999)

Index of twisted Dirac operators

Even if M is not spin, the twisted spinor bundle Σ⊗ Rp,q isglobally defined, as well as the corresponding twisted Diracoperator DRp,q , whenever p + q + k is even.

Theorem 6 (C. LeBrun-S. Salamon 1994)

The index of DRp,q is then given by

ind DRp,q =

0 if p + q < k

(−1)q(b2q−2(M) + b2q(M)

)if p + q = k

Proof of Theorem 6 (Penrose correspondence)

Todd class td(Z ) of Z versus A-class A(M):

td(Z ) =c1(L) e

k2c1(L)

1− e−c1(L)π∗A(M). (2)

from which we infer:

ind DRp,q = χq(Z , L−r )− χq−2(Z , L−(r+1)), (3)

with r = k−p−q2 , whereas:

χq(Z , L−r ) = 0, if r > 0,

χq(Z ,O) = (−1)qb2q(Z ) = (−1)q (b2q(M) + b2q−2(M))

(4)

Proof of Theorem 5, Step 1Let M be any positive qK manifold, of dimension 4k , k > 1, andassume that M 6= Gr2(2 + k ,C), so that, by Theorem 3:

b2(M) = 0

Apply the Atiyah–Singer index theorem for twisted dirac operatorwith auxiliary bundleV = TCM ⊗ Symk−2H ' H ⊗ Symk−2H ⊗ E . ByClebsch–Gordan:

1. V = R1,1 if k = 2, or

2. V = Rk−1,1 ⊕ Rk−3,1 if k ≥ 3.

In both cases, the twisted spinor bundle Σ⊗V is well defined, and:

1. ind V = ind DR1,1 if k = 2, and

2. ind DV = ind DRk−1,1 + ind DRk−3,1 if k ≥ 3.

In both cases, we get (Theorem 6):

indV = −1 .

Proof of Theorem 5, Step 2

On the other hand, by the Atiyah–Singer index theorem:

ind DV =(ch(TCM) ch(Symk−2H) A(TM)

)[M].

Notice that the Chern character ch(Symk−2H) is well-defined inH∗(MK ,Q), even if k is odd, by

ch(Symk−2H) =(ch(Symk−2H)⊗2

) 12 .

We now assume, for a contradiction, that M admits an almostcomplex structure, so that TCM = T ⊕ T = T ⊕ T ∗, whereT = TM is a complex vector bundle. We thus have

ind DV =((

ch(T ) + ch(T ∗))

ch(Symk−2H) A(TM))

[M].

Proof of Theorem 5, Step 3

Observe:

1. H,E are (complex) symplectic bundle, hence self-dual, aswell as their tensor powers: ch(Symk−2H) A(TM) then onlyinvolves elements of H4`(M,Q).

2. Only components in H4`(M,Q) in ch(T ) + ch(T ∗) contributenon-trivally in the index formula. Sincech(T ∗) =

∑(−1)i chi (T ), we can replace T ∗ by T , to get

ind DV = 2(

ch(T ) ch(Symk−2H) A(TM))

[M].

3.(

ch(T ) ch(Symk−2H) A(TM))

[M] is the index of the twisted

Dirac operator DSymk−2H⊗T = DRk−2,0⊗T , acting on the

twisted spinor bundle Σ⊗ Rk−2,0 ⊗ T (globally defined, sncek − 2 + k = 2k − 2 is even).

Proof of Theorem 5: Final step

Putting all this together, we get:

ind DV = −1 = 2 ind DRk−2,0⊗T ,

which is clearly a contradiction, as both indices are integers. Itfollows that TM cannot be a complex vector bundle.

To complete the proof of Theorem 5, we have to consider thestable case, when, for a contradiction, TM ⊕ R2` is supposed tobe a complex vector bundle for some integer `, so that

TCM ⊕ C2` = T ⊕ T ∗, (5)

where T = TM ⊕ R2` is a complex vector. We thus get

ind DV = −1 = 2 ind DRk−2,0⊗T − 2` ind DRk−2,0 .

Contradiction again.

Final considerations

In the same paper, a similar argument was used to establish thenon-existence of (stable) almost complex structures on anycompact inner symmetric space, namely compact symmetric spaceG/H, with rankG = rankH), with the possible exception of

E7SU(8)/Z2

.

Notice that Wolf spaces are inner symmetric spaces.

Later, Andrei Moroianu and Uwe Semmelmann establisheda similar non-existence theorem of (stable) almost complexstructures on a larger class of homogeneous spaces (including themissing E7

SU(8)/Z2)