Journal of Geometry and Physics 60 (2010) 661–663

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Journal of Geometry and Physics

journal homepage: www.elsevier.com/locate/jgp

Holomorphic Cartan geometries and Calabi–Yau manifoldsIndranil Biswas a,∗, Benjamin McKay ba School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, Indiab School of Mathematical Sciences, University College Cork, National University of Ireland, Ireland

a r t i c l e i n f o

Article history:Received 16 February 2009Received in revised form 17 December 2009Accepted 31 December 2009Available online 11 January 2010

MSC:53C1514M17

Keywords:Cartan geometryCalabi–Yau manifoldConnection

a b s t r a c t

LetM be a connected complex projective manifold such that c1(T (1,0)M) = 0. IfM admitsa holomorphic Cartan geometry, then we show that M is holomorphically covered by anabelian variety.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Let G be a complex Lie group and H ⊂ G a closed complex subgroup. The Lie algebras of G and H will be denoted by g andh respectively.LetM be a connected complex manifold. Let

EH −→ M

be a holomorphic principal right H-bundle. For each point h ∈ H , let

τh : EH −→ EH (1.1)

be the translation by the action of h. For any v ∈ h, let ζv be the holomorphic vector field on EH that assigns to z ∈ EH thetangent vector defined by the analytic curve

t 7−→ z exp(tv).

A holomorphic Cartan geometry onM of type G/H is a holomorphic principal H-bundle

EH −→ M (1.2)

together with a holomorphic 1-form on EH with values in g

ω ∈ H0(EH ,Ω1EH ⊗C g) (1.3)

satisfying the following three conditions:

∗ Corresponding author.E-mail addresses: indranil@math.tifr.res.in (I. Biswas), B.McKay@ucc.ie (B. McKay).

0393-0440/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.geomphys.2009.12.011

662 I. Biswas, B. McKay / Journal of Geometry and Physics 60 (2010) 661–663

• ω(z)(ζv(z)) = v for all v ∈ h and z ∈ EH , where ζv is the vector field defined above,• τ ∗hω = Ad(h

−1) ω for all h ∈ H , where τh is defined in (1.1), and• for each point z ∈ EH , the homomorphism from the holomorphic tangent space

ω(z) : TzEH −→ g (1.4)

is an isomorphism of vector spaces.

All Chern classes will have coefficients in Q.By a Calabi–Yau manifold we mean a compact Kähler manifoldM with c1(M) = 0.In [1] the second author conjectured that ifM is a Calabi–Yau manifold admitting a holomorphic Cartan geometry, then

M is holomorphically covered by a complex torus. He proved that if M is a Calabi–Yau manifold admitting a holomorphicparabolic geometry, then M is holomorphically covered by a complex torus (see [1, Theorem 1]). We recall that a Cartangeometry of type G/H is called a parabolic geometry if G is simple and H is a parabolic subgroup of G.Our aim here is to prove this conjecture under the assumption that M is complex projective. We prove the following

theorem (see Theorem 3.1):

Theorem 1.1. Any Calabi–Yau complex projective manifold M bearing a holomorphic Cartan geometry is holomorphicallycovered by an abelian variety.

This paper is based upon works supported by the Science Foundation Ireland under Grant No. MATF634.

Added in Proofs:We have now proved the conjecture of the second author (‘‘Holomorphic Cartan geometries, Calabi-Yaumanifolds, and rational curves’’, Diff. Geom. Appl. (to appear)).

2. Flat vector bundles on a Calabi–Yau manifold

LetM be a Calabi–Yau manifold. We will later impose the condition thatM is projective.

Lemma 2.1. Let (E,∇) be a flat vector bundle over M. Let

E −→ Q −→ 0

be a quotient holomorphic vector bundle such that c1(Q ) = 0. Then

ci(Q ) = 0

for all i ≥ 1.

Proof. It is well known [2, p. 764, Théorème 2(2)] that there is a finite étale Galois covering

M −→ M (2.1)

by a product M = MCY × T n of a simply connected Calabi–Yau manifold with a complex torus. (Note that Xj in [2, Théoréme2(2)] are simply connected [2, p. 763, Proposition 4(2)].) Clearly we can replace M by M without loss of generality, and soassume that the fundamental group π1(M) is Zn. The flat vector bundle E is given by a linear representation of Zn. SinceZn is abelian, any complex linear representation of Zn admits a filtration such that each successive quotient is a complexone-dimensional Zn-module. Therefore, we have a filtration of E by holomorphic sub-bundles

0 = E0 ⊂ E1 ⊂ · · · ⊂ E`−1 ⊂ E` = E (2.2)

such that each successive quotient Ei/Ei−1, i ∈ [1, `], is a flat line bundle.Therefore, E is numerically flat [3, p. 311, Theorem 1.18]. In particular, E is numerically effective. Hence Q is numerically

effective [3, p. 308, Proposition 1.15(i)]. Since c1(Q ) = 0, this implies that E is numerically flat (see [3, p. 311, Definition1.17]). Consequently, ci(Q ) = 0 for all i ≥ 1 [3, p. 311, Corollary 1.19].

Remark 2.2. The referee pointed out that Lemma 2.1 is valid for vector bundles over any complex manifold M (not justCalabi–Yau).

Fix any Kähler class onM . By Yau’s proof of Calabi’s conjecture [4],M admits a Ricci flat Kähler metric, say θ , in the givenKähler class. Note that the condition that θ is Ricci flat implies that the holomorphic tangent bundle TM is polystable ofdegree zero with respect to θ (i.e., a direct sum of stable vector bundles of degree zero).

Lemma 2.3. Let M be a Calabi–Yau complex projective manifold. Let E −→ M be a holomorphic vector bundle with aholomorphic connection. Then E admits a holomorphic flat connection.

Proof. This was proved in [5] (see [5, p. 2827, Theorem A(1)]). Note that TM is semistable because it is polystable.Therefore, [5, Theorem A(1)] applies.

I. Biswas, B. McKay / Journal of Geometry and Physics 60 (2010) 661–663 663

3. Cartan geometries and Calabi–Yau manifolds

As before, M is a Calabi–Yau projective manifold. Let G be a complex Lie group, and let H ⊂ G be a closed complexsubgroup. Let (EH , ω) be a pair as in (1.2) and (1.3) defining a Cartan geometry of type G/H onM .Let

EG := EH ×H G −→ M (3.1)

be the holomorphic principal G-bundle obtained by extending the structure group of the principal H-bundle EH using theinclusionmap ofH in G. The formω defines a holomorphic connection on EG. We recall the construction of this holomorphicconnection.Let ωMC : TG −→ G× g be the g-valued Maurer–Cartan one-form on G constructed using the left invariant vector fields.

Consider the g-valued holomorphic one-form

ω := p∗1ω + p∗

2ωMC

on EH × G, where p1 (respectively, p2) is the projection of EH × G to EH (respectively, G). This form ω descends to a g-valuedholomorphic 1-form on the quotient space EG in (3.1). This descended 1-form defines a holomorphic connection on EG.

Theorem 3.1. Any Calabi–Yau complex projective manifold M bearing a holomorphic Cartan geometry is holomorphicallycovered by an abelian variety.

Proof. Let ad(EH) and ad(EG) be the adjoint vector bundles of EH and EG respectively. Since the homomorphism in (1.4) isan isomorphism, it follows that

TM = ad(EG)/ad(EH). (3.2)

We noted above that EG is equipped with a holomorphic connection. A holomorphic connection on EG defines aholomorphic connection on ad(EG). Hence ad(EG) admits a holomorphic connection. So from Lemma 2.3 it follows thatad(EG) admits a flat holomorphic connection. In view of Lemma 2.1, from (3.2) we conclude that c2(TM) = 0. Now the proofis completed by Lemma 1 of [1].

References

[1] B. McKay, Holomorphic parabolic geometries and Calabi–Yau manifolds, e-print: http://arxiv.org/abs/0812.1749.[2] A. Beauville, Variétés kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983) 755–782.[3] J.-P. Demailly, T. Peternell, M. Schneider, Compact complexmanifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994) 295–345.[4] S.-T. Yau, On the Ricci curvature of a compact Kählermanifold and the complexMonge–Ampère equation. I, Comm. Pure Appl.Math. 31 (1978) 339–411.[5] I. Biswas, Vector bundles with holomorphic connection over a projective manifold with tangent bundle of nonnegative degree, Proc. Amer. Math. Soc.126 (1998) 2827–2834.