Holomorphic Cartan geometries, Calabi–Yau manifolds and rational curves

Abstract

We prove that if a Calabi-Yau manifold M admits a holomorphic Cartan geometry, then M is covered by a complex torus. This is done by establishing the Bogomolov inequality for semistable sheaves on compact Kähler manifolds. We also classify all holomorphic Cartan geometries on rationally connected complex projective manifolds.2000 Mathematics Subject Classification. 53C15, 14M17.

Holomorphic Parabolic Geometries and Calabi-Yau Manifolds

“…Let us contrast our results in this paper with those of [3,5,10]. In [3], we proved that if a smooth complex projective variety with c 1 = 0 bears a holomorphic Cartan geometry, then the smooth projective variety has holomorphic unramified covering map by an Abelian variety.…”

“…In [3], we proved that if a smooth complex projective variety with c 1 = 0 bears a holomorphic Cartan geometry, then the smooth projective variety has holomorphic unramified covering map by an Abelian variety. In [5], we generalized this result to prove that if a compact Kähler manifold with c 1 = 0 bears a holomorphic Cartan geometry, then the compact Kähler manifold has a holomorphic unramified covering map by a complex torus. A special case of this result (for parabolic geometries) will be proven here in the first part of Theorem 1.…”

Holomorphic projective connections on compact complex threefolds

“…In this case it is known that M admits a finite unramified cover which is a compact complex torus [IKO] (the pull-back, to the torus, of such a global representative affine connection is a translation invariant holomorphic torsionfree affine connection). This type of results are also valid in the broader context of holomorphic Cartan geometries [BM2,BM3,Du3,BD4]; see [Sha] for holomorphic Cartan geometries.…”

“…The following proposition (statement (ii)) studies holomorphic projective connections on compact complex tori. Statement (i), which was already known in the broader context of holomorphic Cartan geometries (see for example, [BM2,BM3,BD4,Du3]), shows that compact complex tori cover the case of Kähler manifolds with trivial first Chern class.…”

Logarithmic Cartan geometry on complex manifolds

“…In higher dimension, it is a very stringent condition for a compact complex manifold to admit a holomorphic Cartan geometry. In this direction, several authors proved classifications results for compact complex manifolds bearing holomorphic Cartan geometries (see, for example, [BD1,BD2,BM,Du,IKO,JR1,KO1,KO2,KO3,JR2]).…”