Holomorphic affine connections on non-K\"ahler manifolds

TL;DR

This paper explores the properties of holomorphic affine connections on non-Kähler compact complex manifolds, establishing conditions for local homogeneity and implications for fundamental groups.

Contribution

It proves that certain holomorphic geometric structures on non-Kähler manifolds are locally homogeneous and links rigidity to the infiniteness of the fundamental group.

Findings

Holomorphic affine structures on Calabi-Yau manifolds are locally homogeneous.

Rigid geometric structures imply infinite fundamental group.

Manifolds with holomorphic Riemannian metrics and algebraic dimension one have infinite fundamental group.

Abstract

Our aim here is to investigate the holomorphic geometric structures on compact complex manifolds which may not be K\"ahler. We prove that holomorphic geometric structures of affine type on compact Calabi-Yau manifolds with polystable tangent bundle (with respect to some Gauduchon metric on it) are locally homogeneous. In particular, if the geometric structure is rigid in Gromov's sense, then the fundamental group of the manifold must be infinite. We also prove that compact complex manifolds of algebraic dimension one bearing a holomorphic Riemannian metric must have infinite fundamental group.