Non-K\"ahler Calabi-Yau manifolds and holomorphic geometric structures

TL;DR

This paper investigates holomorphic geometric structures on non-Kähler Calabi-Yau manifolds, establishing local homogeneity results and exploring the implications of rigidity and fundamental group properties.

Contribution

It proves that all affine-type holomorphic geometric structures on Vaisman Calabi-Yau manifolds are locally homogeneous and characterizes rigid structures as existing only on Kodaira manifolds.

Findings

Holomorphic geometric structures of affine type are locally homogeneous on Vaisman Calabi-Yau manifolds.

Rigid structures imply the Vaisman manifold is a Kodaira manifold.

Manifolds with self-dual tangent bundle and rigid structures have infinite fundamental group.

Abstract

We study holomorphic geometric structures on non-K\"ahler compact complex manifolds with trivial canonical line bundle. For Vaisman Calabi-Yau manifolds we prove that all holomorphic geometric structures of affine type on them are locally homogeneous. Moreover, if the geometric structure is rigid, then the Vaisman manifold must be a Kodaira manifold. The proof uses a Beauville-Bogomolov type decomposition from [Is] together with a weak form of Bochner principle for Vaisman Calabi-Yau manifolds that we prove here. Other results show that a compact complex manifold with self-dual holomorphic tangent bundle bearing a rigid holomorphic geometric structure of affine type have infinite fundamental group. We prove the same result for compact complex manifolds with trivial canonical line bundle having semistable holomorphic tangent bundle, with respect to some Gauduchon metric. We exhibit (non-K\"ahler) compact complex simply connected manifolds with trivial canonical line bundle that admit non-closed holomorphic one-forms.