Holomorphic Cartan geometries, Calabi--Yau manifolds and rational curves

TL;DR

This paper proves that Calabi--Yau manifolds with holomorphic Cartan geometries are covered by complex tori and classifies such geometries on rationally connected projective manifolds, advancing understanding of geometric structures on complex manifolds.

Contribution

It establishes the link between holomorphic Cartan geometries and the structure of Calabi--Yau manifolds, and classifies geometries on rationally connected manifolds.

Findings

Calabi--Yau manifolds with such geometries are covered by complex tori.

Classification of holomorphic Cartan geometries on rationally connected projective manifolds.

Established Bogomolov inequality for semistable sheaves on compact K"ahler manifolds.

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 K\"ahler manifolds. We also classify all holomorphic Cartan geometries on rationally connected complex projective manifolds.