Killing Fields of Holomorphic Cartan Geometries

TL;DR

This paper investigates local automorphisms of holomorphic Cartan geometries on compact complex manifolds, leading to classification results and showing that certain Calabi-Yau manifolds admit finite covers that are complex tori.

Contribution

It provides new classification results for compact complex manifolds with holomorphic Cartan geometries, especially relating to Calabi-Yau manifolds and their covers.

Findings

Compact Calabi-Yau manifolds with algebraic holomorphic Cartan geometries have finite covers that are complex tori.

Classification of manifolds admitting holomorphic Cartan geometries.

Analysis of local automorphisms of these geometries.

Abstract

We study local automorphisms of holomorphic Cartan geometries. This leads to classification results for compact complex manifolds admitting Cartan geometries. We prove that a compact Calabi-Yau manifold bearing a holomorphic Cartan geometry of algebraic type admits a finite unramified cover which is a complex torus.