# Branched holomorphic Cartan geometries and Calabi-Yau manifolds

## TL;DR

This paper introduces branched holomorphic Cartan geometries, generalizing complex projective structures, and explores their existence on various compact complex manifolds, including Calabi-Yau manifolds, with new examples and non-existence results.

## Contribution

It defines branched holomorphic Cartan geometries, proves all compact complex projective manifolds admit such structures, and shows non-projective Calabi-Yau manifolds do not, providing new insights and examples.

## Key findings

- All compact complex projective manifolds admit branched flat holomorphic projective structures.
- A non-flat branched holomorphic normal projective structure exists on some compact complex surfaces.
- Non-projective simply connected Kähler Calabi-Yau manifolds do not admit branched holomorphic projective structures.

## Abstract

We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum. This new framework is much more flexible than that of the usual holomorphic Cartan geometries. We show that all compact complex projective manifolds admit branched flat holomorphic projective structure. We also give an example of a non-flat branched holomorphic normal projective structure on a compact complex surface. It is known that no compact complex surface admits such a structure with empty branching locus. We prove that non-projective compact simply connected K\"ahler Calabi-Yau manifolds do not admit branched holomorphic projective structures. The key ingredient of its proof is the following result of independent interest: If E is a holomorphic vector bundle over a compact simply connected K\"ahler Calabi-Yau manifold, and E admits a holomorphic connection, then E is a trivial holomorphic vector bundle equipped with the trivial connection.