# Generalized Holomorphic Cartan geometries

## TL;DR

This survey explores branched holomorphic Cartan geometries, a flexible generalization of classical structures, and classifies them on certain compact complex manifolds, expanding understanding of complex geometric structures.

## Contribution

Introduces and studies branched holomorphic Cartan geometries, generalizing classical concepts and enabling classification on specific compact complex manifolds.

## Key findings

- All compact complex projective manifolds admit branched holomorphic projective structures.
- Classified branched holomorphic Cartan geometries on compact simply connected Calabi-Yau manifolds.
- Generalized the notion of complex projective structures to higher dimensions.

## Abstract

This is largely a survey paper, dealing with Cartan geometries in the complex analytic category. We first remind some standard facts going back to the seminal works of F. Klein, E. Cartan and C. Ehresmann. Then we present the concept of a branched holomorphic Cartan geometry which was introduced by the authors in [BD]. It generalizes to higher dimension the notion of a 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 (e.g. all compact complex projective manifolds admit branched holomorphic projective structures). At the same time, this new definition is rigid enough to enable us to classify branched holomorphic Cartan geometries on compact simply connected Calabi-Yau manifolds.