Holomorphic Cartan geometries and rational curves

TL;DR

This paper investigates the relationship between holomorphic Cartan geometries on compact Kähler manifolds and the existence of rational curves, establishing conditions under which rational curves are present based on inherited geometries.

Contribution

It characterizes when a compact Kähler manifold with a holomorphic Cartan geometry contains rational curves, linking this to inherited geometries from lower-dimensional manifolds.

Findings

Rational curves exist if the Cartan geometry is inherited from a lower-dimensional manifold.

Provides a criterion for the presence of rational curves based on inherited geometries.

Connects the geometry of the manifold with the existence of rational curves.

Abstract

We prove that any compact K\"ahler manifold bearing a holomorphic Cartan geometry contains a rational curve just when the Cartan geometry is inherited from a holomorphic Cartan geometry on a lower dimensional compact K\"ahler manifold.