Cartan geometries on complex manifolds of algebraic dimension zero

TL;DR

This paper proves that compact complex manifolds with algebraic dimension zero that admit a holomorphic Cartan geometry of algebraic type must have infinite fundamental group, extending previous results for special geometric structures.

Contribution

It generalizes earlier theorems by showing the result holds for all holomorphic Cartan geometries of algebraic type on such manifolds.

Findings

Manifolds of algebraic dimension zero with these geometries have infinite fundamental group.

Extends previous results from affine connections and conformal structures to general Cartan geometries.

Provides new insights into the topology of complex manifolds with special geometric structures.

Abstract

We show that compact complex manifolds of algebraic dimension zero bearing a holomorphic Cartan geometry of algebraic type have infinite fundamental group. This generalizes the main Theorem in [DM] where the same result was proved for the special cases of holomorphic affine connections and holomorphic conformal structures.