Holomorphic Cartan geometries on complex tori

TL;DR

This paper proves that all flat holomorphic Cartan geometries of affine type on complex tori are translation invariant, confirming a specific case of a previously posed question.

Contribution

It establishes that holomorphic Cartan geometries with affine complex Lie groups on complex tori are necessarily translation invariant.

Findings

All flat holomorphic Cartan geometries of affine type on complex tori are translation invariant.

The result confirms the conjecture for the case when G is an affine complex Lie group.

The paper provides a complete characterization of such geometries on complex tori.

Abstract

In [DM] it was asked whether all flat holomorphic Cartan geometries (G,H) on a complex torus are translation invariant. We answer this affimatively under the assumption that the complex Lie group G is affine. More precisely, we show that every holomorphic Cartan geometry of type (G,H), with G a complex affine Lie group, on any complex torus is translation invariant.