Principal bundles on compact complex manifolds with trivial tangent bundle

TL;DR

This paper characterizes when holomorphic principal bundles over compact quotients of complex Lie groups admit holomorphic or flat connections, linking invariance and homogeneity to the existence of such connections.

Contribution

It establishes necessary and sufficient conditions for the existence of holomorphic and flat holomorphic connections on principal bundles over certain complex manifolds.

Findings

Holomorphic principal bundles admit a holomorphic connection iff they are invariant.

In simply connected cases, flat holomorphic connections exist iff the bundle is homogeneous.

Provides a complete characterization of connections on bundles over quotients of complex Lie groups.

Abstract

Let $G$ be a connected complex Lie group and $\Gamma\subset G$ a cocompact lattice. Let $H$ be a complex Lie group. We prove that a holomorphic principal $H$-bundle $E_H$ over $G/\Gamma$ admits a holomorphic connection if and only if $E_H$ is invariant. If $G$ is simply connected, we show that a holomorphic principal $H$-bundle $E_H$ over $G/\Gamma$ admits a flat holomorphic connection if and only if $E_H$ is homogeneous.