Equivariant bundles and adapted connections

TL;DR

This paper studies connections on holomorphic principal bundles over complex manifolds with group actions, focusing on those adapted to given connections, with applications to bundles with flat partial connections on foliated manifolds.

Contribution

It introduces the concept of strongly adapted connections on principal bundles with group actions and explores their properties and examples.

Findings

Characterization of strongly adapted connections

Examples involving flat partial connections on foliated manifolds

Insights into the structure of equivariant principal bundles

Abstract

Given a complex manifold $M$ equipped with a holomorphic action of a connected complex Lie group $G$, and a holomorphic principal $H$--bundle $E_H$ over $X$ equipped with a $G$--connection $h$, we investigate the connections on the principal $H$--bundle $E_H$ that are (strongly) adapted to $h$. Examples are provided by holomorphic principal $H$--bundles equipped with a flat partial connection over a foliated manifold.