Lie Algebroid Connections on Principal Bundles

TL;DR

This paper introduces the concept of Lie algebroid valued connections on holomorphic principal bundles over complex varieties, exploring their properties and conditions for existence, especially on smooth projective curves.

Contribution

It defines Lie algebroid connections on principal bundles and analyzes their properties and existence criteria in the context of complex algebraic geometry.

Findings

Defined Lie algebroid valued connections on principal G-bundles

Studied extension and reduction of structure groups for these connections

Established criteria for existence on smooth projective curves

Abstract

Let $X$ be an irreducible smooth complex projective variety. Let $G$ be a linear algebraic group over $\mathbb{C}$. We define the notion of Lie algebroid valued connection on holomorphic principal $G$--bundles on $X$, and study their basic properties under extension and reduction of structure group. Finally we investigate criterions for existence of a Lie algebroid connection on principal $G$--bundles over smooth complex projective curves.