On connections on principal bundles

TL;DR

This paper explains a new universal connection construction and discusses criteria for holomorphic connections on principal bundles over compact Riemann surfaces, generalizing classical results.

Contribution

It provides an explanation of a new universal connection construction and extends Atiyah-Weil criteria to principal bundles.

Findings

Universal connection construction explained

Criteria for holomorphic connections on principal bundles established

Generalization of Atiyah-Weil criterion to principal bundles

Abstract

A new construction of a universal connection was given in \cite{BHS}. The main aim here is to explain this construction. A theorem of Atiyah and Weil says that a holomorphic vector bundle $E$ over a compact Riemann surface admits a holomorphic connection if and only if the degree of every direct summand of $E$ is degree. In \cite{AB}, this criterion was generalized to principal bundles on compact Riemann surfaces. This criterion for principal bundles is also explained.