Complex Structures on Principal Bundles

TL;DR

This paper explores the relationship between holomorphic principal G-bundles over complex manifolds and coadjoint orbits in the dual of a central extension of the Lie algebra, providing integrability conditions for complex structures.

Contribution

It reviews key results connecting holomorphic bundles and coadjoint orbits, and details an integrability condition for almost complex structures on trivial bundles.

Findings

Connection between holomorphic bundles and coadjoint orbits

Integrability condition for almost complex structures

Clarification of the correspondence on Riemann surfaces

Abstract

Holomorphic principal G-bundles over a complex manifold M can be studied using non-abelian cohomology groups H^1(M,G). On the other hand, if M=\Sigma is a closed Riemann surface, there is a correspondence between holomorphic principal G-bundles over \Sigma and coadjoint orbits in the dual of a central extension of the Lie algebra C^\infty(\Sigma, \g). We review these results and provide the details of an integrability condition for almost complex structures on smoothly trivial bundles. This article is a shortened version of the author's Diplom thesis.