On holomorphic principal bundles over a compact Riemann surface admitting a flat connection
type: Item
instance of: scholarly article
title: On holomorphic principal bundles over a compact Riemann surface admitting a flat connection
zbMATH Open document ID: 0999.32010
author: Indranil Biswas
author: Hassan Azad
published in: Mathematische Annalen
publication date: 2002-05-29T00:00:00Z
review text: Let \(X\) be a compact connected Riemann surface, and \(E\) be a holomorphic vector bundle over \(X.\) A theorem of A. Weil says that \(E\) admits a flat connection, compatible with its holomorphic structure, if and only if every direct summand of \(E\) is of degree zero. The aim of this article is to establish an analog of the criterion for principal bundles. Let \(G\) be a connected reductive linear algebraic group over \({\mathbb C},\) and \(L \subset G\) be a Levi factor of some parabolic subgroup of \(G,\) with \(L_{0} = L/[L, L]\) its maximal abelian quotient. Theorem 4.1. A holomorphic \(G\)-bundle \(E_{G}\) over \(X\) admits a flat connection, compatible with its holomorphic structure, if and only if for every parabolic subgroup \(P\) containing the fixed Borel \(B\) and every holomorphic reduction \(E_{L} \subset E_{G}\) of its structure group to \(L,\) the principal \(L_{0}\)-bundle, obtained by extending the structure group of \(E_{G},\) is topologically trivial.
Mathematics Subject Classification ID: 32L05
Mathematics Subject Classification ID: 14H60
zbMATH DE Number: 1747293
zbMATH Keywords: compact connected Riemann surface
zbMATH Keywords: flat connection
zbMATH Keywords: principal bundles
reviewed by: V. V. Chueshev
MaRDI profile type: Publication
full work available at URL: s002080100273
OpenAlex ID: W1970080485
DOI: 10.1007/S002080100273
description: scientific article; zbMATH DE number 1747293