A criterion for flatness of sections of adjoint bundle of a holomorphic principal bundle over a Riemann surface
type: Item
instance of: scholarly article
title: A criterion for flatness of sections of adjoint bundle of a holomorphic principal bundle over a Riemann surface
zbMATH Open document ID: 1271.14043
author: Indranil Biswas
published in: Documenta Mathematica
publication date: 2013-04-17T00:00:00Z
full work available at URL: 05.html
review text: Let \(X\) be a compact connected Riemann surface and \(G\) a connected reductive affine group defined over \(\mathbb C\). Let \(E_G\) be a holomorphic principal \(G\)-bundle over \(X\). Take \(\text{ad}(E_G):=E_G \times ^G \mathfrak g\), the adjoint vector bundle, where \(\mathfrak g\) is the Lie algebra of \(G\). In this paper one gives sufficient conditions for a section \(\beta\) in \(H^0(X, \text{ad}(E_G))\), which has the property that all \(\beta (x)\) (\(x \in X\)) have the same image in \(\mathfrak{g}/G: =\) \{ {\text{ the set of all conjugacy classes in}} \(\mathfrak g\) \}, to have the property that there exists a holomorphic connection on \(E_G\) such that \(\beta\) is flat with respect to the connection induced on \(\text{ad}(E_G)\). The condition is given, among others, in terms of a Levi subgroup of a parabolic subgroup of \(G\), defined naturally as the centralizer of an element in \(\mathfrak g\) whose image in \(\mathfrak{g}/G\) coincide with the image of \(\beta _s(x)\), for \(\beta _s\) a semisimple section. An interesting form for \(G=\mathrm{GL}(r,\mathbb C)\) is given.
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 53C07
zbMATH DE Number: 6155129
zbMATH Keywords: holomorphic connection
zbMATH Keywords: adjoint bundle
zbMATH Keywords: flatness
zbMATH Keywords: canonical connection
reviewed by: Nicolae Manolache
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6155129