On the principal bundles with parabolic structure
type: Item
instance of: scholarly article
title: On the principal bundles with parabolic structure
zbMATH Open document ID: 1062.14022
author: Indranil Biswas
published in: Journal of Mathematics of Kyoto University
publication date: 2004-12-29T00:00:00Z
review text: Let \(G\) be a semisimple complex algebraic group. For a parabolic principal \(G\)-bundle \(E_*\) on a smooth curve, the author proves the following analogue of Weil's theorem: \(E_*\) admits a flat connection if and only if every direct summand of the adjoint bundle \(E_*({\mathfrak g})\) has parabolic degree zero (\({\mathfrak g}\) being the Lie algebra of \(G\)). Given a polynomial \(P(x)\) with non-negative integral coefficients and a parabolic vector bundle \(F_*\), define \(P(F_*)\) to be the parabolic vector bundle obtained by putting \(F_*\) for \(x\) and replacing addition and multiplication by direct sum and tensor product respectively. Then \(F_*\) is called finite if there exist two such distinct polynomials \(P_1\) and \(P_2\) with \(P_1(F_*) \cong P_2(F_*)\). A parabolic principal \(G\)-bundle \(E_*\) on a projective manifold \(X\) is called finite if all the parabolic vector bundles associated to \(E_*\) via representations are finite. The author shows that \(E_*\) is finite if and only if it admits a flat connection with a finite monodromy group.
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14J60
zbMATH DE Number: 2124039
zbMATH Keywords: parabolic principal bundles
zbMATH Keywords: flat connections
zbMATH Keywords: manifolds
reviewed by: Usha N. Bhosle
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250283729
OpenAlex ID: W1554141132
DOI: 10.1215/KJM/1250283729
description: scientific article; zbMATH DE number 2124039