On the existence of unitary flat connections over the punctured sphere with given local monodromy around the punctures
type: Item
instance of: scholarly article
title: On the existence of unitary flat connections over the punctured sphere with given local monodromy around the punctures
zbMATH Open document ID: 0982.14022
author: Indranil Biswas
published in: The Asian Journal of Mathematics
publication date: 2001-06-28T00:00:00Z
review text: A parabolic structure at a point \(x\) on a vector bundle \(E\) on a smooth curve \(X\) consists of a flag on the fiber \(E_x\) together with real numbers called weights attached to it. A parabolic vector bundle is a vector bundle with such a parabolic data at finitely many points of \(X\). Let \(S\) be the set of parabolic points. In case \(X = \mathbb{P}^1\), the author gives sufficient and necessary conditions for the existence of (semi)stable parabolic bundles with given parabolic data. These conditions are inequalities involving parabolic data. This generalises the author's results in the rank two case [\textit{I. Biswas}, Int. J. Math. 9, 523-533 (1998; Zbl 0939.14015)]. There is a one to one correspondence between the parabolic bundles and representations of the fundamental group \(\pi ({\mathbb{P}}^1 - S)\) with the parabolic data at a point \(s\in S\) corresponding to the conjugacy class of the local monodromy at \(s\). Then the above conditions can be interpreted as sufficient and necessary conditions for realising conjugacy classes \(C_s, s\in S\), as the local monodromies of a unitary local system, the conditions being in terms of eigenvalues of \(C_s\) and their multiplicities.
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 1563747
zbMATH Keywords: parabolic bundles
zbMATH Keywords: unitary flat connections
zbMATH Keywords: local monodromy
zbMATH Keywords: representations of the fundamental group
reviewed by: Usha N. Bhosle
MaRDI profile type: Publication
DOI: 10.4310/AJM.1999.V3.N2.A1
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description: scientific article; zbMATH DE number 1563747