Parabolic vortex equations and instantons of infinite energy
type: Item
instance of: scholarly article
title: Parabolic vortex equations and instantons of infinite energy
zbMATH Open document ID: 0882.32013
DOI: 10.1016/S0393-0440(96)00020-4
author: Olivier Biquard
author: Oscar García-Prada
published in: Journal of Geometry and Physics
publication date: 1998-03-02T00:00:00Z
review text: For a compact Kähler manifold \(M\) there is a question about the existence of a Kähler-Einstein metric. Given a holomorphic vector bundle \(E\) on \(M\) an appropriate generalization of a Kähler-Einstein metric on \(M\) appears to be a Hermitian-Einstein metric on \(E\). This question has been studied by the famous works of Donaldson, Narasimhan-Seshadri, and Uhlenbeck-Yau, culminating in the Hitchin-Kobayashi correspondence that the existence of such a metric is equivalent to a notion of the stability of the bundle. This type of theorem can be generalized to bundles with a parabolic structure. On Riemann surfaces such a structure consists of a finite set of points and filtrations by vector spaces of the fibers at those points, together with some real numbers called weights. The desired metric correspondingly is the one adapted to the parabolic structure; in particular, its connection is singular along the parabolic points. In a related direction given two parabolic vector bundles on a compact Riemann surface \(X\) and a morphism or a meromorphic map \(f\) between them there is defined a notion of stability for this triple. The resulting metrics on two bundles satisfy a system of equations called parabolic vortex equations which also involves the section and a real parameter. The present paper studies the Hitchin-Kobayashi correspondence for solutions of parabolic vortex equations and the stable parabolic triples. In the case where there is no parabolic structures this question was studied in [\textit{S. B. Bradlow} and \textit{O. García-Prada}, Math. Ann. 304, No. 2, 225-252 (1996; Zbl 0852.32016) and \textit{O. García-Prada}, Int. J. Math. 5, No. 1, 1-52 (1994; Zbl 0799.32022)]. The present proof is based on a modification of the dimensional reduction arguments used in the ordinary case, which in turns is based on the relation between the usual Hitchin-Kobayashi correspondence on \(X\times \mathbb{P}^1\) and that for coupled vortex equations on \(X\). This step o…
reviewed by: I-Hsun Tsai
Mathematics Subject Classification ID: 32L05
Mathematics Subject Classification ID: 53C07
zbMATH DE Number: 997711
zbMATH Keywords: vortex equation
zbMATH Keywords: parabolic bundle
zbMATH Keywords: Hermitian-Einstein metric
zbMATH Keywords: Hitchin-Kobayashi correspondence
MaRDI profile type: Publication
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full work available at URL: s0393-0440(96)00020-4
OpenAlex ID: W2015914493
description: scientific article; zbMATH DE number 997711