Einstein metrics and stability for flat connections on compact Hermitian manifolds, and a correspondence with Higgs operators in the surface case
type: Item
instance of: scholarly article
title: Einstein metrics and stability for flat connections on compact Hermitian manifolds, and a correspondence with Higgs operators in the surface case
zbMATH Open document ID: 0977.53022
published in: Documenta Mathematica
publication date: 1999-10-20T00:00:00Z
full work available at URL: https://eudml.org/doc/227993
full work available at URL: 15.html
review text: Summary: A flat complex vector bundle \((E,D)\) on a compact Riemannian manifold \((X,g)\) is stable (resp. polystable) in the sense of \textit{K. Corlette} [J. Differ. Geom. 28, 361-382 (1988; Zbl 0676.58007)] if it has no \(D\)-invariant subbundle (resp. if it is the \(D\)-invariant direct sum of stable subbundles). It has been shown in [loc. cit.] that the polystability of \((E,D)\) in this sense is equivalent to the existence of a so-called harmonic metric in \(E\). In this paper we consider flat complex vector bundles on compact Hermitian manifolds \((X,g)\). We propose new notions of \(g\)-(poly-)stability of such bundles, and of \(g\)-Einstein metrics in them; these notions coincide with (poly-)stability and harmonicity in the sense of Corlette if \(g\) is a Kähler metric, but are different in general. Our main result is that the \(g\)-polystability in our sense is equivalent to the existence of a \(g\)-Hermitian-Einstein metric. Our notion of a \(g\)-Einstein metric in a flat bundle is motivated by a correspondence between flat bundles and Higgs bundles over compact surfaces, analogous to the correspondence in the case of Kähler manifolds (C. T. Simpson (1988), (1992)).
Mathematics Subject Classification ID: 53C07
Mathematics Subject Classification ID: 32Q20
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 1350255
zbMATH Keywords: flat complex vector bundles
zbMATH Keywords: compact Hermitian manifolds
zbMATH Keywords: stability
zbMATH Keywords: Einstein metrics
author: M. Lübke
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1350255