Hermitian-Einstein connections and stable vector bundles over compact complex surfaces
type: Item
instance of: scholarly article
title: Hermitian-Einstein connections and stable vector bundles over compact complex surfaces
zbMATH Open document ID: 0617.32044
DOI: 10.1007/BF01450081
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164389
review text: A theorem of P. Gauduchon states that an arbitrary hermitian metric on a compact complex surface has a conformal rescaling such that the associated Kähler form is then \({\bar \partial}\partial\)-closed. Given such a form, the degree of a holomorphic line bundle can be defined in the usual way and with that, the notion of stability in the sense of Mumford and Takemoto for torsion-free sheaves. It is proved here that an indecomposable holomorphic vector bundle on the surface is stable iff it admits an irreducible Hermitian-Einstein connection, where ''stable'' and ''Hermitian-Einstein'' are both with respect to a given positive \({\bar \partial}\partial\)-closed (1,1)-form. This generalizes a result of Donaldson, who proved this theorem in the case of algebraic surfaces in \({\mathbb{P}}_ N\) equipped with a Kähler metric whose Kähler form is cohomologous to that of the Fubini-Study metric.
Mathematics Subject Classification ID: 32L10
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 32J15
zbMATH DE Number: 4000385
zbMATH Keywords: compact complex surface
zbMATH Keywords: Hermitian-Einstein connection
zbMATH Keywords: stable holomorphic vector bundle
author: Nicholas P. Buchdahl
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4000385