Special metrics and stability for holomorphic bundles with global sections

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In this paper we describe canonical metrics on holomorphic bundles in which there are global holomorphic sections. Such metrics are defined by a constraint on the curvature of the corresponding metric connection. The constraint is in the form of a P.D.E which looks like the Hermitian-Yang-Mills equation with an extra zeroth order term. We identify the necessary and sufficient condition for the existence of solutions to this equation. This condition is given in terms of the slopes of subsheaves of the bundle and defines a property similar to stability. We show that if a holomorphic bundle meets this stability-like criterion, then its Chern classes are constrained by an inequality similar to the Bogomolov-Gieseker inequality for stable bundles.

Original languageEnglish (US)
Pages (from-to)169-213
Number of pages45
JournalJournal of Differential Geometry
Issue number1
StatePublished - Jan 1991
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

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