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Special metrics and stability for holomorphic bundles with global sections. (English) Zbl 0697.32014
We describe canonical metrics on holomorphic bundles in which there are global holomorphic sections. Such metrics are defined by 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.
Reviewer: S.B.Bradlow

32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
53C05 Connections, general theory
53B35 Local differential geometry of Hermitian and Kählerian structures
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
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