2007/04/19 by Andrei Teleman, Teleman, Andrei
Mathematics · #32G13 #53C07 #53C55 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #math.AG #math.CV #math.DG #msc:32G13 #msc:53C07 #msc:53C55
paper · pdf · doi:10.48550/arxiv.0704.2629
LaTeX, 26 pages
arxiv created 2007/05/16 · arxiv updated 2009/12/01
The first goal of the article is to solve several fundamental problems in the theory of holomorphic bundles over non-algebraic manifolds: For instance we prove that stability and semi-stability are Zariski open properties in families when the Gauduchon degree map is a topological invariant, or when the parameter manifold is compact. Second we show that, for a generically stable family of bundles over a Kähler manifold, the Petersson-Weil form extends as a closed positive current on the whole parameter space of the family. This extension theorem uses classical tools from Yang-Mills theory developed by Donaldson (e.g. the Donaldson functional and the heat equation for Hermitian metrics on a holomorphic bundle). We apply these results to study families of bundles over a Kählerian manifold Y parameterized by a non-Kählerian surface X, proving that such families must satisfy very restrictive conditions. These results play an important role in our program to prove existence of curves on class VII surfaces.