Families of holomorphic bundles

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 Kahler 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 (e. g., the Donaldson functional on the space of Hermitian metrics and its properties). We apply these results to study families of bundles over a Kahlerian manifold Y parametrized by a non-Kahlerian 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 [22-24].