  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 K\"ahler 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
K\"ahlerian manifold $Y$ parameterized by a non-K\"ahlerian 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.
