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                <term xml:lang="en">Holomorphic bundle</term>
                <term xml:lang="en">stable bundle</term>
                <term xml:lang="en">bundle families</term>
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                <term xml:lang="en">Hermite-Einstein metric</term>
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              <p>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.</p>
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