EXISTENCE OF APPROXIMATE HERMITIAN-EINSTEIN STRUCTURES ON SEMI-STABLE BUNDLES

The purpose of this paper is to investigate canonical metrics on a semi-stable vector bundle E over a compact Kahler manifold X. It is shown that if E is semi-stable, then Donaldson's functional is bounded from below. This implies that E admits an approximate Hermitian-Einstein structure, generalizing a classic result of Kobayashi for projective manifolds to the Kahler case. As an application some basic properties of semi-stable vector bundles over compact Kahler manifolds are established, such as the fact that semi-stability is preserved under certain exterior and symmetric products.