Canonical metrics on stable vector bundles

The problem of constructing moduli space of vector bundles over a projective manifold has attracted many mathematicians for decades. In mid 60’s Mumford first constructed the moduli space of vector bundles over algebraic curves via his celebrated GIT machinery. Later, in early 80’s Atiyah and Bott found an infinite dimensional symplectic quotient description of this moduli space. Since then, we have learned quiet a lot from the work of Kirwan, Guillemin and Sternberg in 80’s that finite dimensional GIT quotient is equivalent to symplectic quotient. A question that is remaining is how Atiyah-Bott’s infinite dimensional symplectic quotient is approximated by Mumford’s finite dimensional GIT quotient. This is the question we are studying in this paper. To state our main result, let (X,OX (1)) be a projective manifold polarized by an ample line bundle OX(1) and E be an irreducible holomorphic vector bundle of rank r on X. Then by Kodaira embedding theorem, we know that for k sufficient large a basis {Sα} of dimH0(X, E(k) := E⊗OX(k)) will give rise to an embedding