Vortex type equations and canonical metrics

We introduce a notion of Gieseker stability for a filtered holomorphic vector bundle over a projective manifold. We relate it to an analytic condition in terms of hermitian metrics on coming from a construction of the Geometric Invariant Theory (G.I.T). We prove that if there is a τ-Hermite-Einstein metric h _HE on , then there exists a sequence of such balanced metrics that converges and its limit is h _HE . As a corollary, we obtain an approximation theorem for quiver Vortex equations and other classical equations.