Einstein type metrics and stability on vector bundles (Q1383655)

The author proves that the Gieseker stability of a holomorphic vector bundle \(E\) over a compact Kähler manifold is equivalent to the existence of an almost Hermitian-Einstein metric on \(E\). The curvature of such a metric satisfies a certain system of Monge-Ampère equations parametrized by a parameter \(k\). He solves this fully nonlinear elliptic system by a singular perturbation technique and shows that the vanishing of obstructions for the perturbation is given by the stability condition. this can be interpreted as an infinite-dimensional analog of the equivalence between geometric invariant theory and symplectic reduction for moduli spaces of vector bundles.