Singular Hermitian-Yang-Mills connections and reflexive sheaves

The Donaldson-Uhlenbeck-Yau theorem relates the existence of Hermitian-Yang-Mills connections over a compact Kahler manifold with algebraic stability of a holomorphic vector bundle. This has been extended by Bando-Siu in 1994 to a class of singular Hermitian-Yang-Mills connections on reflexive sheaves. We study tangent cones of these singular connections in the geometric analytic sense, and show that they can be characterized in terms of certain algebro-geometric invariants of reflexive sheaves. In a sense, this can be viewed as a local” version of the Donaldson-Uhlenbeck-Yau correspondence. Based on joint work with Xuemiao Chen (University of Maryland).