Singularities of Hermitian–Yang–Mills connections and Harder–Narasimhan–Seshadri filtrations

Abstract

This is the first of a series of papers where we relate tangent cones of Hermitian–Yang–Mills connections at a singularity to the complex algebraic geometry of the underlying reflexive sheaf. In this paper we work on the case when the sheaf is locally modeled on the pullback of a holomorphic vector bundle from the projective space, and we shall impose an extra assumption that the graded sheaf determined by the Harder–Narasimhan–Seshadri filtrations of the vector bundle is reflexive. In general, we conjecture that the tangent cone is uniquely determined by the double dual of the associated graded object of a Harder–Narasimhan–Seshadri filtration of an algebraic tangent cone, which is a certain torsion-free sheaf on the projective space. In this paper we also prove this conjecture when there is an algebraic tangent cone which is locally free and stable.