Analytic tangent cones of admissible Hermitian-Yang-Mills connections

Abstract

In this paper we study the analytic tangent cones of admissible Hermitian-Yang-Mills connections near a homogeneous singularity of a reflexive sheaf, and relate it to the Harder-Narasimhan-Seshadri filtration. We also give an algebro-geometric characterization of the bubbling set. This strengthens our previous result in [3].

A note on singular Hermitian Yang-Mills connections

“…On a reflexive sheaf with a conical HYM connection, there is a convexity estimate (cf. Proposition 3.5 in [2])…”

“…An application of [9] shows that our HYM connection on E is asymptotic to the tangent cone at the origin with polynomial decay rate. Some additional insights can be gained by studying the growth rate of holomorphic sections as in [2]. On a reflexive sheaf with a conical HYM connection, there is a convexity estimate (cf.…”

“…A basic heuristic in [2] is that if a singular HYM connection is sufficiently close to being conical on a certain scale, then the convexity behaviour transfers to the HYM connection, so that log ∫ B(r) s 2 log r has some monotonicity property, and one can define the local growth degree…”

“…This has been extended to reflexive sheaves by Bando and Siu [1], and by now there is a developed theory comparing the compactified moduli spaces of HYM connections versus the stable vector bundles [7]. In a more local setting, recent works [9] [2] show that for a large class of reflexive sheaf singularities, the local tangent cones of the HYM connections can be read off algebro-geometrically from the Harder-Narasimhan filtration.…”

Algebraic Tangent Cones of Reflexive Sheaves

“…The goal of this paper is to study a complex-algebraic object that comes out of our study of singularities of Hermitian-Yang-Mills connections ( [2,3]). The discussion here will be purely complex-algebraic, and the connection with the previous results will be given by Conjecture 1.7.…”