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

TL;DR

This paper explores the relationship between tangent cones of Hermitian-Yang-Mills connections at singularities and the algebraic geometry of reflexive sheaves, proposing a conjecture and proving it under specific conditions.

Contribution

It introduces a conjecture linking tangent cones to the double dual of graded sheaves from Harder-Narasimhan-Seshadri filtrations and proves it in the case of locally free, stable tangent cones.

Findings

Conjecture relating tangent cones to graded sheaves is proposed.

Proof of the conjecture when the tangent cone is locally free and stable.

Establishes a connection between differential geometry and algebraic geometry in this context.

Abstract

This is the first of a series of papers where we relate tangent cones of Hermitian-Yang-Mills connections at an isolated singularity to the complex algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally modelled on the pull-back of a holomorphic vector bundle from the projective space. In this paper 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.