Algebraic tangent cones of reflexive sheaves

TL;DR

This paper investigates algebraic tangent cones at singularities of reflexive sheaves, establishing existence and uniqueness of optimal extensions, and connects these results to Hermitian-Yang-Mills connection singularities.

Contribution

It introduces a constructive method for optimal extensions of reflexive sheaves across negative divisors and proves their uniqueness, advancing the understanding of singularities in algebraic geometry.

Findings

Existence of optimal extensions of reflexive sheaves established

Uniqueness of these extensions proved in a suitable sense

Connections to singularities of Hermitian-Yang-Mills connections demonstrated

Abstract

We study the notion of algebraic tangent cones at singularities of reflexive sheaves. These correspond to extensions of reflexive sheaves across a negative divisor. We show the existence of optimal extensions in a constructive manner, and we prove the uniqueness in a suitable sense. The results here are an algebro-geometric counterpart of our previous study on singularities of Hermitian-Yang-Mills connections.