Local holomorphic Euler characteristic and instanton decay

TL;DR

This paper investigates the local holomorphic Euler characteristic near surface singularities, demonstrating how certain line self-intersection numbers obstruct instanton decay, supported by a computational algorithm.

Contribution

It establishes non-existence results for sheaves with specific invariants and links these to obstructions in instanton decay, introducing a computational tool for Euler characteristic calculations.

Findings

Certain sheaves with prescribed invariants do not exist.

Line self-intersection number $ ext{ell}^2$ obstructs instanton decay.

A Macaulay 2 algorithm for computing $ ext{chi}$ is provided.

Abstract

We study the local holomorphic Euler characteristic $\chi(x,\mathcal{F})$ of sheaves near a surface singularity obtained from contracting a line $\ell$ inside a smooth surface $Z$. We prove non-existence of sheaves with certain prescribed numerical invariants. Non-existence of instantons on $Z$ with certain charges follows, and we conclude that $\ell^2$ poses an obstruction to instanton decay. A Macaulay 2 algorithm to compute $\chi$ is made available at http://www.maths.ed.ac.uk/~s0571100/Instanton/