Sheaf counting on local K3 surfaces

TL;DR

This paper explores two methods of counting stable pairs on noncompact local K3 surfaces, establishing a relationship between their generating series and proving several conjectures related to Vafa-Witten invariants.

Contribution

It demonstrates that the two counting methods' generating series are related exponentially and proves conjectures of Toda and Tanaka-Thomas in this context.

Findings

Generated series are related by an exponential formula.

Confirmed conjectures of Toda regarding stable pairs.

Validated Tanaka-Thomas conjecture on Vafa-Witten invariants.

Abstract

There are two natural ways to count stable pairs or Joyce-Song pairs on $X=\mathrm{K3}\times\mathbb C$; one via weighted Euler characteristic and the other by virtual localisation of the reduced virtual class. Since $X$ is noncompact these need not be the same. We show their generating series are related by an exponential. As applications we prove two conjectures of Toda, and a conjecture of Tanaka-Thomas defining Vafa-Witten invariants in the semistable case.