Sheaf counting on local K3 surfaces

Abstract

© 2018, International Press of Boston, Inc.. All rights reserved. There are two natural ways to count stable pairs or Joyce-Song pairs on X =K3× 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.