Sheaf counting on local K3 surfaces

There are two natural ways to count stable pairs or Joyce-Song pairs on $X=\mathrm{K}3\times \mathbb{C}$; 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.