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# Pure and Applied Mathematics Quarterly

## Volume 14 (2018)

### Number 3-4

### Sheaf counting on local $\mathrm{K}3$ surfaces

Pages: 419 – 441

DOI: https://dx.doi.org/10.4310/PAMQ.2018.v14.n3.a1

#### Authors

#### Abstract

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.

The first-named author is supported by NSF grants DMS-1645082 and DMS-1564458. The second-named author acknowledges partial support from EPSRC grant EP/R013349/1.

Received 2 January 2018

Received revised 22 August 2019

Accepted 23 August 2019

Published 5 November 2019