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

Davesh Maulik (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Richard P. Thomas (Department of Mathematics, Imperial College London, United Kingdom)

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.

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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