Pure and Applied Mathematics Quarterly

Volume 13 (2017)

Number 3

Special Issue in Honor of Simon Donaldson

Guest Editors: Kefeng Liu, Richard Thomas, and Shing-Tung Yau

Vafa–Witten invariants for projective surfaces II: semistable case

Pages: 517 – 562

DOI: https://dx.doi.org/10.4310/PAMQ.2017.v13.n3.a6


Yuuji Tanaka (Mathematical Institute, University of Oxford, United Kingdom)

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


We propose a definition of Vafa–Witten invariants counting semistable Higgs pairs on a polarised surface. We use virtual localisation applied to Mochizuki/Joyce–Song pairs.

For $K_S \leq 0$ we expect our definition coincides with an alternative definition using weighted Euler characteristics. We prove this for $\deg K_S \lt 0$ here, and it is proved for $S$ a K3 surface in “Sheaf counting on local K3 surfaces” [D. Maulik and R. P. Thomas, arXiv:1806.02657].

For K3 surfaces we calculate the invariants in terms of modular forms which generalise and prove conjectures of Vafa and Witten.

Dedicated to Simon Donaldson, with admiration and thanks.

Y.T. was partially supported by JSPS Grant-in-Aid for Scientific Research numbers JP15H02054 and JP16K05125, and a Simons Collaboration Grant on “Special holonomy in Geometry, Analysis and Physics”.

Received 20 November 2017

Published 12 November 2018