Vafa–Witten invariants for projective surfaces II: semistable case

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.

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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