Vafa-Witten invariants for projective surfaces II: semistable case
Abstract
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 KS0 we expect our definition coincides with an alternative definition using weighted Euler characteristics. We prove this for deg KS<0 here, and it is proved for S a K3 surface in MT. 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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