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arXiv · 1702.08488

Vafa-Witten invariants for projective surfaces II: semistable case

Yuuji Tanaka·Richard P. Thomas
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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 $K_S\le0$ we expect our definition coincides with an alternative definition using weighted Euler characteristics. We prove this for deg $K_S<0$ here, and it is proved for $S$ a K3 surface in \cite{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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Yuuji Tanaka, Richard P. Thomas. 2018-11-02. Vafa-Witten invariants for projective surfaces II: semistable case. https://doi.org/10.4310/pamq.2017.v13.n3.a6

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