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 $K_S\le 0$ 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.

Error found by Laarakker fixed. To appear in Pure Appl. Math. Quart. (2017), issue in honour of 60th birthday of Simon Donaldson. 37 pages

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• Cobordism invariants from BPS q-series
• Rank $r$ DT theory from rank $1$
• Degeneracy loci, virtual cycles and nested Hilbert schemes II
• Rank $r$ DT theory from rank $0$
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• On the construction of moduli stack of projective Higgs bundles over surfaces
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• A Noether-Lefschetz Theorem for Spectral Varieties with Applications