Vafa-Witten invariants for projective surfaces II: semistable case

Understand

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}.