Title: 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 $K_S\le0$ 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.