# Vafa-Witten invariants for projective surfaces II: semistable case

## TL;DR

This paper defines Vafa-Witten invariants for semistable Higgs pairs on polarized surfaces, proves their equivalence to weighted Euler characteristics in certain cases, and computes them explicitly for K3 surfaces using modular forms.

## Contribution

It introduces a new definition of Vafa-Witten invariants for semistable cases and verifies their consistency with existing theories for specific surfaces.

## Key findings

- Proved the equivalence of definitions for surfaces with negative canonical bundle.
- Calculated invariants for K3 surfaces using modular forms.
- Confirmed conjectures of Vafa and Witten for K3 surfaces.

## 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.