# Vafa-Witten invariants for projective surfaces I: stable case

## TL;DR

This paper develops Vafa-Witten invariants for projective surfaces in the stable case, using virtual localisation and symmetric obstruction theory, connecting to modular forms and instanton moduli spaces.

## Contribution

It introduces a new framework for defining Vafa-Witten invariants on surfaces with stable Higgs pairs, incorporating rational contributions beyond the vanishing theorem case.

## Key findings

- Invariants are constant under deformations.
- Calculations recover modular form predictions.
- Results extend understanding of instanton moduli spaces.

## Abstract

On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a $\mathbb C^*$ action with compact fixed locus. Applying virtual localisation we define invariants constant under deformations. When the vanishing theorem of Vafa-Witten holds, the result is the (signed) Euler characteristic of the moduli space of instantons. In general there are other, rational, contributions. Calculations of these on surfaces with positive canonical bundle recover the first terms of modular forms predicted by Vafa and Witten.