Vafa-Witten invariants for projective surfaces II: semistable case

<p>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.</p> <br/> <p>For KS ≤ 0 we expect our definition coincides with an alternative definition using weighted Euler characteristics. We prove this for degKS &lt; 0 here, and it is proved for S a K3 surface in [MT].</p> <br/> <p>For K3 surfaces we calculate the invariants in terms of modular forms which generalise and prove conjectures of Vafa and Witten.</p>