The blow-up formula for the instanton part of Vafa–Witten invariants on projective surfaces

We prove a blow-up formula for the generating series of virtual χ y -genera for moduli spaces of sheaves on projective surfaces, which is related to a conjectured formula of Göttsche for topological χ y -genera.Our formula is a refinement of one by Vafa-Witten relating to S-duality.We prove the formula simultaneously in the setting of Gieseker stable sheaves on polarised surfaces and also in the setting of framed sheaves on P 2 .The proof is based on the blow-up algorithm of Nakajima-Yoshioka for framed sheaves on P 2 , which has recently been extended to the setting of Gieseker H-stable sheaves on H-polarised surfaces by Kuhn-Tanaka.