A blowup formula for virtual enumerative invariants on projective surfaces

Abstract

We apply virtual localization to the problem of finding blowup formulae for virtual sheaf-theoretic invariants on a smooth projective surface. This leads to a general procedure that can be used to express virtual enumerative invariants on the blowup in terms of those on the original surface. We use an enhanced master space construction over the moduli spaces of $m$-stable sheaves introduced by Nakajima and Yoshioka. Our work extends their analogous results for the equivariant moduli spaces of framed sheaves on $\mathbb{P}^2$. In contrast to their work, we make no use of GIT methods and work with an arbitrary smooth complex projective surface, assuming only the absence of strictly semistable sheaves. The main examples to keep in mind are Mochizuki's virtual analogue of the Donaldson invariant and the virtual $χ_y$-genus of the moduli space of Gieseker semistable sheaves on the surface.