Perverse coherent sheaves on blowup, III: Blow-up formula from wall-crossing

Abstract

In earlier papers of this series we constructed a sequence of intermediate moduli spaces $\{{\widehat{M}}^{m}(c)\}_{m=0,1,2,\ldots}$ connecting a moduli space $M(c)$ of stable torsion-free sheaves on a nonsingular complex projective surface $X$ and ${\widehat{M}}(c)$ on its one-point blow-up $\widehat {X}$. They are moduli spaces of perverse coherent sheaves on $\widehat{X}$. In this paper we study how Donaldson-type invariants (integrals of cohomology classes given by universal sheaves) change from ${\widehat{M}}^{m}(c)$ to ${\widehat{M}}^{m+1}(c)$ and then from $M(c)$ to ${\widehat{M}}(c)$. As an application we prove that Nekrasov-type partition functions satisfy certain equations that determine invariants recursively in second Chern classes. They are generalizations of the blow-up equation for the original Nekrasov deformed partition function for the pure $N=2$ supersymmetric gauge theory, found and used to derive the Seiberg-Witten curves.