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 $\left\{{\mathrm{M̂}}^m\right(c){\}}_{m=0,1,2,\dots }$ connecting a moduli space $M\left(c\right)$ of stable torsion-free sheaves on a nonsingular complex projective surface $X$ and $\mathrm{M̂}\left(c\right)$ on its one-point blow-up $\mathrm{X̂}$. They are moduli spaces of perverse coherent sheaves on $\mathrm{X̂}$. In this paper we study how Donaldson-type invariants (integrals of cohomology classes given by universal sheaves) change from ${\mathrm{M̂}}^m\left(c\right)$ to ${\mathrm{M̂}}^{m+1}\left(c\right)$ and then from $M\left(c\right)$ to $\mathrm{M̂}\left(c\right)$. 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.