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 {M̂m(c)}m=0,1,2,... connecting a moduli space M(c) of stable torsion-free sheaves on a uonsingular complex projective surface X and M̂(c) on its one-point blowup X̂. They are moduli spaces of perverse coherent sheaves on X.this paper we study how Donaldson-type invariants (integrals of cohomology classes given by universal sheaves) change from M̂m(c) to M̂m+1(c) and then from M(c) to M̂(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. © 2011 by Kyoto University.