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

摘要

In earlier papers of this series we constructed a sequence of intermediate mod-uli spaces {M~m (c)}_m=0,1,2,... connecting a moduli space M(c) of stable torsion-free sheaves on a nonsingular complex projective surface X and M(c) on its one-point blow-up X. They are moduli spaces of perverse coherent sheaves on X. In this paper we study how Donaldson-type invariants (integrals of cohomology classes given by universal sheaves) change from M~m (c) to M~m+1(c) and then from M(c) to M(c). As an appli-cation 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.