In earlier papers of this series we constructed a sequence of intermediate moduli spaces connecting a moduli space of stable torsion-free sheaves on a nonsingular complex projective surface and on its one-point blow-up . They are moduli spaces of perverse coherent sheaves on . In this paper we study how Donaldson-type invariants (integrals of cohomology classes given by universal sheaves) change from to and then from to . 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 supersymmetric gauge theory, found and used to derive the Seiberg-Witten curves.
"Perverse coherent sheaves on blowup, III: Blow-up formula from wall-crossing." Kyoto J. Math. 51 (2) 263 - 335, Summer 2011. https://doi.org/10.1215/21562261-1214366