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.
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