  In earlier papers arXiv:0802.3120, arXiv:0806.0463 of this series we
constructed a sequence of intermediate moduli spaces $\bM^m(c)$ connecting a
moduli space $M(c)$ of stable torsion free sheaves on a nonsingular complex
projective surface and $\bM(c)$ on its one point blow-up. They are moduli
spaces of perverse coherent sheaves on the blow-up.
  In this paper we study how Donaldson-type invariants (integrals of cohomology
classes given by universal sheaves) change from $\bM^m(c)$ to $\bM^{m+1}(c)$,
and then from $M(c)$ to $\bM(c)$.
  As an application we prove that Nekrasov-type partition functions satisfy
certain equations which determine invariants recursively in second Chern
classes. They are generalization of the blow-up equation for the original
Nekrasov deformed partition function for the pure N=2 SUSY gauge theory, found
and used to derive the Seiberg-Witten curves in arXiv:math/0306198.
