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arXiv · 0911.1773

Perverse coherent sheaves on blow-up. III. Blow-up formula from wall-crossing

Hiraku Nakajima·Kota Yoshioka
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Abstract

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.

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Hiraku Nakajima, Kota Yoshioka. 2009-11-09. Perverse coherent sheaves on blow-up. III. Blow-up formula from wall-crossing. https://doi.org/10.1215/21562261-1214366

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