$O(-2)$ Blow-up Formula via Instanton Calculus on $\widehat{C^2/Z_2}$ and Weil Conjecture

Abstract

We calculate Betti numbers of the framed moduli space of instantons on $\widehat{{\mathbf{C}}^2/{\mathbf{Z}}_2}$, under the assumption that the corresponding torsion free sheaves $E$ have vanishing properties ($Hom(E,E(-l_{\infty }))=Ex{t}^2(E,E(-l_{\infty }))=0$). Moreover we derive the generating function of Betti numbers and obtain closed formulas. On the other hand, we derive a universal relation between the generating function of Betti numbers of the moduli spaces of stable sheaves on $X$ with an $A_1$-singularity and that on $\hat{X}$ blow-uped at the singularity, by using Weil conjecture. We call this the $O(-2)$ blow-up formula. Applying this to $X={\mathbf{C}}^2/{\mathbf{Z}}_2$ case, we reproduce the formula given by instanton calculus.

31 pages, reference and appendix added

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