C/Z2 and Weil Conjecture

We calculate Betti numbers of the framed moduli space of instantons on Ĉ/Z2, under the assumption that the corresponding torsion free sheaves E have vanishing properties(Hom(E, E(−l∞)) = Ext (E, E(−l∞)) = 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 A1-singularity and that on X̂ blow-uped at the singularity, by using Weil conjecture. We call this the O(−2) blow-up formula. Applying this to X = C2/Z2 case, we reproduce the formula given by instanton calculus.