Orbifold Cohomology of ADE-singularities

We study Ruan’s cohomological crepant resolution conjecture [39] for orbifolds with transversal ADE singularities. In the An-case we compute both the orbifold cohomology ring H ∗ orb([Y ]) and the quantum corrected cohomology ring H(Z)(q1, ..., qn). The former is achieved in general, the later up to some additional, technical assumptions. We construct an explicit isomorphism between H orb([Y ]) and H ∗(Z)(−1) in the A1-case, verifying Ruan’s conjecture. In the Ancase, the family H(Z)(q1, ..., qn) is not defined for q1 = ... = qn = −1. This implies that the conjecture should be slightly modified. We propose a new conjecture in the An-case (Conj. 1.9). Finally, we prove Conj. 1.9 in the A2-case by constructing an explicit isomorphism. Introduction Orbifold cohomology was defined by Chen and Ruan [10] for almost complex orbifolds. This was extended to a non-commutative ring by Fantechi and Göttsche [17] in the case where the orbifold is a global quotient. Abramovich, Graber and Vistoli defined the orbifold cohomology in the algebraic case [1]. Let [Y ] be a complex Gorenstein orbifold such that the coarse moduli space Y admits a crepant resolution ρ : Z → Y . Then, under some technical assumptions on Z, Ruan’s cohomological crepant resolution conjecture [39] predicts the existence of an isomorphism between the orbifold cohomology ring H orb([Y ],C) and the so called quantum corrected cohomology ring of Z. The later is a deformation of the ring H(Z,C) obtained using certain Gromov-Witten invariants of rational curves in Z which are contracted under the resolution map ρ. Notice that if Z carries an holomorphic symplectic structure, then this conjecture also predicts the existence of an isomorphism between the orbifold cohomology ring of [Y ] and the cohomology ring of Z. An interesting testing case for the conjecture is the one of the Hilbert scheme HilbM of r points on a projective surface M . It is a crepant resolution of the symmetric product SymM via the Chow morphism. In this case the conjecture was proved by W.-P. Li and Z. Qin for r = 2 [26], for r general and M with numerically trivial canonical class by Fantechi and Göttsche [17] (using the explicit computation of the ring H∗(HilbM) given by Lehn and Sorger [24]), and independently by Uribe [43]. A different and self-contained proof of this result was given by Z. Qin and W. Wang [35]. In the same situation but with M quasi-projective with a holomorphic symplectic form, the conjecture was proved by W.-P. Li, Z. Qin and W. Wang [27]. In particular this result generalizes the case of the affine plane obtained by Lehn and Sorger [25] and Vasserot [44] independently. The general case where Y = V/G with V complex symplectic vector space and G ⊂ Sp(V ) finite subgroup was proved by Ginzburg and Kaledin [20]. Let us point out that in the previous cases (except [26]) the resolution Z carries a holomorphic symplectic structure, ∗The author was partially supported by SNF, No 200020-107464/1