Orbifold Cohomology of ADE-singularities

Abstract

We study Ruan's "cohomological crepant resolution conjecture" (see math.AG/0108195) for orbifolds with transversal ADE singularities. Let [Y] be such an orbifold, Y its coarse moduli space and Z the crepant resolution of Y. Following Ruan [math.AG/0108195], we have a deformation of the cohomology ring H^*(Z) using some Gromov-Witten invariants of Z. The resulting family of rings will be denoted by H^*(Z)(q_1,...,q_n), where q_1,...,q_n are complex parameters. In the A_n case we compute both the orbifold cohomology ring H^*_{orb}([Y]) and the family H^*(Z)(q_1,...,q_n). The former is achieved in general, the later up to an explicit conjecture on the Gromov-Witten invariants which is verified under additional, technical assumptions. We construct an explicit isomorphism between H^*_{orb}([Y]) and H^*(Z)(-1) in the A_1 case, verifying Ruan's conjecture. In the A_2 case, the family H^*(q_1,q_2) is not defined for q_1=q_2=-1, so the conjecture should be slightly modified. However we give an explicit isomorphism between H^*_{orb}([Y]) and H^*(Z)(q_1,q_2) with q_1=q_2 be a primitive third root of the unity, thus proving a slightly modified version of Ruan's conjecture. It is natural to conjecture that, in the A_n case, H^*_{orb}([Y]) is isomorphic to H^*(Z)(q_1,...,q_n) with q_1=...=q_n be a primitive (n+1)-th root of the unity.