Abstract:
Let [Y] be a Gorenstein orbifold such that its coarse moduli space Y has a crepant resolution \rho: Z \rightarrow Y. In general the orbifold cohomology ring H^*_{orb}([Y]) of [Y] and the cohomology ring of Z are not isomorphic. The cohomological crepant resolution conjecture (by
Y.Ruan) states that the difference between the two rings can be expressed in terms of Gromov-Witten invariants of Z of rational curves which are contracted by the resolution morphism \rho :Z\rightarrow Y.
We study this conjecture in the case where Y is a variety with transversal ADE-singularities and [Y] is the associated reduced orbifold. In the A_n-case we compute both the orbifold cohomology ring and the Gromov-Witten invariants of Z. Finally we verify the conjecture in the A_1-case. In the A_2 case, we prove a slightly modified version of the conjecture. In both cases we give an explicit isomorphism between the orbifold cohomology ring of [Y] and the quantum corrected cohomology ring of Z.
