Orbifold Cohomology of ADE-singularities

In Chapter 1 we collect some basic definitions on orbifolds, morphisms of orbifolds and orbifold vector bundles. In Chapter 3 we first review the definition of orbifold cohomology ring for a complex orbifold, then we state the cohomological crepant resolution conjecture as given by Ruan in [52]. In Chapter 4 we define orbifolds with transversal ADE-singularities, see Definition 4.2.5. Then we give a description of the twisted sectors in general. Finally we specialize to orbifolds with transversal A_n-singularities and, under the technical assumption of trivial monodromy, we compute the orbifold cohomology ring. In Chapter 5 we study the crepant resolution. We first show that any variety with transversal ADE-singularities Y has a unique crepant resolution p : Z --> Y, Proposition 5. 2 .1. Then we restrict our attention to the case of transversal An-singularities and trivial monodromy and we give an explicit description of the cohomology ring of Z. Chapter 6 contains the computations of the Gromov-Witten invariants of Z in the A_n case. We also give a description of the quantum corrected cohomology ring of Z. In Chapter 7 we prove Ruan's conjecture in the Ai case and, in the A2 case with minor modifications.