The Crepant Resolution Conjecture

Abstract

For orbifolds admitting a crepant resolution and satisfying a hard Lefschetz condition, we formulate a conjectural equivalence between the Gromov-Witten theories of the orbifold and the resolution. We prove the conjecture for the equivariant Gromov-Witten theories of the nth symmetric product of the complex plane and the Hilbert scheme of n points in the plane.

The relationship between our conjecture and Ruan's original conjecture is clarified. We have also added the Hard Lefschetz hypothesis for our orbifolds, a condition whose necessity was made clear by the very nice recent paper of Coates, Corti, Iritani, and Tseng (math.AG/0611550)