The Crepant Resolution Conjecture for Type A Surface Singularities

Abstract

Let X be an orbifold with crepant resolution Y. The Crepant Resolution Conjectures of Ruan and Bryan-Graber assert, roughly speaking, that the quantum cohomology of X becomes isomorphic to the quantum cohomology of Y after analytic continuation in certain parameters followed by the specialization of some of these parameters to roots of unity. We prove these conjectures in the case where X is a surface singularity of type A. The key ingredient is mirror symmetry for toric orbifolds.

19 pages. v2: references updated; corrected our description of the work of Davesh Maulik. v3: please note that this preprint has been superseded by arXiv:math/0702234v3. The material here, with various typos corrected, appears as Appendix A there

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• Wall-Crossings in Toric Gromov-Witten Theory II: Local Examples
• Root Systems and the Quantum Cohomology of ADE resolutions
• Hurwitz-Hodge Integrals, the E6 and D4 root systems, and the Crepant Resolution Conjecture
• Crepant resolutions of weighted projective spaces and quantum deformations
• Gerby Localization, Z_3-Hodge Integrals and the GW Theory of C^3/Z_3
• The cohomological crepant resolution conjecture for P(1,3,4,4)
• Cubic Extremal Transition and Gromov-Witten Theory
• An example of crepant resolution conjecture in two steps