Journal of Algebraic Geometry

The orbifold quantum cohomology of $\mathbb{C}^{2}/\mathbb{Z}_3$ and Hurwitz-Hodge integrals

Author(s): Jim Bryan; Tom Graber; Rahul Pandharipande
Journal: J. Algebraic Geom. 17 (2008), 1-28.
Posted: July 9, 2007
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Abstract: Let $\mathbb{Z}_3$ act on $\mathbb{C}^2$ by non-trivial opposite characters. Let $\mathcal{X}=[\mathbb{C}^{2}/\mathbb{Z}_3]$ be the orbifold quotient, and let

be the unique crepant resolution. We show that the equivariant genus 0 Gromov-Witten potentials $F^{\mathcal{X}}$ and $F^{Y}$ are equal after a change of variables--verifying the Crepant Resolution Conjecture for the pair $(\mathcal{X},Y)$ . Our computations involve Hodge integrals on trigonal Hurwitz spaces, which are of independent interest. In a self-contained Appendix, we derive closed formulas for these Hurwitz-Hodge integrals.

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Jim Bryan
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z4 Canada
Email: jbryan@math.ubc.ca

Tom Graber
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Email: graber@caltech.edu

Rahul Pandharipande
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000
Email: rahulp@math.princeton.edu

PII: S 1056-3911(07)00467-5
Received by editor(s): November 30, 2005
Posted: July 9, 2007

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