Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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


Authors: Jim Bryan, Tom Graber and Rahul Pandharipande
Journal: J. Algebraic Geom. 17 (2008), 1-28
DOI: https://doi.org/10.1090/S1056-3911-07-00467-5
Published electronically: July 9, 2007
MathSciNet review: 2357679
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Abstract | References | Additional Information

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 $Y$ 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
ORCID: 0000-0003-2541-5678
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
MR Author ID: 357813
Email: rahulp@math.princeton.edu

Received by editor(s): November 30, 2005
Published electronically: July 9, 2007