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
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.

References [Enhancements On Off] (What's this?)

  • 1. Jim Bryan and Tom Graber.
    The crepant resolution conjecture. arXiv: math. AG/ 0610129.
  • 2. Jim Bryan, Sheldon Katz, and Naichung Conan Leung.
    Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds.
    J. Algebraic Geom., 10(3):549-568, 2001.
    Preprint version: math.AG/9911056. MR 1832332 (2002j:14047)
  • 3. Weimin Chen and Yongbin Ruan.
    Orbifold Gromov-Witten theory.
    In Orbifolds in mathematics and physics (Madison, WI, 2001), volume 310 of Contemp. Math., pages 25-85. Amer. Math. Soc., Providence, RI, 2002. MR 1950941 (2004k:53145)
  • 4. Hélène Esnault and Eckart Viehweg.
    Logarithmic de Rham complexes and vanishing theorems.
    Invent. Math., 86(1):161-194, 1986. MR 853449 (87j:32088)
  • 5. C. Faber and R. Pandharipande.
    Logarithmic series and Hodge integrals in the tautological ring.
    Michigan Math. J., 48:215-252, 2000.
    With an appendix by Don Zagier, Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786488 (2002e:14041)
  • 6. Phillip Griffiths, editor.
    Topics in transcendental algebraic geometry, volume 106 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1984. MR 756842 (86b:14004)
  • 7. Sheldon Katz.
    Small resolutions of Gorenstein threefold singularities.
    In Algebraic geometry: Sundance 1988, pages 61-70. Amer. Math. Soc., Providence, RI, 1991. MR 1108632 (92f:14001)
  • 8. David Mumford.
    Towards an enumerative geometry of the moduli space of curves.
    In Arithmetic and geometry, Vol. II, pages 271-328. Birkhäuser Boston, Boston, Mass., 1983. MR 717614 (85j:14046)
  • 9. Cumrun Vafa.
    String vacua and orbifoldized LG models.
    Modern Phys. Lett. A, 4(12):1169-1185, 1989. MR 1016963 (91g:81128)
  • 10. Eric Zaslow.
    Topological orbifold models and quantum cohomology rings.
    Comm. Math. Phys., 156(2):301-331, 1993. MR 1233848 (94i:32045)

Additional Information

Jim Bryan
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z4 Canada

Tom Graber
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125

Rahul Pandharipande
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000

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

American Mathematical Society