Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Note on a Conjecture of Gopakumar-Vafa

  • 120 Accesses

  • 4 Citations

Abstract

We rephrase the Gopakumar-Vafa conjecture on genus zero Gromov-Witten invariants of Calabi-Yau threefolds in terms of the virtual degree of the moduli of pure dimension one stable sheaves and investigate the conjecture for K3 fibred local Calabi-Yau threefolds.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Behrend, K., Gromov-Witten invariants in algebraic geometry, Invent. Math., 127(3), 1997, 601-617.

  2. 2.

    Behrend, K. and Fantechi, B., The intrinsic normal cone, Invent. Math., 128(1), 1997, 45-88.

  3. 3.

    Bryan, J., Katz, S. and Leung, N. C., Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds, J. Algebraic Geom., 10(3), 2001, 549-568.

  4. 4.

    Bryan, J. and Leung, N. C., The enumerative geometry of K3 surfaces and modular forms, J. Amer. Math. Soc., 13(2), 2000, 371-410.

  5. 5.

    Bryan, J. and Pandharipande, R., BPS states of curves in Calabi-Yau 3-folds, Geom. Topol., 5, 2001, 287-318. math.AG/0009025

  6. 6.

    Gopakumar, R. and Vafa, C., M-Theory and Topological Strings-II. hep-th/9812127

  7. 7.

    Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, New York-Heidelberg, 1977.

  8. 8.

    Hosono, S., Saito, M.-H. and Takahashi, A., Relative Lefschetz Action and BPS State Counting, Internat. Math. Res. Notices 2001, No. 15, 783-816. math.AG/0105148

  9. 9.

    Karp, D., Liu, C.-C. M. and Marino, M., The local Gromov-Witten invariants of configurations of rational curves. math.AG/0506488

  10. 10.

    Katz, S., Genus zero Gopakumar-Vafa invariants of contractible curves. math.AG/0601193

  11. 11.

    Katz, S., Klemm, A. and Vafa, C., M-theory, topological strings and spinning black holes, Adv. Theor. Math. Phys., 3, 1999, 1445-1537.

  12. 12.

    Lee, J.-H. and Leung, N.-C., Yau-Zaslow formula on K3 surfaces for non-primitive classes. math.SG/0404537

  13. 13.

    Li, J., A note on enumerating rational curves in a K3 surface, Geometry and Nonlinear Partial Differential Equations (Hangzhou, 2001), 53-62; AMS/IP Stud. Adv. Math., 29, Amer. Math. Soc., Providence, RI, 2002.

  14. 14.

    Li, J. and Tian, G., Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc., 11(1), 1998, 119-174.

  15. 15.

    Maruyama, M., Moduli of stable sheaves. II, J. Math. Kyoto Univ., 18(3), 1978, 557-614.

  16. 16.

    Maulik, D., Nekrasov, N., Okounkov, A. and Pandharipande, R., Gromov-Witten theory and Donaldson-Thomas theory, I. math.AG/0312059; Gromov-Witten theory and Donaldson-Thomas theory, II. math.AG/0406092

  17. 17.

    Mukai, S., Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math., 77(1), 1984, 101-116.

  18. 18.

    Mumford, D., Fogarty, J. and Kirwan, F., Geometric Invariant Theory, 3rd edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 34, Springer-Verlag, Berlin, 1994.

  19. 19.

    Simpson, C., Moduli of representations of the fundamental group of a smooth projective variety I, Inst. Hautes Études Sci. Publ. Math., 79, 1994, 47-129.

  20. 20.

    Thomas, R. P., A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations, J. Diff. Geom., 54, 2000, 367-438.

  21. 21.

    Wu, B. S., The number of rational curves on K3 surfaces, preprint. math.AG/0602280

  22. 22.

    Yau, S.-T. and Zaslow, E., BPS states, string duality, and nodal curves on K3, Nuclear Phys. B, 471, 1996, 503-512.

Download references

Author information

Correspondence to Jun Li*.

Additional information

(Dedicated to the memory of Shiing-Shen Chern)

*Partially supported by NSF grants DMS-0200477 and DMS-0233550.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Li*, J., Wu, B. Note on a Conjecture of Gopakumar-Vafa. Chin. Ann. Math. Ser. B 27, 219–242 (2006). https://doi.org/10.1007/s11401-006-0038-2

Download citation

Keywords

  • Calabi-Yau threefold
  • Gromov-Witten invariants
  • Moduli of stable sheaves

2000 MR Subject Classification

  • 14D20
  • 14D21
  • 14J30
  • 14N35