Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

The enumerative geometry of $K3$ surfaces and modular forms


Authors: Jim Bryan and Naichung Conan Leung
Journal: J. Amer. Math. Soc. 13 (2000), 371-410
MSC (2000): Primary 14N35, 53D45, 14J28
DOI: https://doi.org/10.1090/S0894-0347-00-00326-X
Published electronically: January 31, 2000
MathSciNet review: 1750955
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $X$ be a $K3$ surface, and let $C$ be a holomorphic curve in $X$ representing a primitive homology class. We count the number of curves of geometric genus $g$ with $n$ nodes passing through $g$ generic points in $X$ in the linear system $\left\vert C\right\vert $ for any $g$ and $n$ satisfying $C\cdot C=2g+2n-2$.

When $g=0$, this coincides with the enumerative problem studied by Yau and Zaslow who obtained a conjectural generating function for the numbers. Recently, Göttsche has generalized their conjecture to arbitrary $g$ in terms of quasi-modular forms. We prove these formulas using Gromov-Witten invariants for families, a degeneration argument, and an obstruction bundle computation. Our methods also apply to $\mathbf{P}^{2}$ blown up at 9 points where we show that the ordinary Gromov-Witten invariants of genus $g$ constrained to $g$ points are also given in terms of quasi-modular forms.


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

  • 1. W. Barth, C. Peters, and A. Van de Ven.
    Compact Complex Surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete.
    Springer-Verlag, 1984. MR 86c:32026
  • 2. Victor Batyrev.
    On the Betti numbers of birationally isomorphic projective varieties with trivial canonical bundles.
    Preprint, alg-geom//9710020.
  • 3. A. Beauville.
    Counting rational curves on $K3$ surfaces.
    Preprint alg-geom/9701019, 1997.
  • 4. K. Behrend.
    Personal communication, 1998.
  • 5. K. Behrend.
    Gromov-Witten invariants in algebraic geometry.
    Invent. Math., 127(3):601-617, 1997. MR 98i:14015
  • 6. K. Behrend and B. Fantechi.
    The intrinsic normal cone.
    Invent. Math., 128(1):45-88, 1997. MR 98e:14022
  • 7. K. Behrend and Yu. Manin.
    Stacks of stable maps and Gromov-Witten invariants.
    Duke Math. J., 85(1):1-60, 1996. MR 98i:14014
  • 8. M. Bershadsky, C. Vafa, and V. Sadov.
    $D-$branes and topological field theories.
    Nuclear Phys. B, 463(2-3):420-434, 1996. MR 97h:81213
  • 9. A. Besse.
    Einstein Manifolds, volume 10 of Ergebnisse der Mathematik und ihrer Grenzgebiete.
    Springer-Verlag, 1987. MR 88f:53087
  • 10. Xi Chen.
    Counting curves on $K3$.
    Ph.D. thesis, Harvard, 1997.
  • 11. Xi Chen.
    Singularities of rational curves on $K3$ surfaces.
    Preprint, math.AG/9812050, 1998.
  • 12. Xi Chen.
    Personal communication, 1999.
  • 13. David A. Cox and Sheldon Katz.
    Mirror Symmetry and Algebraic Geometry.
    American Mathematical Society, Providence, RI, 1999. CMP 99:09
  • 14. S. K. Donaldson.
    Yang-Mills invariants of 4-manifolds.
    In S. K. Donaldson and C. B. Thomas, editors, Geometry of Low-Dimensional Manifolds: Gauge Theory and Algebraic Surfaces, number 150 in London Mathematical Society Lecture Note Series. Cambridge University Press, 1989. MR 93f:57040
  • 15. B. Fantechi, L. Göttsche, and D. van Straten.
    Euler number of the compactified Jacobian and multiplicity of rational curves.
    J. Algebraic Geom., 8(1):115-133, 1999. MR 99i:14065
  • 16. R. Friedman and J. Morgan.
    Smooth Four-manifolds and Complex Surfaces, volume 27 of Ergebnisse der Mathematik und ihrer Grenzgebiete.
    Springer-Verlag, 1994. MR 95m:57046
  • 17. Alexander Givental.
    Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture.
    (alg-geom/9612001).
  • 18. Lothar Göttsche.
    A conjectural generating function for numbers of curves on surfaces.
    Comm. Math. Phys., 196(3):523-533, 1998. CMP 99:01
  • 19. L. Göttsche.
    The Betti numbers of the Hilbert scheme of points on a smooth projective surface.
    Math. Ann., pages 193-207, 1990. MR 91h:14007
  • 20. L. Göttsche and R. Pandharipande.
    The quantum cohomology of blow-ups of ${\mathbf {P}}^{2}$ and enumerative geometry.
    J. Differential Geom., 48(1):61-90, 1998. MR 99d:14057
  • 21. T. Graber and R. Pandharipande.
    Localization of virtual classes.
    Invent. Math., 135(2):487-518, 1999. CMP 99:07
  • 22. Robin Hartshorne.
    Residues and duality.
    Springer-Verlag, Berlin, 1966.
    Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20. MR 36:5145
  • 23. Daniel Huybrechts.
    Compact hyper-Kähler manifolds: basic results.
    Invent. Math., 135(1):63-113, 1999. CMP 99:06
  • 24. Andrew Kresch.
    Cycle groups for Artin stacks.
    math.AG/9810166.
  • 25. P. B. Kronheimer.
    Some non-trivial families of symplectic structures.
    Preprint, 1997.
  • 26. Jun Li and Gang Tian.
    Comparison of the algebraic and the symplectic Gromov-Witten invariants.
    (alg-geom/9712035).
  • 27. Jun Li and Gang Tian.
    Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties.
    J. Amer. Math. Soc., 11(1):119-174, 1998. MR 99d:14011
  • 28. Jun Li and Gang Tian.
    Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds.
    In Topics in symplectic $4$-manifolds (Irvine, CA, 1996), pages 47-83. Internat. Press, Cambridge, MA, 1998. CMP 98:17
  • 29. T.J. Li and A. Liu.
    Family Seiberg-Witten invariant.
    Preprint., 1997.
  • 30. Bong H. Lian, Kefeng Liu, and Shing-Tung Yau.
    Mirror principle. I.
    Asian J. Math., 1(4):729-763, 1997. MR 99e:14062
  • 31. Y. Ruan and G. Tian.
    A mathematical theory of quantum cohomology.
    J. Differential Geometry, 42(2), 1995. MR 96m:58033
  • 32. Yongbin Ruan.
    Virtual neighborhoods and pseudo-holomorphic curves.
    Preprint alg-geom/9611021., 1996.
  • 33. Bernd Siebert.
    Gromov-Witten invariants of general symplectic manifolds.
    Preprint math.DG/9608105., 1996.
  • 34. Bernd Siebert.
    Algebraic and symplectic Gromov-Witten invariants coincide.
    Preprint math.AG/9804108., 1998.
  • 35. Charles A. Weibel.
    An introduction to homological algebra.
    Cambridge University Press, Cambridge, 1994. MR 95f:18001
  • 36. S.-T. Yau.
    On the Ricci curvature of a compact Kähler manifold and the complex Monge Ampère equation I.
    Comm. Pure and Appl. Math, 31:339-411, 1978. MR 81d:53045
  • 37. S.-T. Yau and E. Zaslow.
    BPS states, string duality, and nodal curves on $K3$.
    Nuclear Physics B, 471(3):503-512, 1996.
    Also: hep-th/9512121. MR 97e:14066

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 14N35, 53D45, 14J28

Retrieve articles in all journals with MSC (2000): 14N35, 53D45, 14J28


Additional Information

Jim Bryan
Affiliation: Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, Louisiana 70118
Email: jbryan@math.tulane.edu

Naichung Conan Leung
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: leung@math.umn.edu

DOI: https://doi.org/10.1090/S0894-0347-00-00326-X
Received by editor(s): January 5, 1998
Received by editor(s) in revised form: October 18, 1999
Published electronically: January 31, 2000
Additional Notes: The first author is supported by a Sloan Foundation Fellowship and NSF grant DMS-9802612 and the second author is supported by NSF grant DMS-9626689.
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society