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Noether-Lefschetz theory and the Yau-Zaslow conjecture

Authors: A. Klemm, D. Maulik, R. Pandharipande and E. Scheidegger
Journal: J. Amer. Math. Soc. 23 (2010), 1013-1040
MSC (2010): Primary 14N35
Published electronically: June 9, 2010
MathSciNet review: 2669707
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Abstract: The Yau-Zaslow conjecture predicts the genus 0 curve counts of $ K3$ surfaces in terms of the Dedekind $ \eta$ function. The classical intersection theory of curves in the moduli of $ K3$ surfaces with Noether-Lefschetz divisors is related to 3-fold Gromov-Witten invariants via the $ K3$ curve counts. Results by Borcherds and Kudla-Millson determine these classical intersections in terms of vector-valued modular forms. Proven mirror transformations can often be used to calculate the 3-fold invariants which arise.

Via a detailed study of the STU model (determining special curves in the moduli of $ K3$ surfaces), we prove the Yau-Zaslow conjecture for all curve classes on $ K3$ surfaces. Two modular form identities are required. The first, the Klemm-Lerche-Mayr identity relating hypergeometric series to modular forms after mirror transformation, is proven here. The second, the Harvey-Moore identity, is proven by D. Zagier and presented in the paper.

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  • 1. K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), 601-617. MR 1431140 (98i:14015)
  • 2. K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), 45-88. MR 1437495 (98e:14022)
  • 3. A. Beauville, Counting rational curves on $ K3$ surfaces, Duke Math. J. 97 (1999), 99-108. MR 1682284 (2000c:14073)
  • 4. R. Borcherds, The Gross-Kohnen-Zagier theorem in higher dimensions, Duke J. Math. 97 (1999), 219-233. MR 1682249 (2000f:11052)
  • 5. J. Bruinier, On the rank of Picard groups of modular varieties attached to orthogonal groups, Compositio Math. 133, (2002), 49-63. MR 1918289 (2003h:11053)
  • 6. J. Bryan and C. Leung, The enumerative geometry of $ K3$ surfaces and modular forms, J. AMS 13 (2000), 371-410. MR 1750955 (2001i:14071)
  • 7. X. Chen, Rational curves on $ K3$ surfaces, J. Alg. Geom. 8 (1999), 245-278. MR 1675158 (2000d:14057)
  • 8. D. Cox, The homogeneous coordinate ring of a toric variety, J. Alg. Geom. 4 (1995), 17-50. MR 1299003 (95i:14046)
  • 9. D. Cox and S. Katz,, Mirror symmetry and algebraic geometry, AMS: Providence, RI, 1999. MR 1677117 (2000d:14048)
  • 10. I. Dolgachev and S. Kondo, Moduli spaces of $ K3$ surfaces and complex ball quotients, Lectures in Istanbul, math.AG/0511051.
  • 11. B. Fantechi, L. Göttsche, and Duco van Straten, Euler number of the compactified Jacobian and multiplicity of rational curves, J. Alg. Geom. 8 (1999), 115-133. MR 1658220 (99i:14065)
  • 12. W. Fulton, Introduction to toric varieties, Princeton University Press: Princeton, 1993. MR 1234037 (94g:14028)
  • 13. A. Gathmann, The number of plane conics $ 5$-fold tangent to a smooth curve, Comp. Math. 141 (2005), 487-501. MR 2134277 (2006b:14099)
  • 14. A. Givental, Equivariant Gromov-Witten invariants, Int. Math. Res. Notices 13 (1996), 613-663. MR 1408320 (97e:14015)
  • 15. A. Givental, A mirror theorem for toric complete intersections, math.AG/9701016. Progr. Math. 160, Birkhäuser, Boston, MA, 1998. MR 1653024 (2000a:14063)
  • 16. R. Gopakumar and C. Vafa, M-theory and topological strings I, hep-th/9809187.
  • 17. R. Gopakumar and C. Vafa, M-theory and topological strings II, hep-th/9812127.
  • 18. J. Harvey and G. Moore, Algebras, BPS states, and strings, Nucl. Phys. B463 (1996), 315-368. MR 1393643 (97h:81163)
  • 19. J. Harvey and G. Moore, Exact gravitational threshold correction in the FHSV model, Phys. Rev. D57 (1998), 2329-2336. MR 1607775 (2000c:81262)
  • 20. S. Kachru and C. Vafa, Exact results for $ N=2$ compactifications of heterotic strings, Nucl. Phys. B 450 (1995), 69-89. MR 1349793 (96j:81100)
  • 21. S. Katz, A. Klemm, C. Vafa, M-theory, topological strings, and spinning black holes, Adv. Theor. Math. Phys. 3 (1999), 1445-1537. MR 1796683 (2002b:81124)
  • 22. T. Kawai and K Yoshioka, String partition functions and infinite products, Adv. Theor. Math. Phys. 4 (2000), 397-485. MR 1838446 (2002g:11054)
  • 23. A. Klemm, M. Kreuzer, E. Riegler, and E. Scheidegger, Topological string amplitudes, complete intersections Calabi-Yau spaces, and threshold corrections, hep-th/0410018. J. High Energy Phys. 2005, no. 5, 116 pp. MR 2155395 (2006h:81263)
  • 24. A. Klemm, W. Lerche, and P. Mayr, $ K3$-fibrations and heterotic-type II string duality, Physics Lett. B 357 (1995), 313-322. MR 1352443 (96j:81101)
  • 25. S. Kudla and J. Millson, Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients on holomorphic modular forms in several complex variables, Pub. IHES 71 (1990), 121-172. MR 1079646 (92e:11035)
  • 26. J. Lee and C. Leung, Yau-Zaslow formula for non-primitive classes in $ K3$ surfaces, Geom. Topol. 9 (2005), 1977-2012. MR 2175162 (2006e:14073)
  • 27. J. Li, A degeneration formula for Gromov-Witten invariants, J. Diff. Geom. 60 (2002), 199-293. MR 1938113 (2004k:14096)
  • 28. J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. AMS 11 (1998), 119-174. MR 1467172 (99d:14011)
  • 29. B. Lian, K. Liu, and S.-T. Yau, Mirror principle I, Asian J. Math. 4 (1997), 729-763. MR 1621573 (99e:14062)
  • 30. M. Mariño and G. Moore, Counting higher genus curves in a Calabi-Yau manifold, Nucl. Phys. B453 (1999), 592-614. MR 1683802 (2000f:14086)
  • 31. D. Maulik and R. Pandharipande, Gromov-Witten theory and Noether-Lefschetz theory, arXiv/0705.1653.
  • 32. T. Oda, Convex bodies and algebraic geometry, Springer-Verlag: Berlin, 1988. MR 922894 (88m:14038)
  • 33. R. Pandharipande, Rational curves on hypersurfaces [after A. Givental], Séminaire Bourbaki, 50ème année, 1997-1998, no. 848. MR 1685628 (2000e:14094)
  • 34. R. Pandharipande and R. Thomas, Stable pairs and BPS invariants, arXiv/0711.3899. J. Amer. Math. Soc. 23 (2010), 267-297. MR 2552254
  • 35. B. Wu, The number of rational curves on $ K3$ surfaces, math.AG/0602280. Asian J. Math. 11 (2007), 635-650. MR 2402942 (2009e:14101)
  • 36. S.-T. Yau and E. Zaslow, BPS states, string duality, and nodal curves on $ K3$, Nucl. Phys. B457 (1995), 484-512. MR 1398633 (97e:14066)
  • 37. D. Zagier, Elliptic modular forms and their applications, in The 1-2-3 of modular forms: Lectures at a summer school in Nordfjordeid, Norway, J. Bruinier, G. van der Geer, G. Harder, D. Zagier, eds., Springer: Berlin, 2008. MR 2409678 (2010b:11047)
  • 38. D. Zagier, Letter on the Harvey-Moore identity, October 2007.

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Additional Information

A. Klemm
Affiliation: Department of Physics, University of Bonn, Endenicher Allee 11-13, Bonn, 53115 Germany, and Department of Physics, University of Wisconsin, 1150 University Avenue, Madison, WI 53706-1390

D. Maulik
Affiliation: Department of Mathematics, Columbia University, New York, NY 10027

R. Pandharipande
Affiliation: Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000

E. Scheidegger
Affiliation: Department of Mathematics, University of Augsburg, 86135 Augsburg, Germany

Received by editor(s): December 28, 2008
Received by editor(s) in revised form: April 13, 2010
Published electronically: June 9, 2010
Additional Notes: The first author was partially supported by DOE grant DE-FG02-95ER40896
The second author was partially supported by a Clay research fellowship
The third author was partially support by NSF grant DMS-0500187
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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