Noether-Lefschetz theory and the Yau-Zaslow conjecture

Klemm, A.; Maulik, D.; Pandharipande, R.; Scheidegger, E.

doi:10.1090/S0894-0347-2010-00672-8

Noether-Lefschetz theory and the Yau-Zaslow conjecture
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by A. Klemm, D. Maulik, R. Pandharipande and E. Scheidegger

J. Amer. Math. Soc. 23 (2010), 1013-1040

DOI: https://doi.org/10.1090/S0894-0347-2010-00672-8

Published electronically: June 9, 2010

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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.

References

K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), no. 3, 601–617. MR 1431140, DOI 10.1007/s002220050132
K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88. MR 1437495, DOI 10.1007/s002220050136
Arnaud Beauville, Counting rational curves on $K3$ surfaces, Duke Math. J. 97 (1999), no. 1, 99–108. MR 1682284, DOI 10.1215/S0012-7094-99-09704-1
Richard E. Borcherds, The Gross-Kohnen-Zagier theorem in higher dimensions, Duke Math. J. 97 (1999), no. 2, 219–233. MR 1682249, DOI 10.1215/S0012-7094-99-09710-7
Jan Hendrik Bruinier, On the rank of Picard groups of modular varieties attached to orthogonal groups, Compositio Math. 133 (2002), no. 1, 49–63. MR 1918289, DOI 10.1023/A:1016357029843
Jim Bryan and Naichung Conan Leung, The enumerative geometry of $K3$ surfaces and modular forms, J. Amer. Math. Soc. 13 (2000), no. 2, 371–410. MR 1750955, DOI 10.1090/S0894-0347-00-00326-X
Xi Chen, Rational curves on $K3$ surfaces, J. Algebraic Geom. 8 (1999), no. 2, 245–278. MR 1675158
David A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), no. 1, 17–50. MR 1299003
David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, vol. 68, American Mathematical Society, Providence, RI, 1999. MR 1677117, DOI 10.1090/surv/068
I. Dolgachev and S. Kondo, Moduli spaces of $K3$ surfaces and complex ball quotients, Lectures in Istanbul, math.AG/0511051.
B. Fantechi, L. Göttsche, and D. van Straten, Euler number of the compactified Jacobian and multiplicity of rational curves, J. Algebraic Geom. 8 (1999), no. 1, 115–133. MR 1658220
William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
Andreas Gathmann, The number of plane conics that are five-fold tangent to a given curve, Compos. Math. 141 (2005), no. 2, 487–501. MR 2134277, DOI 10.1112/S0010437X04001083
Alexander B. Givental, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices 13 (1996), 613–663. MR 1408320, DOI 10.1155/S1073792896000414
Alexander Givental, A mirror theorem for toric complete intersections, Topological field theory, primitive forms and related topics (Kyoto, 1996) Progr. Math., vol. 160, Birkhäuser Boston, Boston, MA, 1998, pp. 141–175. MR 1653024
R. Gopakumar and C. Vafa, M-theory and topological strings I, hep-th/9809187.
R. Gopakumar and C. Vafa, M-theory and topological strings II, hep-th/9812127.
Jeffrey A. Harvey and Gregory Moore, Algebras, BPS states, and strings, Nuclear Phys. B 463 (1996), no. 2-3, 315–368. MR 1393643, DOI 10.1016/0550-3213(95)00605-2
Jeffrey A. Harvey and Gregory Moore, Exact gravitational threshold correction in the Ferrara-Harvey-Strominger-Vafa model, Phys. Rev. D (3) 57 (1998), no. 4, 2329–2336. MR 1607775, DOI 10.1103/PhysRevD.57.2329
Shamit Kachru and Cumrun Vafa, Exact results for $N=2$ compactifications of heterotic strings, Nuclear Phys. B 450 (1995), no. 1-2, 69–89. MR 1349793, DOI 10.1016/0550-3213(95)00307-E
Sheldon Katz, Albrecht Klemm, and Cumrun Vafa, M-theory, topological strings and spinning black holes, Adv. Theor. Math. Phys. 3 (1999), no. 5, 1445–1537. MR 1796683, DOI 10.4310/ATMP.1999.v3.n5.a6
Toshiya Kawai and K\B{o}ta Yoshioka, String partition functions and infinite products, Adv. Theor. Math. Phys. 4 (2000), no. 2, 397–485. MR 1838446, DOI 10.4310/ATMP.2000.v4.n2.a7
Albrecht Klemm, Maximilian Kreuzer, Erwin Riegler, and Emanuel Scheidegger, Topological string amplitudes, complete intersection Calabi-Yau spaces and threshold corrections, J. High Energy Phys. 5 (2005), 023, 116. MR 2155395, DOI 10.1088/1126-6708/2005/05/023
A. Klemm, W. Lerche, and P. Mayr, $K_3$-fibrations and heterotic–type II string duality, Phys. Lett. B 357 (1995), no. 3, 313–322. MR 1352443, DOI 10.1016/0370-2693(95)00937-G
Stephen S. Kudla and John J. Millson, Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables, Inst. Hautes Études Sci. Publ. Math. 71 (1990), 121–172. MR 1079646, DOI 10.1007/BF02699880
Junho Lee and Naichung Conan Leung, Yau-Zaslow formula on $K3$ surfaces for non-primitive classes, Geom. Topol. 9 (2005), 1977–2012. MR 2175162, DOI 10.2140/gt.2005.9.1977
Jun Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199–293. MR 1938113
Jun Li and Gang Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), no. 1, 119–174. MR 1467172, DOI 10.1090/S0894-0347-98-00250-1
Bong H. Lian, Kefeng Liu, and Shing-Tung Yau, Mirror principle. I, Asian J. Math. 1 (1997), no. 4, 729–763. MR 1621573, DOI 10.4310/AJM.1997.v1.n4.a5
Marcos Mariño and Gregory Moore, Counting higher genus curves in a Calabi-Yau manifold, Nuclear Phys. B 543 (1999), no. 3, 592–614. MR 1683802, DOI 10.1016/S0550-3213(98)00847-5
D. Maulik and R. Pandharipande, Gromov-Witten theory and Noether-Lefschetz theory, arXiv/0705.1653.
Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR 922894
Rahul Pandharipande, Rational curves on hypersurfaces (after A. Givental), Astérisque 252 (1998), Exp. No. 848, 5, 307–340. Séminaire Bourbaki. Vol. 1997/98. MR 1685628
R. Pandharipande and R. P. Thomas, Stable pairs and BPS invariants, J. Amer. Math. Soc. 23 (2010), no. 1, 267–297. MR 2552254, DOI 10.1090/S0894-0347-09-00646-8
Baosen Wu, The number of rational curves on $K3$ surfaces, Asian J. Math. 11 (2007), no. 4, 635–650. MR 2402942, DOI 10.4310/AJM.2007.v11.n4.a6
Shing-Tung Yau and Eric Zaslow, BPS states, string duality, and nodal curves on $K3$, Nuclear Phys. B 471 (1996), no. 3, 503–512. MR 1398633, DOI 10.1016/0550-3213(96)00176-9
Don Zagier, Elliptic modular forms and their applications, The 1-2-3 of modular forms, Universitext, Springer, Berlin, 2008, pp. 1–103. MR 2409678, DOI 10.1007/978-3-540-74119-0_{1}
D. Zagier, Letter on the Harvey-Moore identity, October 2007.