Noether-Lefschetz theory and the Yau-Zaslow conjecture

Klemm, A.¹ ; Maulik, D.² ; Pandharipande, R.³ ; Scheidegger, E.⁴

¹ 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
² Department of Mathematics, Columbia University, New York, NY 10027
³ Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000
⁴ Department of Mathematics, University of Augsburg, 86135 Augsburg, Germany

Journal of the American Mathematical Society, Tome 23 (2010) no. 4, pp. 1013-1040

Résumé

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.

Détail

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

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