## 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.

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## Bibliographic 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
- MR Author ID: 357813
**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 - © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**23**(2010), 1013-1040 - MSC (2010): Primary 14N35
- DOI: https://doi.org/10.1090/S0894-0347-2010-00672-8
- MathSciNet review: 2669707