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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

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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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

DOI: http://dx.doi.org/10.1090/S0894-0347-2010-00672-8
PII: S 0894-0347(2010)00672-8
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.