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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Noether-Lefschetz theory and the Yau-Zaslow conjecture
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by A. Klemm, D. Maulik, R. Pandharipande and E. Scheidegger PDF
J. Amer. Math. Soc. 23 (2010), 1013-1040 Request permission

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
  • 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