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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 mathematics.

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

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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The reduced genus $1$ Gromov-Witten invariants of Calabi-Yau hypersurfaces
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by Aleksey Zinger
J. Amer. Math. Soc. 22 (2009), 691-737
Published electronically: October 2, 2008


We compute the reduced genus 1 Gromov-Witten invariants of Calabi-Yau hypersurfaces. As a consequence, we confirm the 1993 Bershadsky-Cecotti-Ooguri-Vafa (BCOV) prediction for the standard genus 1 GW-invariants of a quintic threefold. We combine constructions from a series of previous papers with the classical localization theorem to relate the reduced genus 1 invariants of a CY-hypersurface to previously computed integrals on moduli spaces of stable genus 0 maps into projective space. The resulting, rather unwieldy, expressions for a genus 1 equivariant generating function simplify drastically, using a regularity property of a genus 0 equivariant generating function in half of the cases. Finally, by disregarding terms that cannot effect the non-equivariant part of the former, we relate the answer to an explicit hypergeometric series in a simple way. The approach described in this paper is systematic. It is directly applicable to computing reduced genus 1 GW-invariants of other complete intersections and should apply to higher-genus localization computations.
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Bibliographic Information
  • Aleksey Zinger
  • Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651
  • Email:
  • Received by editor(s): July 23, 2007
  • Published electronically: October 2, 2008
  • Additional Notes: The author was partially supported by a Sloan Fellowship and DMS Grant 0604874
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 22 (2009), 691-737
  • MSC (2000): Primary 14N35, 53D45
  • DOI:
  • MathSciNet review: 2505298