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THE KATZ–KLEMM–VAFA CONJECTURE FOR $K3$ SURFACES

Part of: Projective and enumerative geometry

Published online by Cambridge University Press:  06 June 2016

R. PANDHARIPANDE  and
R. P. THOMAS
R. PANDHARIPANDE
Affiliation:
Departement Mathematik, ETH Zürich, Switzerland; [email protected]
R. P. THOMAS
Affiliation:
Department of Mathematics, Imperial College, UK; [email protected]

Abstract

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We prove the KKV conjecture expressing Gromov–Witten invariants of $K3$ surfaces in terms of modular forms. Our results apply in every genus and for every curve class. The proof uses the Gromov–Witten/Pairs correspondence for $K3$ -fibered hypersurfaces of dimension 3 to reduce the KKV conjecture to statements about stable pairs on (thickenings of) $K3$ surfaces. Using degeneration arguments and new multiple cover results for stable pairs, we reduce the KKV conjecture further to the known primitive cases. Our results yield a new proof of the full Yau–Zaslow formula, establish new Gromov–Witten multiple cover formulas, and express the fiberwise Gromov–Witten partition functions of $K3$ -fibered 3-folds in terms of explicit modular forms.

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 4 , 2016 , e4
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2016

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