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Curves on K3 surfaces in divisibility 2

Part of: Discontinuous groups and automorphic forms Projective and enumerative geometry Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  01 February 2021

Younghan Bae  and
Tim-Henrik Buelles
Younghan Bae
Affiliation:
ETH Zürich, Department of Mathematics, Zürich, Switzerland; E-mail: younghan.bae@math.ethz.ch
Tim-Henrik Buelles
Affiliation:
ETH Zürich, Department of Mathematics, Zürich, Switzerland; E-mail: buelles@math.ethz.ch

Abstract

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We prove a conjecture of Maulik, Pandharipande and Thomas expressing the Gromov–Witten invariants of K3 surfaces for divisibility 2 curve classes in all genera in terms of weakly holomorphic quasi-modular forms of level 2. Then we establish the holomorphic anomaly equation in divisibility 2 in all genera. Our approach involves a refined boundary induction, relying on the top tautological group of the moduli space of smooth curves, together with a degeneration formula for the reduced virtual fundamental class with imprimitive curve classes. We use double ramification relations with target variety as a new tool to prove the initial condition. The relationship between the holomorphic anomaly equation for higher divisibility and the conjectural multiple cover formula of Oberdieck and Pandharipande is discussed in detail and illustrated with several examples.

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Secondary: 14J28: $K3$ surfaces and Enriques surfaces 11F03: Modular and automorphic functions
Type
Mathematical Physics
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e9
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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), 2021. Published by Cambridge University Press

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