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Curves on \(K3\) surfaces in divisibility 2. (English) Zbl 1468.14094

Summary: We prove a conjecture of D. Maulik et al. [J. Topol. 3, No. 4, 937–996 (2010; Zbl 1207.14058)] 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 G. Oberdieck and R. Pandharipande [Prog. Math. 315, 245–278 (2016; Zbl 1349.14176)] is discussed in detail and illustrated with several examples.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14J28 \(K3\) surfaces and Enriques surfaces
11F03 Modular and automorphic functions
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References:

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