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Gromov-Witten theory and Noether-Lefschetz theory.(English)Zbl 1317.14126

Hassett, Brendan (ed.) et al., A celebration of algebraic geometry. A conference in honor of Joe Harris’ 60th birthday, Harvard University, Cambridge, MA, USA, August 25–28, 2011. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute (ISBN 978-0-8218-8983-1/pbk). Clay Mathematics Proceedings 18, 469-507 (2013).
Summary: Noether-Lefschetz divisors in the moduli of \(K3\) surfaces are the loci corresponding to Picard rank at least 2. We relate the degrees of the Noether-Lefschetz divisors in 1-parameter families of \(K3\) surfaces to the Gromov-Witten theory of the 3-fold total space. The reduced \(K3\) theory and the Yau-Zaslow formula play an important role. We use results of Borcherds and Kudla-Millson for \(O(2,19)\) lattices to determine the Noether-Lefschetz degrees in classical families of \(K3\) surfaces of degrees 2, 4, 6 and 8. For the quartic \(K3\) surfaces, the Noether-Lefschetz degrees are proven to be the Fourier coefficients of an explicitly computed modular form of weight \(21/2\) and level \(8\). The interplay with mirror symmetry is discussed. We close with a conjecture on the Picard ranks of moduli spaces of \(K3\) surfaces.
For the entire collection see [Zbl 1275.14003].

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14J10 Families, moduli, classification: algebraic theory
14J28 \(K3\) surfaces and Enriques surfaces
14D22 Fine and coarse moduli spaces
14J33 Mirror symmetry (algebro-geometric aspects)

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