Gromov-Witten theory and Noether-Lefschetz theory
By Davesh Maulik, R. Pandharipande. 2007
134 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/801c1ba38534ecd1c6512b26433a10487073ef52
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In Geometry and Topology
- D. Huybrechts
- R. Pandharipande
- Bertrand Toën
- Richard P. Thomas
- Davesh Maulik
- G. Vezzosi
- Timo Schürg
- Yukinobu Toda
Citations in the corpus
- Curves on K3 surfaces and modular forms (2010, methodology): “ce of curves Mg(S,β) → Mg when gis at least 2. The top Chern class λg of Eis the most beautiful and well-behaved integrand in the reduced theory of S. Define (1) Rg,β = Z [Mg(S,β)]red (−1)gλ g . 1 See [6, 36] for foundational discussions. 2Primitive implies nonzero. CURVES ON K3 SURFACES AND MODULAR ”
- The Noether-Lefschetz conjecture and generalizations (2014, background): “Besides the Hodge line bundle, there are actually many other natural divisors on Kg coming from Noether-Lefschetz theory developed by Griffiths and Harris in [21] (see also [41]).”
- Lectures on K3 Surfaces (2016, background): “The more recent article of Maulik and Pandharipande [397] investigates so called Noether–Lefschetz loci which produce explicit divisors.”
- Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes (2011, methodology): “This reduced class is one dimension larger that the one obtained from the standard perfect obstruction theory and leads to many interesting enumerative results (see [20, 21, 25]).”