Gromov-Witten theory and Noether-Lefschetz theory
Abstract
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.
Cite
@article{arxiv.0705.1653,
title = {Gromov-Witten theory and Noether-Lefschetz theory},
author = {D. Maulik and R. Pandharipande},
journal= {arXiv preprint arXiv:0705.1653},
year = {2012}
}