Gromov-Witten theory and Noether-Lefschetz theory

TL;DR

This paper connects Noether-Lefschetz divisors in K3 surface moduli to Gromov-Witten theory, using lattice results and modular forms to compute degrees and explore mirror symmetry implications.

Contribution

It establishes a novel relationship between Noether-Lefschetz degrees and Gromov-Witten invariants, providing explicit calculations for classical K3 families and linking to modular forms.

Findings

Noether-Lefschetz degrees for degrees 2, 4, 6, 8 K3 surfaces determined

Noether-Lefschetz degrees for quartic K3 surfaces are Fourier coefficients of a specific modular form

Discussion of mirror symmetry and a conjecture on Picard ranks of K3 moduli spaces

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.