A note about special cycles on moduli spaces of K3 surfaces

TL;DR

This paper explores the connection between special cycles on moduli spaces of lattice polarized K3 surfaces and Siegel modular forms, extending Kudla-Millson's results to higher Noether-Lefschetz loci.

Contribution

It applies Kudla-Millson's modularity results to K3 surfaces, linking special cycles to classical modular forms and constructing explicit examples related to quadratic spaces over totally real fields.

Findings

Generating series for special cycles are modular forms.

Higher Noether-Lefschetz numbers appear as Fourier coefficients.

Explicit constructions of related families are proposed.

Abstract

We describe the application of the results of Kudla-Millson on the modularity of generating series for cohomology classes of special cycles to the case of lattice polarized K3 surfaces. In this case, the special cycles can be interpreted as higher Noether-Lefschetz loci. These generating series can be paired with the cohomology classes of complete subvarieties of the moduli space to give classical Siegel modular forms with higher Noether-Lefschetz numbers as Fourier coefficients. Examples of such complete families associated to quadratic spaces over totally real number fields are constructed. A more explicit and concrete construction of such families and the resulting modular forms would be of interest.