Humbert surfaces and the moduli of lattice polarized K3 surfaces

摘要

In this article we introduce a collection of partial differential equations in the moduli of lattice polarized K3 surfaces whose algebraic solutions are the loci of K3 surfaces with lattice polarizations of higher rank. In the special case of rank 17 polarization such loci encode the well-known Humbert surfaces. The differential equations treated in the present article are directly derived from the Gauss-Manin connection of families of lattice polarized K3 surfaces. We also introduce some techniques to calculate the Gauss-Manin connection with the presence of isolated singularities.