On the mixed-twist construction and monodromy of associated Picard-Fuchs systems

TL;DR

This paper explores the mixed-twist construction of K3 surfaces, linking Picard-Fuchs systems to hypergeometric and GKZ systems, and explicitly computes monodromy for mirror families.

Contribution

It applies the mixed-twist construction to produce new families of K3 surfaces and connects their Picard-Fuchs systems to known hypergeometric and GKZ systems, including explicit monodromy calculations.

Findings

Picard-Fuchs systems match known hypergeometric systems

Constructed non-resonant GKZ systems with explicit solutions

Computed monodromy of mirror family solutions

Abstract

We use the mixed-twist construction of Doran and Malmendier to obtain a multi-parameter family of K3 surfaces of Picard rank $\rho \ge 16$. Upon identifying a particular Jacobian elliptic fibration on its general member, we determine the lattice polarization and the Picard-Fuchs system for the family. We construct a sequence of restrictions that lead to extensions of the polarization by two-elementary lattices. We show that the Picard-Fuchs operators for the restricted families coincide with known resonant hypergeometric systems. Second, for the one-parameter mirror families of deformed Fermat hypersurfaces we show that the mixed-twist construction produces a non-resonant GKZ system for which a basis of solutions in the form of absolutely convergent Mellin-Barnes integrals exists whose monodromy we compute explicitly.