Kummer sandwiches and Greene-Plesser construction

Abstract

In the context of K3 mirror symmetry, the Greene-Plesser orbifolding method constructs a family of K3 surfaces, the mirror of quartic hypersurfaces in ${\mathbb{P}}^3$, starting from a special one-parameter family of K3 varieties known as the quartic Dwork pencil. We show that certain K3 double covers obtained from the three-parameter family of quartic Kummer surfaces associated with a principally polarized abelian surface generalize the relation of the Dwork pencil and the quartic mirror family. Moreover, for the three-parameter family we compute a formula for the rational point-count of its generic member and derive its transformation behavior with respect to $(2,2)$-isogenies of the underlying abelian surface.

27 pages; minor typos corrected in version 2

References in corpus (1)

• K3 surfaces, modular forms, and non-geometric heterotic compactifications

Cited by in corpus (5)

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• On holomorphic conformal structures associated with lattice polarized K3 surfaces
• Isogenies of certain K3 surfaces of rank 18