Period- and mirror-maps for the quartic K3

TL;DR

This paper provides explicit computations of Hodge structures and period maps for the quartic K3 surface, establishing a detailed mirror symmetry correspondence and verifying compatibility with known mirror maps.

Contribution

It explicitly computes Hodge structures, identifies a mirror map, and demonstrates compatibility with Morrison's mirror map for the quartic K3 surface.

Findings

Explicit Hodge structure computations for quartic K3

Identification of a mirror map between deformation parameters

Verification of compatibility with Morrison's mirror map

Abstract

We study in detail mirror symmetry for the quartic K3 surface in P3 and the mirror family obtained by the orbifold construction. As explained by Aspinwall and Morrison, mirror symmetry for K3 surfaces can be entirely described in terms of Hodge structures. (1) We give an explicit computation of the Hodge structures and period maps for these families of K3 surfaces. (2) We identify a mirror map, i.e. an isomorphism between the complex and symplectic deformation parameters, and explicit isomorphisms between the Hodge structures at these points. (3) We show compatibility of our mirror map with the one defined by Morrison near the point of maximal unipotent monodromy. Our results rely on earlier work by Narumiyah-Shiga, Dolgachev and Nagura-Sugiyama.