Period- and mirror-maps for the quartic K3

Abstract

We study in detail mirror symmetry for the quartic K3 surface in $${\mathbb{P}^3}$$ and the mirror family obtained by the orbifold construction. As explained by Aspinwall and Morrison (Mirror symmetry II, 1997), 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 (Essays on mirror manifolds, 1992) near the point of maximal unipotent monodromy. Our results rely on earlier work by Narumiyah–Shiga (Proceedings on Moonshine and related topics, 2001), Dolgachev (J. Math. Sci., 1996) and Nagura–Sugiyama (Int. J. Mod. Phys. A, 1995).