Epimorphic period mapping for K3-surfaces that can be represented as double planes

Abstract

In [i] the global Torelli theorem has been proved for K3-surfaces. Let us recall some definitions and notation. Definition i. A simply connected surface with a zero canonical class is called a K3surface. It is shown in [2] that for each K3-surface the group H 2 (X, Z) is torsion-free and an even unimodular lattice and the signature of H 2 (X, Z)~ Q is equal to (3, 19). We know that even unimodular lattices L, for which L Q Q has the signature (3, 19), are isomorphic. Definition 2. A triple (X, ~, ~), where X is a K3-surface, ~: H~-(X, Z)-+ L is an isomorphism such that ~ (~)= Z, L being a fixed lattice, l ~ L, ands is the class of very ample divisor on X, is called a distinguished K3-surface. Let us describe briefly the space of Hodge structures for a polarized K3-surface. Let be the space of all complex linear functionals on L. Let us define the scalar product on in the usual manner; viz., at first let us extend the scalar product existing on L to L ~ C, and then we use the natural isomorphism between ~ and L ~ C. Let now ~ denote the set of all straight lines ~ ~ ~, that pass through the origin of coordinates and are such that ~ = O, (1)