Study on the family of K3 surfaces induced from the lattice $(D_4)^3 \oplus < -2 > \oplus < 2 >

Abstract

Let us consider the rank 14 lattice $P=D_4^3\oplus < -2> \oplus < 2>$. We define a K3 surface S of type P with the property that $P\subset {\rm Pic}(S) $, where ${\rm Pic}(S) $ indicates the Picard lattice of S. In this article we study the family of K3 surfaces of type P with a certain fixed multipolarization. We note the orthogonal complement of P in the K3 lattice takes the form $$ U(2)\oplus U(2)\oplus (-2I_4). $$ We show the following results: \item{(1)} A K3 surface of type P has a representation as a double cover over ${\bf P}^1\times {\bf P}^1$ as the following affine form in (s,t,w) space: $$ S=S(x): w^2=\prod_{k=1}^4 (x_{1}^{(k)}st+x_{2}^{(k)}s+x_{3}^{(k)}t+x_{4}^{(k)}), \ x_k=\pmatrix{x_{1}^{(k)}&x_{2}^{(k)}\cr x_{3}^{(k)}&x_{4}^{(k)}} \in M(2,{\bf C}). $$ We make explicit description of the Picard lattice and the transcendental lattice of S(x). \item{(2)} We describe the period domain for our family of marked K3 surfaces and determine the modular group. \par \noindent \item{(3)} We describe the differential equation for the period integral of S(x) as a function of $x\in (GL(2,{\bf C}))^4$. That bocomes to be a certain kind of hypergeometric one. We determine the rank, the singular locus and the monodromy group for it. \par \noindent \item{(4)} It appears a family of 8 dimensional abelian varieties as the family of Kuga-Satake varieties for our K3 surfaces. The abelian variety is characterized by the property that the endomorphism algebra contains the Hamilton quarternion field over ${\bf Q}$.