Treffer: Study on the family of \(K3\) surfaces induced from the lattice \((D_ 4)^ 3\oplus\langle-2\rangle\oplus\langle 2\rangle\).

Summary: 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 \(P\subset \text{Pic}(S) \), where \(\text{Pic}(S) \) 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: (1) A \(K3\) surface of type \(P\) has a representation as a double cover over \({\mathbb{P}}^1\times {\mathbb{P}}^1\) as the following affine form in \((s,t,w)\) space: \[ S=S(x): w^2=\prod_{k=1}^4 (x_{1}^{(k)}st+x_{2}^{(k)}s+x_{3}^{(k)}t+x_{4}^{(k)}), \;x^{(k)} \in M(2,{\mathbb{C}}). \] We make explicit description of the Picard lattice and the transcendental lattice of \(S(x)\). (2) We describe the period domain for our family of marked \(K3\) surfaces and determine the modular group. (3) We describe the differential equation for the period integral of \(S(x)\) as a function of \(x\in (\text{GL}(2,{\mathbb{C}}))^4\). That becomes to be a certain kind of hypergeometric one. We determine the rank, the singular locus and the monodromy group for it. (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 \({\mathbb{Q}}\).