{
  "id": "math/0003145",
  "version": "v1",
  "published": "2000-03-24T03:41:44.000Z",
  "updated": "2000-03-24T03:41:44.000Z",
  "title": "Study on the family of K3 surfaces induced from the lattice $(D_4)^3 \\oplus < -2 > \\oplus < 2 >",
  "authors": [
    "K Koike",
    "H Shiga",
    "N Takayama",
    "T Tsutsui"
  ],
  "comment": "26 pages, 12 figures",
  "categories": [
    "math.AG"
  ],
  "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 ${\\bf Q}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-03-24T03:41:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J28",
      "33C70",
      "14K99"
    ],
    "keywords": [
      "k3 surfaces",
      "picard lattice",
      "abelian variety",
      "hamilton quarternion field",
      "endomorphism algebra contains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......3145K"
    }
  }
}