{
  "id": "1109.5632",
  "version": "v4",
  "published": "2011-09-26T16:43:36.000Z",
  "updated": "2013-02-25T17:18:39.000Z",
  "title": "Semi-algebraic horizontal subvarieties of Calabi-Yau type",
  "authors": [
    "Robert Friedman",
    "Radu Laza"
  ],
  "comment": "53 pages, final version, to appear in Duke Math. J.; changes from v3: new references added; changes from v2: for Hermitian VHS of CY 3-fold type with real multiplication, we discuss the case SU(3,3) for arbitrary totally real number fields; the case SO^*(12) is discussed in arXiv:1301.2582; changes from v1: some inaccuracies corrected, Section 3 substantially expanded",
  "journal": "Duke Math. J. 162 (2013), no. 12, 2077-2148",
  "categories": [
    "math.AG"
  ],
  "abstract": "We study horizontal subvarieties $Z$ of a Griffiths period domain $\\mathbb D$. If $Z$ is defined by algebraic equations, and if $Z$ is also invariant under a large discrete subgroup in an appropriate sense, we prove that $Z$ is a Hermitian symmetric domain $\\mathcal D$, embedded via a totally geodesic embedding in $\\mathbb D$. Next we discuss the case when $Z$ is in addition of Calabi-Yau type. We classify the possible VHS of Calabi-Yau type parametrized by Hermitian symmetric domains $\\mathcal D$ and show that they are essentially those found by Gross and Sheng-Zuo, up to taking factors of symmetric powers and certain shift operations. In the weight three case, we explicitly describe the embedding $Z\\hookrightarrow \\mathbb D$ from the perspective of Griffiths transversality and relate this description to the Harish-Chandra realization of $\\mathcal D$ and to the Kor\\'anyi-Wolf tube domain description. There are further connections to homogeneous Legendrian varieties and the four Severi varieties of Zak.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-02-25T17:18:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14D07",
      "32G20",
      "14C30",
      "14G35",
      "32M15"
    ],
    "keywords": [
      "calabi-yau type",
      "semi-algebraic horizontal subvarieties",
      "hermitian symmetric domain",
      "koranyi-wolf tube domain description",
      "large discrete subgroup"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.5632F"
    }
  }
}