{
  "id": "0812.4317",
  "version": "v1",
  "published": "2008-12-23T00:17:14.000Z",
  "updated": "2008-12-23T00:17:14.000Z",
  "title": "On varieties whose universal cover is a product of curves",
  "authors": [
    "Fabrizio Catanese",
    "Marco Franciosi"
  ],
  "comment": "22 pages, dedicated to Sommese's 60-th birthday. Greatly improves, expands and supersedes arXiv:0803.3008, of which also corrects a mistake",
  "categories": [
    "math.AG",
    "math.CV"
  ],
  "abstract": "We investigate a necessary condition for a compact complex manifold X of dimension n in order that its universal cover be the Cartesian product $C^n$ of a curve $C = \\PP^1 or \\HH$: the existence of a semispecial tensor $\\omega$. A semispecial tensor is a non zero section $ 0 \\neq \\omega \\in H^0(X, S^n\\Omega^1_X (-K_X) \\otimes \\eta) $), where $\\eta$ is an invertible sheaf of 2-torsion (i.e., $\\eta^2\\cong \\hol_X$). We show that this condition works out nicely, as a sufficient condition, when coupled with some other simple hypothesis, in the case of dimension $n= 2$ or $ n= 3$; but it is not sufficient alone, even in dimension 2. In the case of K\\\"ahler surfaces we use the above results in order to give a characterization of the surfaces whose universal cover is a product of two curves, distinguishing the 6 possible cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-12-23T00:17:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J25",
      "32Q30",
      "14J29"
    ],
    "keywords": [
      "universal cover",
      "semispecial tensor",
      "compact complex manifold",
      "non zero section",
      "simple hypothesis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0812.4317C"
    }
  }
}