{
  "id": "2007.07572",
  "version": "v1",
  "published": "2020-07-15T09:44:27.000Z",
  "updated": "2020-07-15T09:44:27.000Z",
  "title": "Hyperbolicity and Specialness of Symmetric Powers",
  "authors": [
    "Benoit Cadorel",
    "Frédéric Campana",
    "Erwan Rousseau"
  ],
  "categories": [
    "math.AG",
    "math.CV"
  ],
  "abstract": "Inspired by the computation of the Kodaira dimension of symmetric powers Xm of a complex projective variety X of dimension n $\\ge$ 2 by Arapura and Archava, we study their analytic and algebraic hyperbolic properties. First we show that Xm is special if and only if X is special (except when the core of X is a curve). Then we construct dense entire curves in (suf-ficiently hig) symmetric powers of K3 surfaces and product of curves. We also give a criterion based on the positivity of jet differentials bundles that implies pseudo-hyperbolicity of symmetric powers. As an application, we obtain the Kobayashi hyperbolicity of symmetric powers of generic projective hypersur-faces of sufficiently high degree. On the algebraic side, we give a criterion implying that subvarieties of codimension $\\le$ n -- 2 of symmetric powers are of general type. This applies in particular to varieties with ample cotangent bundles. Finally, based on a metric approach we study symmetric powers of ball quotients.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-15T09:44:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hyperbolicity",
      "specialness",
      "construct dense entire curves",
      "symmetric powers xm",
      "jet differentials bundles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}