{
  "id": "math/9607217",
  "version": "v1",
  "published": "1996-07-08T00:00:00.000Z",
  "updated": "1996-07-08T00:00:00.000Z",
  "title": "Rational curves and ampleness properties of the tangent bundle of algebraic varieties",
  "authors": [
    "Frédéric Campana",
    "Thomas Peternell"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "The purpose of this paper is to translate positivity properties of the tangent bundle (and the anti-canonical bundle) of an algebraic manifold into existence and movability properties of rational curves and to investigate the impact on the global geometry of the manifold $X$. Among the results we prove are these: \\quad If $X$ is a projective manifold, and ${\\cal E} \\subset T_X$ is an ample locally free sheaf with $n-2\\ge rk {\\cal E}\\ge n$, then $X \\simeq \\EP_n$. \\quad Let $X$ be a projective manifold. If $X$ is rationally connected, then there exists a free $T_X$-ample family of (rational) curves. If $X$ admits a free $T_X$-ample family of curves, then $X$ is rationally generated.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1996-07-08T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational curves",
      "tangent bundle",
      "algebraic varieties",
      "ampleness properties",
      "translate positivity properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1996math......7217C"
    }
  }
}