{
  "id": "1710.01903",
  "version": "v1",
  "published": "2017-10-05T07:38:26.000Z",
  "updated": "2017-10-05T07:38:26.000Z",
  "title": "Polarized endomorphisms of normal projective threefolds in arbitrary characteristic",
  "authors": [
    "Sheng Meng",
    "De-Qi Zhang"
  ],
  "comment": "Appendix coauthored with Paolo Cascini",
  "categories": [
    "math.AG",
    "math.DS"
  ],
  "abstract": "Let $X$ be a projective variety over an algebraically closed field $k$ of arbitrary characteristic $p \\ge 0$. A surjective endomorphism $f$ of $X$ is $q$-polarized if $f^\\ast H \\sim qH$ for some ample Cartier divisor $H$ and integer $q > 1$. When $f$ is separable and $X$ is $Q$-Gorenstein and normal, we show that the anti-canonical divisor $-K_X$ is numerically equivalent to an effective $Q$-Cartier divisor, strengthening slightly the conclusion of Boucksom, de Fernex and Favre (Theorem C) and also covering singular varieties over an algebraically closed field of arbitrary characteristic. Let $f^{Gal}:\\overline{X}\\to X$ be the Galois closure of $f$. We show that if $p>5$ and co-prime to $deg\\, f^{Gal}$ then one can run the minimal model program (MMP) $f$-equivariantly, after replacing $f$ by a positive power, for a mildly singular threefold $X$ and reach a variety $Y$ with torsion canonical divisor (and also with $Y$ being a quasi-\\'etale quotient of an abelian variety when $dim(Y)\\le 2$). Along the way, we show that a power of $f$ acts as a scalar multiplication on the Neron-Severi group of $X$ (modulo torsion) when $X$ is a smooth and rationally chain connected projective variety of dimension at most three. In the appendix, suppose $X$ is a normal projective variety with a polarized separable endomorphism $f$. We show that the Albanese morphism of $X$ is an algebraic fibre space and $f$ induces polarized endomorphisms on the Albanese and also the Picard variety of $X$, and $K_X$ being pseudo-effective and $Q$-Cartier means being a torsion $Q$-divisor.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-05T07:38:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H30",
      "32H50",
      "14E30",
      "11G10",
      "08A35"
    ],
    "keywords": [
      "arbitrary characteristic",
      "normal projective threefolds",
      "polarized endomorphisms",
      "chain connected projective variety",
      "algebraically closed field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}