{
  "id": "1406.2761",
  "version": "v3",
  "published": "2014-06-11T02:07:49.000Z",
  "updated": "2014-08-19T05:17:42.000Z",
  "title": "Non-liftability of automorphism groups of a K3 surface in positive characteristic",
  "authors": [
    "Hélène Esnault",
    "Keiji Oguiso"
  ],
  "comment": "as compared to v1: Thm 7.5 is new; we corrected section 2.4 and Proposition 6.2",
  "categories": [
    "math.AG"
  ],
  "abstract": "We show that for a K3 surface in characteristic $p\\ge 3$, there is a projective model $X_R\\to {\\rm Spec} \\ R$ in characteristic $0$ with Picard number $1$ over a geometric generic point (Thm. 4.2). In particular, this model essentially kills all automorphisms (Thm. 5.1). We show that there is an automorphism on a supersingular K3 surface in characteristic $3$, which has positive entropy, the logarithm of a Salem number of degree $22$ (Thm. 6.4). In particular it does not lift to characteristic $0$. In addition, we show that in any large characteristic, there is an automorphism of a supersingular K3 which has positive entropy and does not lift to characteristic $0$ (Thm. 7.5).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-06-18T10:27:35.000Z",
      "abstract": "We show that for a K3 surface in characteristic $p\\ge 3$, there is a projective model $X_R\\to {\\rm Spec} R$ in characteristic $0$ with Picard number $1$ over a geometric generic point (Theorem 4.2). In particular, this model essentially kills all automorphisms (Theorem 5.1). We show that there is a supersingular K3 surface in characteristic $3$, with an automorphism of positive entropy, the logarithm of a Salem number of degree $22$, which does not lift to characteristic $0$ at all (Theorem 6.4). We construct elliptic K3 surfaces in characteristic $p\\ge 3$ such that the automorphism group of any lifting to characteristic $0$ does not hit the whole automorphism subgroup of the Mordell-Weil group of the elliptic fibrations and some automorphisms of positive entropy (Theorem 7.3).",
      "comment": "26 pages, some typos are corrected",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-08-19T05:17:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "automorphism group",
      "positive characteristic",
      "non-liftability",
      "construct elliptic k3 surfaces",
      "supersingular k3 surface"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.2761E"
    }
  }
}