{
  "id": "0804.1522",
  "version": "v2",
  "published": "2008-04-09T16:50:20.000Z",
  "updated": "2010-10-26T10:31:15.000Z",
  "title": "The supersingular locus of the Shimura variety of GU(1,n-1) II",
  "authors": [
    "I. Vollaard",
    "T. Wedhorn"
  ],
  "comment": "38 pages, Proof of Proposition 3.6 simplified",
  "categories": [
    "math.AG",
    "math.RT"
  ],
  "abstract": "We complete the study of the supersingular locus in the fiber at $p$ of a Shimura variety attached to a unitary similitude group $\\GU(1,n-1)$ over $\\QQ$ in the case that $p$ is inert. This was started by the first author in \\cite{Vo_Uni} where complete results were obtained for $n =2,3$. The supersingular locus is uniformized by a formal scheme $\\Ncal$ which is a moduli space of so-called unitary $p$-divisible groups. It depends on the choice of a unitary isocrystal $\\Nbf$. We define a stratification of $\\Ncal$ indexed by vertices of the Bruhat-Tits building attached to the reductive group of automorphisms of $\\Nbf$. We show that the combinatorial behaviour of this stratification is given by the simplicial structure of the building. The closures of the strata (and in particular the reduced irreducible components of $\\Ncal$) are identified with (generalized) Deligne-Lusztig varieties. We show that the Bruhat-Tits stratification is a refinement of the Ekedahl-Oort stratification and also relate the Ekedahl-Oort strata to Deligne-Lusztig varieties. We deduce that the supersingular locus is locally a complete intersection, that its irreducible components and each Ekedahl-Oort stratum in every irreducible component is isomorphic to a Deligne-Lusztig variety, and give formulas for the number of irreducible components of every Ekedahl-Oort stratum of the supersingular locus.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-10-26T10:31:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G35",
      "14K10",
      "11E95"
    ],
    "keywords": [
      "supersingular locus",
      "shimura variety",
      "irreducible component",
      "deligne-lusztig variety",
      "ekedahl-oort stratum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.1522V"
    }
  }
}