{
  "id": "1211.1731",
  "version": "v5",
  "published": "2012-11-08T00:16:12.000Z",
  "updated": "2015-05-29T22:24:54.000Z",
  "title": "Toroidal compactifications of integral models of Shimura varieties of Hodge type",
  "authors": [
    "Keerthi Madapusi Pera"
  ],
  "comment": "Completely rewritten version",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We construct projective toroidal compactifications for integral models of Shimura varieties of Hodge type that parameterize isogenies of abelian varieties with additional structure. We also construct integral models of the minimal (Satake-Baily-Borel) compactification. Our results essentially reduce the problem to understanding the integral models themselves. As such, they cover all previously known cases of PEL type, as well as all cases of Hodge type involving parahoric level structures. At primes where the level is hyperspecial, we show that our compactifications are canonical in a precise sense. We also provide a new proof of Y. Morita's conjecture on the everywhere good reduction of abelian varieties whose Mumford-Tate group is anisotropic modulo center. Along the way, we demonstrate an interesting rationality property of Hodge cycles on abelian varieties with respect to p-adic analytic uniformizations.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-02-20T05:30:28.000Z",
      "abstract": "We construct smooth projective toroidal compactifications for the integral canonical models of Shimura varieties of Hodge type constructed by Kisin and Vasiu at primes where the level is hyperspecial. This construction is a consequence of the main result of the paper, which shows, without any unramifiedness conditions on the Shimura datum, that the Zariski closure of a Shimura sub-variety of Hodge type in a Chai-Faltings compactification always intersects the boundary in a relative Cartier divisor. This result also provides a new proof of Y. Morita's conjecture on the everywhere good reduction of abelian varieties (over number fields) whose Mumford-Tate group is anisotropic modulo center. We also construct integral models of the minimal (Satake-Baily-Borel) compactification for Shimura varieties of Hodge type.",
      "comment": "Many corrections",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2015-05-29T22:24:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hodge type",
      "shimura varieties",
      "construct smooth projective toroidal compactifications",
      "anisotropic modulo center",
      "construct integral models"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.1731M"
    }
  }
}