{
  "id": "1312.4869",
  "version": "v3",
  "published": "2013-12-17T17:27:11.000Z",
  "updated": "2017-07-04T14:04:57.000Z",
  "title": "Ekedahl-Oort strata for good reductions of Shimura varieties of Hodge type",
  "authors": [
    "Chao Zhang"
  ],
  "comment": "3rd version, several corrections and simplifications, 27 pages, accepted by Canadian Journal of Math",
  "categories": [
    "math.AG"
  ],
  "abstract": "For a Shimura variety of Hodge type with hyperspecial level structure at a prime $p$, Vasiu and Kisin constructed a smooth integral model (namely the integral canonical model) uniquely determined by a certain extension property. We define and study the Ekedahl-Oort stratifications on the special fibers of those integral canonical models when $p>2$. This generalizes Ekedahl-Oort stratifications defined and studied by Oort on moduli spaces of principally polarized abelian varieties and those defined and studied by Moonen, Wedhorn and Viehmann on good reductions of Shimura varieties of PEL type. We show that the Ekedahl-Oort strata are parameterized by certain elements $w$ in the Weyl group of the reductive group in the Shimura datum. We prove that the stratum corresponding to $w$ is smooth of dimension $l(w)$ (i.e. the length of $w$) if it is non-empty. We also determine the closure of each stratum.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-04-21T18:28:25.000Z",
      "abstract": "Ekedahl-Oort strata were defined and studied by Ekedahl and Oort for Siegel modular variety, and then generalized to PEL Shimura varieties by Moonen, Wedhorn, Viehmann... The main properties of this stratification are 1) all the E-O strata are listed by certain representatives of certain Weyl cosets, and hence finite, 2) each E-O stratum is smooth and equi-dimensional, 3) there is a dimension formula for an E-O stratum, provided it is non-empty, 4) the closure of an E-O stratum is a union of E-O strata, and we can find them using the list in 1). In this paper, using $G$-zips defined by Pink, Wedhorn and Ziegler, we construct E-O strata for good reductions of Shimura varieties of Hodge type, and generalize many results in the PEL cases (for example 1, 2, 3 and 4 above) to Shimura varieties of Hodge type.",
      "comment": "2nd version, several corrections",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2017-07-04T14:04:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G35"
    ],
    "keywords": [
      "hodge type",
      "e-o stratum",
      "ekedahl-oort strata",
      "reductions",
      "pel shimura varieties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.4869Z"
    }
  }
}