{
  "id": "0704.0778",
  "version": "v2",
  "published": "2007-04-05T18:58:44.000Z",
  "updated": "2008-09-10T13:36:49.000Z",
  "title": "Frobenius splitting and geometry of $G$-Schubert varieties",
  "authors": [
    "Xuhua He",
    "Jesper Funch Thomsen"
  ],
  "comment": "Final version, 44 pages",
  "categories": [
    "math.AG",
    "math.AC",
    "math.RT"
  ],
  "abstract": "Let $X$ be an equivariant embedding of a connected reductive group $G$ over an algebraically closed field $k$ of positive characteristic. Let $B$ denote a Borel subgroup of $G$. A $G$-Schubert variety in $X$ is a subvariety of the form $\\diag(G) \\cdot V$, where $V$ is a $B \\times B$-orbit closure in $X$. In the case where $X$ is the wonderful compactification of a group of adjoint type, the $G$-Schubert varieties are the closures of Lusztig's $G$-stable pieces. We prove that $X$ admits a Frobenius splitting which is compatible with all $G$-Schubert varieties. Moreover, when $X$ is smooth, projective and toroidal, then any $G$-Schubert variety in $X$ admits a stable Frobenius splitting along an ample divisors. Although this indicates that $G$-Schubert varieties have nice singularities we present an example of a non-normal $G$-Schubert variety in the wonderful compactification of a group of type $G_2$. Finally we also extend the Frobenius splitting results to the more general class of $\\mathcal R$-Schubert varieties.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-09-10T13:36:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "schubert variety",
      "wonderful compactification",
      "general class",
      "orbit closure",
      "adjoint type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0704.0778H"
    }
  }
}