{
  "id": "1805.11558",
  "version": "v1",
  "published": "2018-05-29T16:07:57.000Z",
  "updated": "2018-05-29T16:07:57.000Z",
  "title": "Białynicki-Birula decomposition for reductive groups",
  "authors": [
    "Joachim Jelisiejew",
    "Łukasz Sienkiewicz"
  ],
  "comment": "30 pages",
  "categories": [
    "math.AG",
    "math.RT"
  ],
  "abstract": "We generalize the Bia{\\l}ynicki-Birula decomposition from actions of $G_m$ on smooth varieties to actions of linearly reductive group ${\\bf G}$ on finite type schemes and algebraic spaces. We also provide a relative version and briefly discuss the case of algebraic stacks. We define the Bia{\\l}ynicki-Birula decomposition functorially: for a fixed ${\\bf G}$-scheme $X$ and a monoid $\\overline{\\bf G}$ which compactifies ${\\bf G}$, the BB decomposition parameterizes ${\\bf G}$-schemes over $X$ for which the ${\\bf G}$-action extends to the $\\overline{\\bf G}$-action. The freedom of choice of $\\overline{\\bf G}$ makes the theory richer than the $G_m$-case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-29T16:07:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14L30",
      "20G15",
      "14D22",
      "20G05"
    ],
    "keywords": [
      "reductive group",
      "białynicki-birula decomposition",
      "finite type schemes",
      "bb decomposition parameterizes",
      "action extends"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}