{
  "id": "0812.2643",
  "version": "v2",
  "published": "2008-12-15T10:26:44.000Z",
  "updated": "2009-12-22T11:52:09.000Z",
  "title": "Compact Kaehler quotients of algebraic varieties and Geometric Invariant Theory",
  "authors": [
    "Daniel Greb"
  ],
  "comment": "33 pages, 1 figure; improved exposition, many of the results are now proven for complete and not only for projective quotients, examples showing the necessity of the assumptions made in the main results added; to appear in Advances in Mathematics",
  "journal": "Adv. Math. 224 (2010), no. 2, 401-431",
  "doi": "10.1016/j.aim.2009.11.013",
  "categories": [
    "math.AG",
    "math.SG"
  ],
  "abstract": "Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kaehler quotient. Additionally, as a byproduct of our discussion we give an example of a complete Kaehlerian non-projective algebraic surface, which may be of independent interest.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-12-22T11:52:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14L30",
      "14L24",
      "32M05",
      "53D20",
      "53C55"
    ],
    "keywords": [
      "compact kaehler quotient",
      "geometric invariant theory",
      "algebraic varieties",
      "compact momentum map quotients",
      "complete kaehlerian non-projective algebraic surface"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier",
      "journal": "Adv. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0812.2643G"
    }
  }
}