{
  "id": "math/0511741",
  "version": "v6",
  "published": "2005-11-30T16:00:17.000Z",
  "updated": "2011-07-26T21:54:45.000Z",
  "title": "Complex Hyperbolic Structures on Disc Bundles over Surfaces",
  "authors": [
    "Sasha Anan'in",
    "Carlos H. Grossi",
    "Nikolay Gusevskii"
  ],
  "comment": "52 pages, 12 pictures, 10 tables, 20 references. Changes: final version",
  "journal": "International Mathematics Research Notices (2011) 2011 (19): 4295-4375",
  "doi": "10.1093/imrn/rnq161",
  "categories": [
    "math.GT",
    "math.DG"
  ],
  "abstract": "We study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group as a subgroup of index 2 or 4. The results obtained provide the first complex hyperbolic disc bundles M->{\\Sigma} that: admit both real and complex hyperbolic structures; satisfy the equality 2(\\chi+e)=3\\tau; satisfy the inequality \\chi/2<e; and induce discrete and faithful representations \\pi_1\\Sigma->PU(2,1) with fractional Toledo invariant; where {\\chi} is the Euler characteristic of \\Sigma, e denotes the Euler number of M, and {\\tau} stands for the Toledo invariant of M. To get a satisfactory explanation of the equality 2(\\chi+e)=3\\tau, we conjecture that there exists a holomorphic section in all our examples. In order to reduce the amount of calculations, we systematically explore coordinate-free methods.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2011-07-26T21:54:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57S30",
      "30F35",
      "51M10",
      "57M50"
    ],
    "keywords": [
      "complex hyperbolic structures",
      "study complex hyperbolic disc bundles",
      "first complex hyperbolic disc bundles",
      "fractional toledo invariant"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....11741A"
    }
  }
}