{
  "id": "1309.0951",
  "version": "v2",
  "published": "2013-09-04T09:33:32.000Z",
  "updated": "2014-01-06T10:07:15.000Z",
  "title": "Torus bundles and 2-forms on the universal family of Riemann surfaces",
  "authors": [
    "Robin de Jong"
  ],
  "comment": "27 pages; minor revisions",
  "categories": [
    "math.GT",
    "math.AG"
  ],
  "abstract": "We revisit three results due to Morita expressing certain natural integral cohomology classes on the universal family of Riemann surfaces C_g, coming from the parallel symplectic form on the universal jacobian, in terms of the Miller-Morita-Mumford classes e and e_1. Our discussion will be on the level of the natural 2-forms representing the relevant cohomology classes, and involves a comparison with other natural 2-forms representing e, e_1 induced by the Arakelov metric on the relative tangent bundle of C_g over M_g. A secondary object called a_g occurs, which was discovered and studied by Kawazumi around 2008. We present alternative proofs of Kawazumi's (unpublished) results on the second variation of a_g on M_g. Also we review some results that were previously obtained on the invariant a_g, with a focus on its connection with Faltings's delta-invariant and Hain-Reed's beta-invariant.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-01-06T10:07:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32G15",
      "14H15"
    ],
    "keywords": [
      "riemann surfaces",
      "torus bundles",
      "universal family",
      "natural integral cohomology classes",
      "parallel symplectic form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1253179,
      "adsabs": "2013arXiv1309.0951D"
    }
  }
}