{
  "id": "math/9910037",
  "version": "v2",
  "published": "1999-10-06T23:28:39.000Z",
  "updated": "1999-12-21T23:45:07.000Z",
  "title": "The Moduli of Flat U(p,1) Structures on Riemann Surfaces",
  "authors": [
    "Eugene Z. Xia"
  ],
  "comment": "12 pages. The revised version corrects a technical mistake in the previous version in section 4.1",
  "categories": [
    "math.AG"
  ],
  "abstract": "For a compact Riemann surface $X$ of genus $g > 1$, $\\Hom(\\pi_1(X), U(p,1))/U(p,1)$ is the moduli space of flat $\\U(p,1)$-connections on $X$. There is an integer invariant, $\\tau$, the Toledo invariant associated with each element in $\\Hom(\\pi_1(X), U(p,1))/U(p,1)$. If $q = 1$, then $-2(g-1) \\le \\tau \\le 2(g-1)$. This paper shows that $\\Hom(\\pi_1(X), U(p,1))/U(p,1)$ has one connected component corresponding to each $\\tau \\in 2Z$ with $-2(g-1) \\le \\tau \\le 2(g-1)$. Therefore the total number of connected components is $2(g-1) + 1$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "1999-12-21T23:45:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14D20",
      "14H60"
    ],
    "keywords": [
      "structures",
      "compact riemann surface",
      "connected component",
      "total number",
      "integer invariant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1999math.....10037X"
    }
  }
}