{
  "id": "0902.2589",
  "version": "v3",
  "published": "2009-02-16T00:50:08.000Z",
  "updated": "2010-08-25T11:53:30.000Z",
  "title": "Character Varieties",
  "authors": [
    "Adam S. Sikora"
  ],
  "comment": "34 pages, to appear in Transactions of AMS",
  "categories": [
    "math.RT",
    "math.GT",
    "math.SG"
  ],
  "abstract": "We study properties of irreducible and completely reducible representations of finitely generated groups Gamma into reductive algebraic groups G in in the context of the geometric invariant theory of the G-action on Hom(Gamma,G) by conjugation. In particular, we study properties of character varieties, X_G(Gamma)=Hom(Gamma,G)//G. We describe the tangent spaces to X_G(Gamma) in terms of first cohomology groups of Gamma with twisted coefficients, generalizing the well known formula. Let M be an orientable 3-manifold with a connected boundary F of genus > 1 and let X_G^g(F) be the subset of the G -character variety of F composed of conjugacy classes of good representations. By a theorem of Goldman, X_G^g(F) is a holomorphic symplectic manifold. We prove that the set of good G-representations of pi_1(F) which extend to representations of pi_1(M) is an isotropic submanifold of X_G^g(F). If these representations correspond to reduced points of the G-character variety of M then this submanifold is Lagrangian.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-08-25T11:53:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14D20",
      "14L24",
      "57M50",
      "57M27"
    ],
    "keywords": [
      "representations",
      "study properties",
      "geometric invariant theory",
      "first cohomology groups",
      "holomorphic symplectic manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.2589S"
    }
  }
}