{
  "id": "1904.03906",
  "version": "v1",
  "published": "2019-04-08T09:31:01.000Z",
  "updated": "2019-04-08T09:31:01.000Z",
  "title": "On the moduli space of holomorphic G-connections on a compact Riemann surface",
  "authors": [
    "Indranil Biswas"
  ],
  "comment": "Final version",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X$ be a compact connected Riemann surface of genus at least two and $G$ a connected reductive complex affine algebraic group. The Riemann--Hilbert correspondence produces a biholomorphism between the moduli space ${\\mathcal M}_X(G)$ parametrizing holomorphic $G$--connections on $X$ and the $G$--character variety $${\\mathcal R}(G):= \\text{Hom}(\\pi_1(X, x_0), G)/\\!\\!/G\\, .$$ While ${\\mathcal R}(G)$ is known to be affine, we show that ${\\mathcal M}_X(G)$ is not affine. The scheme ${\\mathcal R}(G)$ has an algebraic symplectic form constructed by Goldman. We construct an algebraic symplectic form on ${\\mathcal M}_X(G)$ with the property that the Riemann--Hilbert correspondence pulls back to the Goldman symplectic form to it. Therefore, despite the Riemann--Hilbert correspondence being non-algebraic, the pullback of the Goldman symplectic form by the Riemann--Hilbert correspondence nevertheless continues to be algebraic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-08T09:31:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compact riemann surface",
      "moduli space",
      "holomorphic g-connections",
      "complex affine algebraic group",
      "riemann-hilbert correspondence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}