{
  "id": "2203.06854",
  "version": "v1",
  "published": "2022-03-14T04:51:05.000Z",
  "updated": "2022-03-14T04:51:05.000Z",
  "title": "Line bundles on the moduli space of parabolic connections over a compact Riemann surface",
  "authors": [
    "Anoop Singh"
  ],
  "comment": "25 pages, accepted for publication in 'Advances in Mathematics'",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X$ be a compact Riemann surface of genus $g \\geq 3$ and $S$ a finite subset of $X$. Let $\\xi$ be fixed a holomorphic line bundle over $X$ of degree $d$. Let $\\mathcal{M}_{pc}(r, d, \\alpha)$ (respectively, $\\mathcal{M}_{pc}(r, \\alpha, \\xi)$ ) denote the moduli space of parabolic connections of rank $r$, degree $d$ and full flag rational generic weight system $\\alpha$, (respectively, with the fixed determinant $\\xi$) singular over the parabolic points $S \\subset X$. Let $\\mathcal{M}'_{pc}(r, d, \\alpha)$ (respectively, $\\mathcal{M}'_{pc}(r, \\alpha, \\xi)$) be the Zariski dense open subset of $\\mathcal{M}_{pc}(r, d, \\alpha)$ (respectively, $\\mathcal{M}_{pc}(r, \\alpha, \\xi)$ )parametrizing all parabolic connections such that the underlying parabolic bundle is stable. We show that there is a natural compactification of the moduli spaces $\\mathcal{M}'_{pc}(r, d, \\alpha)$, and $\\mathcal{M}'_{pc}(r, \\alpha, \\xi)$ by smooth divisors. We describe the numerically effectiveness of these divisors at infinity. We determine the Picard group of the moduli spaces $\\mathcal{M}_{pc}(r, d, \\alpha)$, and $\\mathcal{M}_{pc}(r, \\alpha, \\xi)$. Let $\\mathcal{C}(L)$ denote the space of holomorphic connections on an ample line bundle $L$ over the moduli space $\\mathcal{M}(r, d, \\alpha)$ of parabolic bundles. We show that $\\mathcal{C}(L)$ does not admit any non-constant algebraic function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-14T04:51:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14D20",
      "14C22",
      "14H05"
    ],
    "keywords": [
      "moduli space",
      "compact riemann surface",
      "parabolic connections",
      "line bundle",
      "full flag rational generic weight"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}