{
  "id": "2002.01797",
  "version": "v1",
  "published": "2020-02-05T14:07:26.000Z",
  "updated": "2020-02-05T14:07:26.000Z",
  "title": "The $\\bar\\partial$-equation for $(p,q)$-forms on a non-reduced analytic space",
  "authors": [
    "Mats Andersson",
    "Richard Lärkäng",
    "Mattias Lennartsson",
    "Håkan Samuelsson Kalm"
  ],
  "categories": [
    "math.CV"
  ],
  "abstract": "On any pure $n$-dimensional, possibly non-reduced, analytic space $X$ we introduce the sheaves $\\mathscr{E}_X^{p,q}$ of smooth $(p,q)$-forms and certain extensions $\\mathscr{A}_X^{p,q}$ of them such that the corresponding Dolbeault complex is exact, i.e., the $\\bar\\partial$-equation is locally solvable in $\\mathscr{A}_X$. The sheaves $\\mathscr{A}_X^{p,q}$ are modules over the smooth forms, in particular, they are fine sheaves. We also introduce certain sheaves $\\mathscr{B}_X^{n-p,n-q}$ of currents on $X$ that are dual to $\\mathscr{A}_X^{p,q}$ in the sense of Serre duality. More precisely, we show that the compactly supported Dolbeault cohomology of $\\mathscr{B}^{n-p,n-q}(X)$ in a natural way is the dual of the Dolbeault cohomology of $\\mathscr{A}^{p,q}(X)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-05T14:07:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-reduced analytic space",
      "corresponding dolbeault complex",
      "smooth forms",
      "fine sheaves",
      "serre duality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}