{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] and let $p, q \\ge 0$ be integers.\nLet $\\mathcal{A}^{p,q}(X)$ be the complex vector space of smooth [[(p,q)-forms|DC81TCR98Y]] on $X$, with $\\mathcal{A}^{p,-1}(X) = 0$.\nLet $\\bar\\partial \\colon \\mathcal{A}^{p,q}(X) \\to \\mathcal{A}^{p,q+1}(X)$ be the [[dbar operator|M1VKCW3RFW]], which satisfies $\\bar\\partial \\circ \\bar\\partial = 0$.\nThe *Dolbeault cohomology group* of bidegree $(p,q)$ is the quotient vector space\n$$H^{p,q}_{\\bar\\partial}(X) = \\frac{\\ker\\big(\\bar\\partial \\colon \\mathcal{A}^{p,q}(X) \\to \\mathcal{A}^{p,q+1}(X)\\big)}{\\im\\big(\\bar\\partial \\colon \\mathcal{A}^{p,q-1}(X) \\to \\mathcal{A}^{p,q}(X)\\big)}.$$\n\n## Remarks\n\nThe class of a $\\bar\\partial$-closed form $\\alpha$ is written $[\\alpha]$; two closed forms define the same class when they differ by $\\bar\\partial \\beta$.\nExample: on $X = \\C$, every smooth $(0,1)$-form $g\\, d\\bar z$ with compact support is $\\bar\\partial$ of the function $u(z) = \\frac{1}{2\\pi i} \\int_\\C \\frac{g(w)}{w - z}\\, dw \\wedge d\\bar w$, so it gives the zero class in $H^{0,1}_{\\bar\\partial}(\\C)$; in fact $H^{0,1}_{\\bar\\partial}(\\C) = 0$.\nExample: $H^{p,0}_{\\bar\\partial}(X)$ is the space of holomorphic $p$-forms on $X$, since there are no $(p,-1)$-forms to divide by.\nA nonzero class: let $X = \\C/\\Lambda$ with $\\Lambda = \\Z + i\\Z$.\nThe form $d\\bar z$ is $\\bar\\partial$-closed on $X$, but it is not $\\bar\\partial u$ for a smooth function $u$ on $X$.\nIndeed, if it were, the lift $\\tilde u$ of $u$ to $\\C$ would satisfy $\\partial \\tilde u / \\partial \\bar z = 1$, so $h = \\tilde u - \\bar z$ would be entire.\nThen $h(z + 1) = h(z) - 1$ and $h(z + i) = h(z) + i$, so $h'$ is a doubly periodic entire function; it is bounded, hence a constant $a$ by Liouville's theorem.\nThe two relations give $a = -1$ and $a = 1$, a contradiction.\nSo $[d\\bar z] \\ne 0$ in $H^{0,1}_{\\bar\\partial}(X)$.\nThe groups are also written $H^{p,q}(X)$ when no confusion with other cohomologies arises.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "Y3900T2EY5",
    "kind": "definition",
    "msc": [
      "32L10",
      "32C35",
      "58A10"
    ],
    "name": "Dolbeault cohomology",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dolbeault_cohomology"
      }
    ],
    "tag": "6GWRCGBGWC",
    "uses": [
      "JSGWAN2RJV",
      "DC81TCR98Y",
      "M1VKCW3RFW"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/6GWRCGBGWC/",
  "links": [
    {
      "field": "defines",
      "target": "Y3900T2EY5"
    },
    {
      "field": "uses",
      "target": "JSGWAN2RJV"
    },
    {
      "field": "uses",
      "target": "DC81TCR98Y"
    },
    {
      "field": "uses",
      "target": "M1VKCW3RFW"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/6GWRCGBGWC.md",
  "msc": [
    "32L10",
    "32C35",
    "58A10"
  ],
  "name": "Dolbeault cohomology",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6GW/6GWRCGBGWC.md",
  "status": null,
  "tag": "6GWRCGBGWC",
  "uri": "m2://6GWRCGBGWC",
  "url": "https://tamnd.github.io/math2/wiki/dolbeault-cohomology-definition/",
  "writing": {
    "tag": "6GWRCGBGWC",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}