{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $U \\subseteq \\C^n$ be an open set, with coordinates $z_j = x_j + i y_j$.\nConsider complex-valued [[differential forms|0E3G0GRPZ6]] on $U$, and put $dz_j = dx_j + i\\, dy_j$ and $d\\bar z_j = dx_j - i\\, dy_j$.\nFor increasing tuples $I = (i_1 \u003c \\cdots \u003c i_p)$ and $J = (j_1 \u003c \\cdots \u003c j_q)$ of elements of $\\set{1, \\dots, n}$, write $dz_I = dz_{i_1} \\wedge \\cdots \\wedge dz_{i_p}$ and $d\\bar z_J = d\\bar z_{j_1} \\wedge \\cdots \\wedge d\\bar z_{j_q}$.\nLet $p, q \\ge 0$ be integers.\nA *$(p, q)$-form* on $U$ is a differential form\n$$\\omega = \\sum_{\\abs{I} = p, \\ \\abs{J} = q} \\omega_{IJ} \\, dz_I \\wedge d\\bar z_J,$$\nwhere the sum runs over increasing tuples and each $\\omega_{IJ} \\colon U \\to \\C$ is a [[smooth function|0V33BNKZ9S]].\n\n## Remarks\n\nThe forms $dz_I \\wedge d\\bar z_J$ with $\\abs{I} + \\abs{J} = k$ form a basis of the complex $k$-th [[exterior power|0MDMZ965VE]] of the cotangent space at each point.\nSo every complex $k$-form is, in a unique way, a sum of $(p, q)$-forms with $p + q = k$.\nExample: $\\omega = \\bar z_1 \\, dz_1 \\wedge d\\bar z_2$ on $\\C^2$ is a $(1, 1)$-form, with $\\omega_{(1)(2)} = \\bar z_1$ and all other coefficients zero.\nNon-example: $dx_1 = \\tfrac12 dz_1 + \\tfrac12 d\\bar z_1$ is a $1$-form whose $(1, 0)$ and $(0, 1)$ parts are both nonzero, so it is not a $(p, q)$-form for any $(p, q)$.\nA holomorphic change of coordinates $w = F(z)$ gives $dw_k = \\sum_j \\frac{\\partial F_k}{\\partial z_j} dz_j$ with no $d\\bar z_j$ terms, so the type $(p, q)$ is preserved; thus $(p, q)$-forms are defined on complex manifolds.\nThe space of $(p, q)$-forms on $U$ is often written $\\mathcal{E}^{p, q}(U)$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "DC81TCR98Y",
    "kind": "definition",
    "msc": [
      "32W05",
      "58A10"
    ],
    "name": "(p,q)-form",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Complex_differential_form"
      }
    ],
    "tag": "P3PJGESDSF",
    "uses": [
      "0E3G0GRPZ6",
      "0MDMZ965VE",
      "0V33BNKZ9S"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/P3PJGESDSF/",
  "links": [
    {
      "field": "defines",
      "target": "DC81TCR98Y"
    },
    {
      "field": "uses",
      "target": "0E3G0GRPZ6"
    },
    {
      "field": "uses",
      "target": "0MDMZ965VE"
    },
    {
      "field": "uses",
      "target": "0V33BNKZ9S"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/P3PJGESDSF.md",
  "msc": [
    "32W05",
    "58A10"
  ],
  "name": "(p,q)-form",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/P3P/P3PJGESDSF.md",
  "status": null,
  "tag": "P3PJGESDSF",
  "uri": "m2://P3PJGESDSF",
  "url": "https://tamnd.github.io/math2/wiki/p-q-form-definition/",
  "writing": {
    "tag": "P3PJGESDSF",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}