{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] of dimension $n$ and let $0 \\le p \\le n$.\nA [[current|360H7MD9RD]] $T$ of bidegree $(p, p)$ on $X$ is *positive* if\n$$T \\wedge \\left(i \\alpha_1 \\wedge \\bar\\alpha_1\\right) \\wedge \\cdots \\wedge \\left(i \\alpha_{n-p} \\wedge \\bar\\alpha_{n-p}\\right) \\ge 0$$\nfor every choice of smooth [[$(1, 0)$-forms|DC81TCR98Y]] $\\alpha_1, \\dots, \\alpha_{n-p}$ on $X$.\nHere an $(n, n)$-current $S$ is identified with a distribution, and $S \\ge 0$ means $S(\\chi) \\ge 0$ for every smooth compactly supported function $\\chi \\ge 0$.\nThen $S$ is a positive measure.\nA current is *closed* if $dT = 0$ in the sense of currents.\n\n## Remarks\n\nThe condition is the same as $T(\\varphi) \\ge 0$ for all compactly supported smooth strongly positive $(n - p, n - p)$-forms $\\varphi$, with the current acting on forms of complementary degree.\nA positive current is real, and its coefficients in local coordinates are complex measures.\nExample: for $X = \\C^n$ and $p = 1$, the smooth form $\\omega = i \\partial\\bar\\partial \\abs{z}^2 = i \\sum_j dz_j \\wedge d\\bar z_j$ is positive, because its coefficient matrix is the identity matrix (see the criterion for $p = 1$ below).\nExample: let $Z \\subseteq X$ be a complex analytic subset of pure dimension $n - p$.\nBy Lelong's theorem, the current of integration $[Z](\\varphi) = \\int_{Z_{\\mathrm{reg}}} \\varphi$ is a closed positive current of bidegree $(p, p)$.\nExample: for $u$ [[plurisubharmonic|FF2P7ER3QT]], $i \\partial\\bar\\partial u$ is a closed positive $(1, 1)$-current.\nNon-example: on $\\C^2$, the form $i \\, dz_1 \\wedge d\\bar z_1 - i \\, dz_2 \\wedge d\\bar z_2$ is not positive; take $\\alpha_1 = dz_1$, then the product is $-i\\,dz_2 \\wedge d\\bar z_2 \\wedge i \\, dz_1 \\wedge d\\bar z_1$, which is $-4$ times Lebesgue measure.\nFor $p = 1$, a $(1,1)$-form $i \\sum h_{jk} \\, dz_j \\wedge d\\bar z_k$ is positive exactly when the Hermitian matrix $(h_{jk})$ is positive semidefinite at each point.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "BYJHP6A906",
    "kind": "definition",
    "msc": [
      "32U40",
      "32C30"
    ],
    "name": "Positive current",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Positive_current"
      }
    ],
    "tag": "T19H7BJ9Y3",
    "uses": [
      "360H7MD9RD",
      "DC81TCR98Y",
      "JSGWAN2RJV"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/T19H7BJ9Y3/",
  "links": [
    {
      "field": "defines",
      "target": "BYJHP6A906"
    },
    {
      "field": "uses",
      "target": "360H7MD9RD"
    },
    {
      "field": "uses",
      "target": "DC81TCR98Y"
    },
    {
      "field": "uses",
      "target": "JSGWAN2RJV"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/T19H7BJ9Y3.md",
  "msc": [
    "32U40",
    "32C30"
  ],
  "name": "Positive current",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/T19/T19H7BJ9Y3.md",
  "status": null,
  "tag": "T19H7BJ9Y3",
  "uri": "m2://T19H7BJ9Y3",
  "url": "https://tamnd.github.io/math2/wiki/positive-current-definition/",
  "writing": {
    "tag": "T19H7BJ9Y3",
    "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
      }
    ]
  }
}