{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "31CV5K5SAJ"
    },
    {
      "field": "about",
      "target": "B2YBWQQMHA"
    },
    {
      "field": "about",
      "target": "THSVFE7YER"
    },
    {
      "field": "uses",
      "target": "31CV5K5SAJ"
    },
    {
      "field": "uses",
      "target": "B2YBWQQMHA"
    },
    {
      "field": "uses",
      "target": "CPC8ZMF1SD"
    },
    {
      "field": "uses",
      "target": "D6JTF3613D"
    },
    {
      "field": "uses",
      "target": "THSVFE7YER"
    },
    {
      "field": "defines",
      "target": "T19H7BJ9Y3"
    }
  ],
  "body": "\n## Summary\n\nA positive current is a [[current|360H7MD9RD]] of bidegree $(p, p)$ on a [[complex manifold|JSGWAN2RJV]] with a positivity property.\nIts wedge product with products of the forms $i \\alpha \\wedge \\bar\\alpha$, for $(1, 0)$-forms $\\alpha$, is a nonnegative measure.\nPierre Lelong introduced closed positive currents, and they are now a central tool of complex geometry.\nThe two basic examples are the current of integration over a complex analytic subvariety and the current $dd^c u$ of a [[plurisubharmonic function|FF2P7ER3QT]] $u$.\nThe coefficients of a positive current are complex measures, and closed positive currents carry [[Lelong numbers|WVB60RKC3Z]] that measure their singularities.\n\n## Explanations\n\n### L1 human\n\nA current is a generalised differential form whose coefficients may be measures or even distributions.\nIn complex geometry there is a natural notion of positivity for forms of type $(p, p)$, which extends the idea of a nonnegative function or of a positive semidefinite Hermitian matrix.\nA positive current is a current that is positive in this sense.\nThe simplest example is integration over a complex curve $C$ in $\\C^2$: it sends a $2$-form to its integral over $C$, and complex curves always carry a positive orientation.\nAnother example is $i \\partial\\bar\\partial \\abs{z}^2$, the standard Kähler form of $\\C^n$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "32U40",
      "32C30"
    ],
    "name": "Positive current",
    "same_as": {
      "wikidata": "Q7233260",
      "wikipedia": "Positive current"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Positive_current"
      }
    ],
    "specializes": [
      "360H7MD9RD"
    ],
    "tag": "BYJHP6A906"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/BYJHP6A906/",
  "links": [
    {
      "field": "specializes",
      "target": "360H7MD9RD"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/BYJHP6A906.md",
  "msc": [
    "32U40",
    "32C30"
  ],
  "name": "Positive current",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BYJ/BYJHP6A906.md",
  "status": null,
  "tag": "BYJHP6A906",
  "uri": "m2://BYJHP6A906",
  "url": "https://tamnd.github.io/math2/wiki/positive-current/",
  "writing": {
    "tag": "BYJHP6A906",
    "kind": "concept",
    "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
      }
    ]
  }
}