{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] of dimension $n$ and let $0 \\le p \\le n$.\nLet $T$ be a closed [[positive current|BYJHP6A906]] on $X$ of bidimension $(p, p)$, that is, of bidegree $(n - p, n - p)$.\nFor $x \\in X$ let $\\nu(T, x)$ be the [[Lelong number|WVB60RKC3Z]] of $T$ at $x$.\nThen for every real $c \u003e 0$ the set\n$$E_c(T) = \\set{x \\in X : \\nu(T, x) \\ge c}$$\nis a closed [[analytic subset|GB7M582M5Y]] of $X$ of [[dimension|AN0BYRKNRA]] at most $p$.\n\n## Remarks\n\nThat $E_c(T)$ is closed is elementary, since $x \\mapsto \\nu(T, x)$ is upper semicontinuous.\nThe content is that $E_c(T)$ is analytic.\nSiu's original proof was long; a shorter proof by Demailly writes $E_c(T)$ locally as a common zero set of holomorphic functions built with [[Hörmander's L2 estimates|TPSH4ETK67]].\n\nExample: let $A \\subseteq X$ be an analytic subset of pure dimension $p$ and $T = [A]$ its current of integration.\nBy [[Thie's theorem|G5JKJEAKZ9]], $\\nu([A], x)$ is the multiplicity of $A$ at $x$, so $E_c([A])$ is the set of points of multiplicity at least $c$, an analytic set.\n\nFor a [[plurisubharmonic function|FF2P7ER3QT]] $u$ on an open set of $\\C^n$, apply the theorem to $T = dd^c u$ with $p = n - 1$: the sets $\\set{x : \\nu(u, x) \\ge c}$ are analytic of dimension at most $n - 1$.\n\n## Explanations\n\n### L1 human\n\nThe Lelong number measures how singular a current is at a point.\nThe theorem says that the points where this singularity is at least a given size are not scattered at random.\nThey form an analytic set, cut out locally by holomorphic equations.\n\n### L2 human\n\nThe theorem is the starting point of Siu's decomposition of a closed positive current.\nIt writes the current as a countable sum of currents of integration over analytic sets plus a residual part.\nFor the residual part, the upper level sets of Lelong numbers have smaller dimension.\nIt is used with multiplier ideals and with regularization of currents in the study of positivity of line bundles.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "WVB60RKC3Z",
      "BYJHP6A906"
    ],
    "kind": "statement",
    "msc": [
      "32U25",
      "32U40"
    ],
    "name": "Siu's semicontinuity theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lelong_number"
      }
    ],
    "tag": "THSVFE7YER",
    "uses": [
      "WVB60RKC3Z",
      "BYJHP6A906",
      "JSGWAN2RJV",
      "GB7M582M5Y",
      "AN0BYRKNRA",
      "FF2P7ER3QT",
      "TPSH4ETK67",
      "G5JKJEAKZ9"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/THSVFE7YER/",
  "links": [
    {
      "field": "about",
      "target": "WVB60RKC3Z"
    },
    {
      "field": "about",
      "target": "BYJHP6A906"
    },
    {
      "field": "uses",
      "target": "WVB60RKC3Z"
    },
    {
      "field": "uses",
      "target": "BYJHP6A906"
    },
    {
      "field": "uses",
      "target": "JSGWAN2RJV"
    },
    {
      "field": "uses",
      "target": "GB7M582M5Y"
    },
    {
      "field": "uses",
      "target": "AN0BYRKNRA"
    },
    {
      "field": "uses",
      "target": "FF2P7ER3QT"
    },
    {
      "field": "uses",
      "target": "TPSH4ETK67"
    },
    {
      "field": "uses",
      "target": "G5JKJEAKZ9"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/THSVFE7YER.md",
  "msc": [
    "32U25",
    "32U40"
  ],
  "name": "Siu's semicontinuity theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/THS/THSVFE7YER.md",
  "status": null,
  "tag": "THSVFE7YER",
  "uri": "m2://THSVFE7YER",
  "url": "https://tamnd.github.io/math2/wiki/sius-semicontinuity-theorem/",
  "writing": {
    "tag": "THSVFE7YER",
    "kind": "statement",
    "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
      }
    ]
  }
}