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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\C^n$ be open, let $u$ be a [[plurisubharmonic function|FF2P7ER3QT]] on $\\Omega$ and let $x \\in \\Omega$.\nThe *Lelong number* of $u$ at $x$ is\n$$\\nu(u, x) = \\lim_{r \\to 0^+} \\frac{\\sup_{\\abs{z - x} \u003c r} u(z)}{\\log r}.$$\nHere $\\abs{\\cdot}$ is the Euclidean norm on $\\C^n$.\n\n## Remarks\n\nThe limit exists and lies in $[0, +\\infty)$, because $\\sup_{\\abs{z - x} \u003c r} u(z)$ is a convex increasing function of $\\log r$.\nAn equivalent form is $\\nu(u, x) = \\liminf_{z \\to x} u(z) / \\log \\abs{z - x}$.\n\nExample: let $f$ be holomorphic on $\\Omega$ with order of vanishing $m$ at $x$, and $u = \\log \\abs{f}$.\nNear $x$ one has $c_1 r^m \\le \\sup_{\\abs{z - x} \u003c r} \\abs{f(z)} \\le c_2 r^m$ for constants $c_1, c_2 \u003e 0$, so $\\nu(u, x) = m$.\nIn particular $\\nu(\\log \\abs{z_1}, 0) = 1$.\n\nIf $u(x) \u003e -\\infty$, then $\\nu(u, x) = 0$.\nThe converse fails.\nOn the unit ball of $\\C^n$ let $u(z) = -\\sqrt{-\\log \\abs{z}}$, a plurisubharmonic function because $t \\mapsto -\\sqrt{-t}$ is convex and increasing on $(-\\infty, 0)$.\nThen $u(0) = -\\infty$, but $\\sup_{\\abs{z} \u003c r} u(z) / \\log r = 1 / \\sqrt{-\\log r} \\to 0$, so $\\nu(u, 0) = 0$.\n\nLet $T$ be a closed [[positive current|BYJHP6A906]] of bidimension $(p, p)$.\nIts Lelong number $\\nu(T, x)$ is the limit as $r \\to 0$ of the trace mass of $T$ on the ball $B(x, r)$, divided by $\\pi^p r^{2p} / p!$.\nThe divisor $\\pi^p r^{2p} / p!$ is the volume of a ball of radius $r$ in $\\C^p$.\nWith a suitable normalisation of $dd^c$, one has $\\nu(u, x) = \\nu(dd^c u, x)$.\n",
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    "defines": "WVB60RKC3Z",
    "kind": "definition",
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    "name": "Lelong number",
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        "url": "https://en.wikipedia.org/wiki/Lelong_number"
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  "name": "Lelong number",
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