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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\C^n$ be open, let $A \\subseteq \\Omega$ be an [[analytic subset|GB7M582M5Y]] of pure [[dimension|AN0BYRKNRA]] $p \\ge 1$ and let $x \\in A$.\nLet $\\pi \\colon \\C^n \\to \\C^p$ be a generic complex linear map.\nThen the restriction of $\\pi$ to $A$ near $x$ is a branched covering of a neighbourhood of $\\pi(x)$.\nLet $m_x(A)$ be its number of sheets, the multiplicity of $A$ at $x$.\nLet $\\operatorname{vol}_{2p}$ denote the $2p$-dimensional volume of the set of [[regular points|ZW5A88PCPK]] of $A$.\nThen\n$$\\lim_{r \\to 0^+} \\frac{\\operatorname{vol}_{2p}\\bigl(A \\cap B(x, r)\\bigr)}{\\pi^p r^{2p} / p!} = m_x(A).$$\nIn other words, the [[Lelong number|WVB60RKC3Z]] at $x$ of the [[current|360H7MD9RD]] of integration $[A]$ equals $m_x(A)$.\n\n## Remarks\n\nThe denominator $\\pi^p r^{2p} / p!$ is the volume of a ball of radius $r$ in $\\C^p = \\R^{2p}$.\nThe proof uses a generic linear projection $\\pi$, under which $A$ covers a neighbourhood of $\\pi(x)$ with $m_x(A)$ sheets.\nIt then compares, as $r \\to 0$, the volume of the part of each sheet near $x$ with the volume of its image in $\\C^p$.\n\nExample: $A = \\set{z_1 z_2 = 0} \\subseteq \\C^2$ is the union of two lines through $0$.\nThe set $A \\cap B(0, r)$ is two discs of area $\\pi r^2$, so the ratio is $2$, and indeed a generic projection to $\\C$ has two sheets.\nAt a regular point the multiplicity is $1$, and the ratio tends to $1$.\n\n## Explanations\n\n### L1 human\n\nNear a singular point, an analytic set may look like several sheets passing through the same point.\nThe multiplicity counts these sheets.\nThe theorem says that one can count them by measuring area.\nThe area of the set inside a small ball is about the number of sheets times the area of a flat disc of the same radius.\n\n### L2 human\n\nThe theorem links an algebraic invariant, the multiplicity of the local ring of $A$ at $x$, with a metric invariant, the density of $[A]$.\nIt is the reason why the points of multiplicity at least $c$ appear as upper level sets of Lelong numbers.\nIt is also the model case for Lelong numbers of general closed positive currents.\n",
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