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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\C^n$ be a bounded [[pseudoconvex|AHYZMSBBNH]] domain and let $\\varphi$ be a [[plurisubharmonic function|FF2P7ER3QT]] on $\\Omega$, not identically $-\\infty$.\nFor each integer $m \\ge 1$, let $\\mathcal{H}_m$ be the Hilbert space of holomorphic functions $f$ on $\\Omega$ with $\\int_\\Omega \\abs{f}^2 e^{-2m\\varphi} \\, dV \u003c \\infty$.\nLet $(\\sigma_{m,l})_{l \\ge 1}$ be an orthonormal basis of $\\mathcal{H}_m$, and let\n$$\\varphi_m = \\frac{1}{2m} \\log \\sum_{l} \\abs{\\sigma_{m,l}}^2 .$$\nThen there are constants $C_1, C_2 \u003e 0$, independent of $m$, such that for all $z \\in \\Omega$ and all $r$ with $0 \u003c r \u003c \\operatorname{dist}(z, \\partial\\Omega)$:\n\n1. $\\varphi(z) - \\dfrac{C_1}{m} \\le \\varphi_m(z) \\le \\sup_{\\abs{\\zeta - z} \u003c r} \\varphi(\\zeta) + \\dfrac{1}{m} \\log \\dfrac{C_2}{r^n}$;\n2. $\\nu(\\varphi, z) - \\dfrac{n}{m} \\le \\nu(\\varphi_m, z) \\le \\nu(\\varphi, z)$, where $\\nu$ denotes the [[Lelong number|WVB60RKC3Z]].\n\nIn particular $\\varphi_m$ converges to $\\varphi$ pointwise and in $L^1_{\\mathrm{loc}}(\\Omega)$ as $m \\to \\infty$.\n\n## Remarks\n\nEach $\\varphi_m$ is plurisubharmonic and has analytic singularities: locally it is $\\frac{1}{2m} \\log \\sum \\abs{g_j}^2$ plus a bounded function, for finitely many holomorphic $g_j$, which generate the [[multiplier ideal|BKM0TQSCD9]] $\\mathcal{I}(m\\varphi)$.\nProof idea: $\\varphi_m(z) = \\frac{1}{m} \\sup\\set{\\log\\abs{f(z)} : f \\in \\mathcal{H}_m, \\norm{f} \\le 1}$, so it is the logarithm of a weighted Bergman kernel on the diagonal.\nThe upper bound follows from the mean value inequality for $\\abs{f}^2$ on the ball of radius $r$ around $z$.\nThe lower bound follows from the [[Ohsawa–Takegoshi extension theorem|2SRQCEB16J]], which gives $f \\in \\mathcal{H}_m$ with $f(z) = e^{m\\varphi(z)}$ and $\\norm{f}$ bounded independently of $z$ and $m$.\nThe Lelong number inequalities follow from these bounds.\nExample: for $\\varphi = \\log\\abs{z}$ on the unit disc of $\\C$, $\\nu(\\varphi, 0) = 1$, and the bounds force $1 - \\frac{1}{m} \\le \\nu(\\varphi_m, 0) \\le 1$.\nThe method can be globalised with a partition of unity.\nOn a compact complex manifold, it approximates a closed [[positive current|BYJHP6A906]] of bidegree $(1, 1)$ by closed currents in the same cohomology class.\nThe approximating currents have analytic singularities, lose only a small amount of positivity, and have Lelong numbers converging to those of the current.\n\n## Explanations\n\n### L1 human\n\nPlurisubharmonic functions can have very complicated singularities.\nThe theorem replaces such a function by a sequence of simpler ones, built from logarithms of sums of squares of holomorphic functions.\nThe simpler functions converge to the original one, and their singularities converge too, measured by Lelong numbers.\n\n### L2 human\n\nThis is the main regularisation tool of analytic methods in algebraic geometry.\nFunctions with analytic singularities can be resolved by blowing up, so statements about them reduce to divisors with normal crossings.\nThe global version for closed positive $(1,1)$-currents is used to study pseudo-effective and nef line bundles.\nIt also gives a proof of Siu's theorem that the upper level sets of Lelong numbers are analytic.\nIt relies on the $L^2$ theory of $\\bar\\partial$, through the Ohsawa–Takegoshi theorem.\n",
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  "name": "Demailly's regularization of currents",
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