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  "body": "\n## Definition\n\nLet $X$ be a complex manifold.\nLet $\\varphi$ be a [[plurisubharmonic function|FF2P7ER3QT]] on $X$, or more generally a function that is locally a plurisubharmonic function plus a smooth function.\nThe *multiplier ideal sheaf* $\\mathcal{I}(\\varphi) \\subseteq \\mathcal{O}_X$ is the subsheaf of $\\mathcal{O}_X$ defined as follows.\nIts stalk at $x \\in X$ is the set of [[germs|7PJ8KVZZCQ]] $f \\in \\mathcal{O}_{X,x}$ that satisfy the following condition.\nThere is a coordinate neighbourhood $U$ of $x$ with $\\int_U \\abs{f}^2 e^{-2\\varphi} \\, dV \u003c \\infty$, where $dV$ is Lebesgue measure in the coordinates of $U$.\n\n## Remarks\n\nThe condition does not depend on the coordinates, since a change of coordinates multiplies $dV$ by a smooth positive function.\nThe stalks are ideals, so $\\mathcal{I}(\\varphi)$ is an [[ideal sheaf|N3PGYJJQ7W]]; Nadel's theorem says it is coherent.\nExample: $X = \\C$, $\\varphi = c \\log\\abs{z}$ with $c \u003e 0$.\nIn polar coordinates, $\\int_{\\abs{z} \u003c r} \\abs{z}^{2k} \\abs{z}^{-2c} \\, dV = 2\\pi \\int_0^r t^{2k - 2c + 1} \\, dt$, which is finite exactly when $k \u003e c - 1$.\nSo the stalk at $0$ is $(z^{\\floor{c}})$; for example $\\mathcal{I}(\\log\\abs{z})_0 = (z)$.\nNon-example: for $\\varphi = \\log\\abs{z}$, the germ $f = 1$ is not in the stalk at $0$, since $\\int \\abs{z}^{-2} \\, dV$ diverges near $0$.\nIf $\\varphi$ is locally bounded, then $\\mathcal{I}(\\varphi) = \\mathcal{O}_X$.\nAlgebraic form: for an ideal $\\mathfrak{a} = (g_1, \\dots, g_k)$ and a real $c \u003e 0$, $\\mathcal{J}(c \\cdot \\mathfrak{a}) = \\mathcal{I}\\left(\\tfrac{c}{2} \\log \\sum_i \\abs{g_i}^2\\right)$, the germs with $\\abs{f}^2 / (\\sum_i \\abs{g_i}^2)^c$ locally integrable.\nSome authors use the weight $e^{-\\varphi}$ instead of $e^{-2\\varphi}$.\n",
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