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  "body": "\n## Summary\n\nThe multiplier ideal of a [[plurisubharmonic function|FF2P7ER3QT]] $\\varphi$ is the sheaf of holomorphic germs $f$ for which $\\abs{f}^2 e^{-2\\varphi}$ is locally integrable.\nIt measures the singularities of $\\varphi$: the more singular $\\varphi$ is, the smaller the ideal.\nNadel proved that it is a [[coherent analytic sheaf|DSJET02WPR]].\nThe Nadel vanishing theorem, a consequence of [[Hörmander's L2 estimates|TPSH4ETK67]], makes it a basic tool in complex and algebraic geometry.\nIn algebraic geometry, the multiplier ideal of an ideal sheaf with a rational exponent is defined through a log resolution.\nIt agrees with the analytic definition.\n\n## Explanations\n\n### L1 human\n\nTake a function $\\varphi$ that tends to $-\\infty$ at some points, such as $\\varphi = c \\log\\abs{z}$ near $0$ in $\\C$.\nThen $e^{-2\\varphi} = \\abs{z}^{-2c}$ blows up at $0$, and may fail to be integrable.\nMultiplying by a holomorphic $f$ that vanishes at $0$ can tame the blow-up.\nThe multiplier ideal is the set of such \"multipliers\" $f$ that make $\\abs{f}^2 e^{-2\\varphi}$ integrable.\nFor $\\varphi = c \\log\\abs{z}$ with $c \u003e 0$, these are the germs that vanish to order at least $\\floor{c}$ at $0$.\n",
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