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  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]], for example a domain in $\\C^n$.\nA *plurisubharmonic exhaustion function* on $X$ is a function $\\varphi \\colon X \\to [-\\infty, \\infty)$ such that:\n\n1. $\\varphi$ is [[plurisubharmonic|FF2P7ER3QT]] on $X$.\n2. $\\varphi$ is an exhaustion: for every $c \\in \\R$, the set $\\set{x \\in X : \\varphi(x) \u003c c}$ is relatively compact in $X$, that is, its closure in $X$ is [[compact|0R6EWFKRCF]].\n\nIt is *smooth and strictly plurisubharmonic* if moreover $\\varphi$ is $C^\\infty$ and its [[Levi form|80MTNXWW80]] is positive definite at every point, in local holomorphic coordinates.\n\n## Remarks\n\nExample: on $\\C^n$, $\\varphi(z) = \\abs{z}^2$ is a smooth strictly plurisubharmonic exhaustion function.\nIts Levi form is $\\abs{w}^2$, and its sublevel sets are open balls.\nNon-example: on the unit ball $B \\subseteq \\C^n$, $\\varphi(z) = \\abs{z}^2$ is still strictly plurisubharmonic but is not an exhaustion, since $\\set{\\varphi \u003c 2} = B$ is not relatively compact in $B$.\nOn $B$, the function $-\\log(1 - \\abs{z}^2)$ is a smooth strictly plurisubharmonic exhaustion function.\nFor a domain $\\Omega \\subsetneq \\C^n$, an equivalent form of clause 2 for continuous $\\varphi$ is that $\\varphi(z) \\to +\\infty$ as $z$ tends to $\\partial\\Omega$ or, when $\\Omega$ is unbounded, as $\\abs{z} \\to \\infty$.\n",
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