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  "body": "\n## Statement\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] that is second countable.\nThen $X$ is a [[Stein manifold|HMCD0JBFST]] if and only if $X$ has a smooth strictly [[plurisubharmonic exhaustion function|FGSKR6ZS3Q]].\n\n## Remarks\n\nThe direction from Stein to the exhaustion function is easy.\nA Stein manifold has a proper holomorphic embedding $F \\colon X \\to \\C^N$, and then $\\varphi = \\abs{F}^2$ is smooth, strictly plurisubharmonic, and an exhaustion because $F$ is proper.\nGrauert proved the converse in 1958.\nHis proof enlarges the sublevel sets $\\set{\\varphi \u003c c}$ step by step and uses a finiteness theorem for cohomology on strongly pseudoconvex domains.\nA later proof by Hörmander uses $L^2$ estimates for $\\bar\\partial$.\nExample: the unit ball $B \\subseteq \\C^n$ is Stein, since $-\\log(1 - \\abs{z}^2)$ is a smooth strictly plurisubharmonic exhaustion function.\nStrictness is needed: on a compact complex manifold $X$ of positive dimension, every constant is a smooth plurisubharmonic exhaustion function, yet $X$ is not Stein.\n\n## Explanations\n\n### L1 human\n\nA Stein manifold is a complex manifold with very many global holomorphic functions.\nThe theorem says that this is the same as having a function that behaves like $\\abs{z}^2$ on $\\C^n$.\nSuch a function grows to infinity at the ends of the manifold and is strictly convex in the complex sense.\nSo a question about the existence of holomorphic functions is reduced to the existence of one real function with a convexity property.\n\n### L2 human\n\nFor domains in $\\C^n$ the theorem gives back [[Oka's solution of the Levi problem|6BJB149MDK]], after one smooths a continuous plurisubharmonic exhaustion and adds $\\abs{z}^2$ to make it strict.\nThe function $\\varphi$ is a Morse function after a small perturbation, and the strict plurisubharmonicity bounds the indices of its critical points by $\\dim_\\C X$.\nThis gives the result of Andreotti and Frankel that a Stein manifold of complex dimension $n$ has the homotopy type of a CW complex of real dimension at most $n$.\nThe Andreotti–Grauert theorem generalizes the theorem to $q$-convex manifolds.\n",
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