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  "body": "\n## Statement\n\nLet $X$ be a [[Stein manifold|HMCD0JBFST]] and let $D \\subseteq X$ be an open subset.\nSuppose that $D$ is locally Stein in $X$.\nThis means that every point $p \\in \\partial D$ has an open neighbourhood $U$ in $X$ such that $U \\cap D$ is Stein.\nThen $D$ is a Stein manifold.\n\n## Remarks\n\nFor $X = \\C^n$ this is Oka's solution of the Levi problem: a locally pseudoconvex domain in $\\C^n$ is a domain of holomorphy.\nThe proof follows the same pattern.\nUsing that $X$ embeds in some $\\C^N$ and has a holomorphic retraction from a neighbourhood, one builds a distance-to-the-boundary function $\\delta$ on $D$.\nThe local Stein condition makes $-\\log \\delta$ [[plurisubharmonic|FF2P7ER3QT]] near $\\partial D$.\nAdding a strictly plurisubharmonic exhaustion of $X$ gives a plurisubharmonic exhaustion function of $D$, and Grauert's solution of the Levi problem then shows that $D$ is Stein.\nThe same holds for unbranched Riemann domains spread over a Stein manifold, which is the form in which Docquier and Grauert proved it.\nExample: if $D_1, D_2$ are Stein open subsets of $X$, then $D_1 \\cap D_2$ is locally Stein, hence Stein.\nThe hypothesis that $X$ is Stein cannot be dropped.\nIf $X$ is the Riemann sphere and $D = X$, then $\\partial D$ is empty, so $D$ is locally Stein, but $D$ is compact and not Stein.\n\n## Explanations\n\n### L1 human\n\nBeing Stein is a global property of a complex manifold, roughly saying that it has many holomorphic functions.\nThe theorem says that, inside a Stein manifold, it is enough to check this property near each boundary point of an open set.\nA local test at the boundary then gives a global conclusion.\n\n### L2 human\n\nThe theorem shows that the class of Stein open sets of a Stein manifold is described by a local boundary condition, pseudoconvexity, just as for domains in $\\C^n$.\nIt is used to prove that intersections, and suitable unions and limits, of Stein domains are Stein.\nIt reduces many questions on Stein manifolds to the case of domains in $\\C^n$ through the embedding theorem.\n",
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  "name": "Docquier–Grauert theorem",
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