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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\C^n$ be a domain.\nThe following are equivalent:\n\n1. $\\Omega$ is a [[domain of holomorphy|4487CSC0MB]].\n2. $\\Omega$ is [[pseudoconvex|AHYZMSBBNH]].\n3. $\\Omega$ has a continuous [[plurisubharmonic exhaustion function|FGSKR6ZS3Q]].\n\n## Remarks\n\nThe implication from 1 to 3 is the easy one.\nLet $d(z)$ be the distance from $z$ to $\\partial\\Omega$; on a domain of holomorphy $\\Omega \\ne \\C^n$, the function $-\\log d$ is [[plurisubharmonic|FF2P7ER3QT]].\nThen $\\max(-\\log d(z), \\abs{z}^2)$ is a continuous plurisubharmonic exhaustion function.\nThe hard implication, from 3 to 1, is the solution of the [[Levi problem|47CVS9RJ9D]].\nFor each boundary point one needs a holomorphic function on $\\Omega$ that cannot be continued past it.\nOka built such functions with his own methods; his proof for general $n$ used his theory of coherent ideal sheaves.\nHörmander later obtained them by solving $\\bar\\partial u = f$ with $L^2$ estimates for weights $e^{-k\\varphi}$, where $\\varphi$ is the exhaustion function.\nExample: any convex domain $\\Omega$ is pseudoconvex, so it is a domain of holomorphy.\nFor convex domains this can also be seen directly with the functions $1/\\ell$.\nHere $\\ell$ is an affine complex-linear function that vanishes at a given boundary point and nowhere on $\\Omega$.\nThe same equivalence holds for unramified Riemann domains over $\\C^n$.\n\n## Explanations\n\n### L1 human\n\nA domain of holomorphy is a domain that is the natural home of some holomorphic function, one that cannot be pushed past any boundary point.\nPseudoconvexity is a convexity condition on the shape of the domain.\nThe theorem says that the two notions are the same.\nSo an analytic question, whether every holomorphic function extends, becomes a geometric question about the boundary.\n\n### L2 human\n\nThe theorem completes the [[Cartan–Thullen theorem|1XCBCB3MD7]], which says that domains of holomorphy are the holomorphically convex domains.\nTogether they give four equivalent conditions on a domain in $\\C^n$: domain of holomorphy, holomorphically convex, pseudoconvex, and Stein.\nFor domains with $C^2$ boundary, pseudoconvexity can be checked with the [[Levi form|80MTNXWW80]], so this gives a computable test.\nGrauert's theorem extends the result from domains to complex manifolds.\n",
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