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  "body": "\n## Summary\n\nA domain of holomorphy is an open set in $\\C^n$ that is the natural domain of its [[holomorphic functions|507C34W0ZT]].\nNo part of its boundary can be crossed by all of these functions at once.\nIn one variable every open set has this property, but for $n \\ge 2$ many open sets do not, by the Hartogs phenomenon.\nDomains of holomorphy are characterised by holomorphic convexity and, by the solution of the Levi problem, by pseudoconvexity.\nThey are the domains in $\\C^n$ on which the global theory of holomorphic functions works best.\n\n## Explanations\n\n### L1 human\n\nSome regions in $\\C^n$ have hidden extensions: every holomorphic function on them automatically extends further.\nA ball with a small closed ball removed from the middle is such a region in $\\C^2$, since every function fills in the hole.\nA domain of holomorphy has no such hidden extension: for each boundary point there is a function that really breaks down there.\nThe open unit ball is a domain of holomorphy; near a boundary point $p$, the function $1/(\\ip{z}{p} - 1)$ blows up, where $\\ip{z}{p} = \\sum_j z_j \\bar p_j$.\n",
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