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  "body": "\n## Statement\n\nLet $n \\ge 2$, let $D \\subseteq \\C^n$ be a connected [[open set|02BH3MCBQB]], and let $K \\subseteq D$ be a [[compact set|0R6EWFKRCF]] such that $D \\setminus K$ is [[connected|0SPE8WNWFV]].\nThen every [[holomorphic function|507C34W0ZT]] $f$ on $D \\setminus K$ is the restriction of a unique holomorphic function $F$ on $D$.\n\n## Remarks\n\nA short proof uses the [[$\\bar\\partial$ operator|M1VKCW3RFW]].\nChoose a smooth function $\\chi$ with compact support in $D$ that equals $1$ near $K$, and let $g = (1 - \\chi) f$, extended by $0$ on $K$.\nThe form $\\bar\\partial g = -f \\, \\bar\\partial \\chi$ is smooth with compact support in $\\C^n$, and for $n \\ge 2$ the equation $\\bar\\partial u = \\bar\\partial g$ has a solution $u$ with compact support.\nThen $F = g - u$ is holomorphic on $D$.\nThe function $u$ vanishes on the unbounded component of the complement of the support of $\\bar\\partial g$.\nSo $F = f$ on a nonempty open subset of $D \\setminus K$, hence on all of $D \\setminus K$ by the identity theorem.\n\nThe hypothesis $n \\ge 2$ is necessary: $1/z$ is holomorphic on $\\C \\setminus \\set{0}$ and does not extend to $\\C$.\nConnectedness of $D \\setminus K$ is also necessary.\nIn the unit ball of $\\C^n$ let $K$ be the sphere $\\abs{z} = 1/2$.\nThe function equal to $0$ inside $K$ and to $1$ outside is holomorphic on the ball minus $K$, but it has no holomorphic extension.\n\nExample: every holomorphic function on the punctured ball $\\set{0 \u003c \\abs{z} \u003c 1}$ in $\\C^2$ extends holomorphically over the origin.\n\n## Explanations\n\n### L1 human\n\nIn several complex variables a holomorphic function cannot have a compact hole of singularities.\nIf it is defined everywhere except on a compact set, it automatically extends across that set.\nThis is very different from one variable, where $1/z$ has an isolated singularity at $0$.\n\n### L2 human\n\nThe theorem is the first sign that domains in $\\C^n$, $n \\ge 2$, carry hidden constraints for holomorphic functions.\nIt implies that the zeros and the singularities of holomorphic functions are never isolated when $n \\ge 2$.\nIt leads to the notions of domain of holomorphy and envelope of holomorphy, which describe how far all holomorphic functions on a domain extend.\nThe proof sketched above shows the role of the compactly supported solution of $\\bar\\partial u = v$, which exists only when $n \\ge 2$.\n",
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  "name": "Hartogs's extension theorem",
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