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  "body": "\n## Statement\n\nLet $M$ be a [[real analytic manifold|Z7H98E1NHP]] of dimension $n$ whose topology has a countable basis.\nThen there is a [[complex manifold|JSGWAN2RJV]] $X$ of complex dimension $n$ that contains $M$ as a closed subset, with the following property.\nEach point $p \\in M$ has a holomorphic chart $\\varphi \\colon U \\to \\C^n$ of $X$ around $p$ with $\\varphi(U \\cap M) = \\varphi(U) \\cap \\R^n$.\nMoreover $\\varphi|_{U \\cap M}$ is a real analytic chart of $M$.\nSuch an $X$ is a complexification of $M$.\nIt is unique as a germ, in the following sense.\nLet $X$ and $X'$ be two complexifications of $M$.\nThen there are open neighbourhoods $V \\subseteq X$ and $V' \\subseteq X'$ of $M$ and a biholomorphic map $V \\to V'$ that is the identity on $M$.\n\n## Remarks\n\nThe existence is due to Whitney and Bruhat.\nLocally, a real analytic chart extends to a holomorphic chart by replacing real variables by complex ones in the convergent power series of the transition maps.\nThe transition maps of $M$ then extend holomorphically to small complex neighbourhoods.\nOne glues these local pieces along a neighbourhood of $M$ that is chosen thin enough for the extensions to agree.\nUniqueness follows from the identity theorem: a holomorphic map is determined by its values on a totally real submanifold of maximal dimension.\n\nExamples: $\\R^n$ has the complexification $\\C^n$.\nThe circle $S^1 = \\set{x^2 + y^2 = 1} \\subseteq \\R^2$ has the complexification $\\set{(z, w) \\in \\C^2 : z^2 + w^2 = 1}$, which is biholomorphic to $\\C \\setminus \\set{0}$ by $(z, w) \\mapsto z + i w$.\n\nBy a theorem of Grauert, the complexification $X$ can be chosen to be a Stein manifold.\n\n## Explanations\n\n### L1 human\n\nA real analytic manifold is described by convergent power series in real variables.\nFeeding complex numbers into these series gives a complex manifold of the same dimension that contains the real one, as $\\C^n$ contains $\\R^n$.\nNear the real manifold this complex manifold is unique.\n\n### L2 human\n\nComplexification moves problems about real analytic objects to complex analysis, where stronger tools are available.\nA real analytic function on $M$ extends to a holomorphic function on a neighbourhood of $M$ in $X$.\nWith a Stein complexification, Cartan's theorems give many such functions, which leads to Grauert's analytic embedding of $M$ in some $\\R^N$.\n",
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