Several complex variables and analytic spaces Analytic spaces Theorem

Complexification of a real analytic manifold

Builds on Real analytic manifold Complex manifold

Statement

Let $M$ be a real analytic manifold of dimension $n$ whose topology has a countable basis. Then there is a complex manifold $X$ of complex dimension $n$ that contains $M$ as a closed subset, with the following property. Each point $p \in M$ has a holomorphic chart $\varphi \colon U \to \C^n$ of $X$ around $p$ with $\varphi(U \cap M) = \varphi(U) \cap \R^n$ . Moreover $\varphi|_{U \cap M}$ is a real analytic chart of $M$ . Such an $X$ is a complexification of $M$ . It is unique as a germ, in the following sense. Let $X$ and $X'$ be two complexifications of $M$ . Then 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 $M$ .

Remarks

The existence is due to Whitney and Bruhat. Locally, 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. The transition maps of $M$ then extend holomorphically to small complex neighbourhoods. One glues these local pieces along a neighbourhood of $M$ that is chosen thin enough for the extensions to agree. Uniqueness follows from the identity theorem: a holomorphic map is determined by its values on a totally real submanifold of maximal dimension.

Examples: $\R^n$ has the complexification $\C^n$ . The 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 $(z, w) \mapsto z + i w$ .

By a theorem of Grauert, the complexification $X$ can be chosen to be a Stein manifold.

Going deeper

For a student

Complexification moves problems about real analytic objects to complex analysis, where stronger tools are available. A real analytic function on $M$ extends to a holomorphic function on a neighbourhood of $M$ in $X$ . With a Stein complexification, Cartan's theorems give many such functions, which leads to Grauert's analytic embedding of $M$ in some $\R^N$ .

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Complex manifold concept
Real analytic manifold concept
Complexification of a real analytic manifold statement

Tag	`E7G1S7Y1MT`
URI	`m2://E7G1S7Y1MT`
Kind	statement, theorem
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MSC	32C09, 32C05
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About	Real analytic manifold
Uses	Real analytic manifold, Complex manifold