Several complex variables and analytic spaces Stein manifolds and coherent sheaves Theorem

Embedding theorem for Stein manifolds

Builds on Stein manifold Proper holomorphic map Holomorphic map

Statement

Let $X$ be a Stein manifold of complex dimension $n$ . Then there is a proper injective holomorphic map $F \colon X \to \C^{2n+1}$ whose differential is injective at every point. So $X$ is biholomorphic to a closed complex submanifold of $\C^{2n+1}$ .

Remarks

The theorem is due to Remmert, Bishop and Narasimhan. The proof starts from the many global holomorphic functions that Cartan's theorems A and B provide on a Stein manifold. With them one builds a proper injective holomorphic map into some $\C^N$ with injective differential. Then one lowers $N$ by generic linear projections, as in the proof of the Whitney embedding theorem. The converse holds: every closed complex submanifold of $\C^N$ is Stein, because the coordinate functions separate points and give compact holomorphically convex hulls. So the Stein manifolds are exactly the closed complex submanifolds of complex Euclidean spaces. Example: a noncompact Riemann surface, which is Stein, embeds properly into $\C^3$ . The dimension $2n + 1$ is not optimal. For $n \ge 2$ , Eliashberg and Gromov, with a final step by Schürmann, showed that $\C^N$ with $N = \floor{3n/2} + 1$ suffices, and this value is optimal.

Going deeper

For a student

Embeddings reduce questions about Stein manifolds to questions about submanifolds of $\C^N$ . For example, $\abs{F}^2$ is a smooth strictly plurisubharmonic exhaustion function, which gives the easy half of Grauert's characterization of Stein manifolds. Whether every open Riemann surface embeds properly in $\C^2$ is an open problem.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Holomorphic map concept
Proper holomorphic map concept
Stein manifold concept
Embedding theorem for Stein manifolds statement

Tag	`6VF47Q35YA`
URI	`m2://6VF47Q35YA`
Kind	statement, theorem
Writing	W0 stub
MSC	32Q40, 32H02, 32Q28
Status	This copy of the site has no build data. Run `m2 status 6VF47Q35YA`.
About	Stein manifold
Uses	Stein manifold, Proper holomorphic map, Holomorphic map