Stein manifold: Difference between revisions

Revision as of 15:12, 10 September 2005

Revision as of 15:15, 10 September 2005

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is a compact subset of <math>X</math> for every compact subset <math>K \subset X</math>. Here

<math>\mathcal O(X)</math> denotes the ring of [[holomorphic]] functions on <math>X</math>.

* If <math>x \neq y</math> are two points in <math>X</math>, then there is a holomorphic function <math>f \in \mathcal O(X)</math>, such that <math>f(x) \neq f(y)</math>.

* For every point <math>fx \in X</math>, there are <math>n</math> holomorphic functions <math>f_1, ..., f_n \in \mathcal O(X)</math>, suchwhich thatform a coordinate system at <math>f(x) \neq f(y)</math>.

* For every point <math>x \in X</math>, there are <math>n</math> holomorphic functions

<math>f_1, ..., f_n \in \mathcal O(X)</math>, which form a coordinate system at <math>x</math>.

== Properties and examples of Stein manifolds ==

* The standard complex space <math>\mathbb C^n</math> is a Stein manifold.

* It can be shown quite easily, that every submanifold of a Stein manifold is a Stein manifold, too.

* The embedding theorem for Stein manifolds states the following: Every Stein manifold <math>X</math> of complex dimension <math>n</math> can be embedded into <math>\mathbb C^{2 n+1}</math>. (The proof of this theorem requires some harder analysis).

dimension <math>n</math> can be embedded into <math>\mathbb C^{2 n+1}</math>. (The proof of this theorem requires some harder analysis).

Gathering these facts, one sees, that Stein manifold is a synonym for a submanifold of complex space.

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* A consequence of the embedding theorem is the following fact: a connected [[Riemann surface]] (i.e. a complex

manifold of dimension 1) is a Stein manifold if and only if it is not compact.

* Being a Stein manifold is equivalent to be a (complex) ''strongly pseudoconvex manifold''. (The latter means, that it has a strongly pseudoconvex exhaustive function, i.e. a smooth real function <math>\psi</math> on <math>X</math> with <math>i \partial \bar \partial \psi >0</math>, such that the subsets <math>\{z \in X, \psi (z)<c \}</math> are compact in <math>X</math> for every real number <math>c</math>). This question was the so-called Levi problem.

the subsets <math>\{z \in X, \psi (z)<c \}</math> are compact in <math>X</math> for every real number <math>c</math>). This question was the so-called Levi problem.

Numerous further characterizations of such manifolds exist, in particular capturing the property of their having "many" [[holomorphic function]]s taking values in the complex numbers. See for example [[Cartan's theorems A and B]], relating to [[sheaf cohomology]]. The initial impetus was to have a description of the properties of the ___domain of definition of the (maximal) [[analytic continuation]] of an [[analytic function]].