Stein manifold

Definition.

A complex manifold $M$ of complex dimension $n$ is a Stein manifold if it satisfies the following properties

1. $M$ is holomorphically convex,

2. if $z,w\in M$ and $z\ne w$ then $f(z)\ne f(w)$ for some function $f$ holomorphic on $M$ (i.e. $M$ is holomorphically separable),

3. for every $z\in M$ there are holomorphic functions $f_1,\mathrm{\dots },f_n$ which form a coordinate system at $z$ (i.e. $M$ is holomorphically spreadable).

Stein manifold is a generalization of the concept of the domain of holomorphy to manifolds. Furthermore, Stein manifolds are the generalizations of Riemann surfaces in higher dimensions. Every noncompact Riemann surface is a Stein manifold by a theorem of Behnke and Stein. Note that every domain of holomorphy in ${ℂ}^n$ is a Stein manifold. It is not hard to see that every closed complex submanifold of a Stein manifold is Stein.

Theorem (Remmert, Narasimhan, Bishop).

If $M$ is a Stein manifold of dimension $n$. There exists a proper (http://planetmath.org/ProperMap) holomorphic embedding of $M$ into ${\mathrm{C}}^{\mathrm{2}\mathit{}n\mathrm{+}\mathrm{1}}$.

Note that no compact complex manifold can be Stein since compact complex manifolds have no holomorphic functions. On the other hand, every compact complex manifold is holomorphically convex.

Title	Stein manifold
Canonical name	SteinManifold
Date of creation	2013-03-22 15:04:37
Last modified on	2013-03-22 15:04:37
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	7
Author	jirka (4157)
Entry type	Definition
Classification	msc 32E10
Related topic	HolomorphicallyConvex
Related topic	DomainOfHolomorphy
Defines	holomorphically separable
Defines	holomorphically spreadable