# Stein manifold

###### Definition.

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

1. 1.

$M$ is holomorphically convex,

2. 2.

if $z,w\in M$ and $z\not=w$ then $f(z)\not=f(w)$ for some function $f$ holomorphic on $M$ (i.e. $M$ is holomorphically separable),

3. 3.

for every $z\in M$ there are holomorphic functions $f_{1},\ldots,f_{n}$ 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 ${\mathbb{C}}^{n}$ 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 ${\mathbb{C}}^{2n+1}$.

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.

## References

• 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
• 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title Stein manifold SteinManifold 2013-03-22 15:04:37 2013-03-22 15:04:37 jirka (4157) jirka (4157) 7 jirka (4157) Definition msc 32E10 HolomorphicallyConvex DomainOfHolomorphy holomorphically separable holomorphically spreadable