Stein manifoldSteinManifold2005-02-22 13:07:392008-03-31 16:52:45Definitionholomorphically separableholomorphically spreadable% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{rmk}{Remark}\begin{defn}
A complex manifold $M$ of complex dimension $n$ is a {\it Stein manifold} if it satisfies the following properties
\begin{enumerate}
\item $M$ is holomorphically convex,
\item 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 {\it holomorphically separable}),
\item 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 {\it holomorphically spreadable}).
\end{enumerate}
\end{defn}
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.
\begin{thm}[Remmert, Narasimhan, Bishop]
If $M$ is a Stein manifold of dimension $n$. There exists a \PMlinkname{proper}{ProperMap} holomorphic embedding of $M$ into ${\mathbb{C}}^{2n+1}$.
\end{thm}
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.
\begin{thebibliography}{9}
\bibitem{Hormander:several}
Lars H\"ormander.
{\em \PMlinkescapetext{An Introduction to Complex Analysis in Several
Variables}},
North-Holland Publishing Company, New York, New York, 1973.
\bibitem{Krantz:several}
Steven~G.\@ Krantz.
{\em \PMlinkescapetext{Function Theory of Several Complex Variables}},
AMS Chelsea Publishing, Providence, Rhode Island, 1992.
\end{thebibliography}