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 manifoldby 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.