Definition 1   A complex manifold of complex dimension is a Stein manifold if it satisfies the following properties

1. is holomorphically convex,
2. if and then for some function holomorphic on (i.e. is holomorphically separable),
3. for every there are holomorphic functions which form a coordinate system at (i.e. 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 is a Stein manifold. It is not hard to see that every closed complex submanifold of a Stein manifold is Stein.

Theorem 1 (Remmert, Narasimhan, Bishop)   If is a Stein manifold of dimension . There exists a proper holomorphic embedding of into .

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.

Bibliography

1

Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.

2

Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.