Stein manifold (original) (raw)

Definition.

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 ℂn 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 C2⁢n+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.