Stein manifold


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

  1. 1.
  2. 2.

    if z,wM and zw then f(z)f(w) for some function f holomorphic on M (i.e. M is holomorphically separable),

  3. 3.

    for every zM there are holomorphic functions f1,,fn which form a coordinate systemMathworldPlanetmath 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 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 ( holomorphic embedding of M into C2n+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.


  • 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
Canonical name SteinManifold
Date of creation 2013-03-22 15:04:37
Last modified on 2013-03-22 15:04:37
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Definition
Classification msc 32E10
Related topic HolomorphicallyConvex
Related topic DomainOfHolomorphy
Defines holomorphically separable
Defines holomorphically spreadable