Stein manifold

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In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a complex submanifold of the vector space of n complex dimensions. The name is for Karl Stein.

[edit] Definition

A complex manifold X of complex dimension n is called a Stein manifold if the following conditions hold:

  • X is holomorphically convex, i.e. the so-called holomorphic convex hull
\bar K = \{z, z \in X, |f(z)| \leq \sup_K |f| \forall f \in \mathcal O(X) \}
is a compact subset of X for every compact subset K \subset X. Here \mathcal O(X) denotes the ring of holomorphic functions on X.
  • X is holomorphically separable, i.e. if x \neq y are two points in X, then there is a holomorphic function
f \in \mathcal O(X)
such that f(x) \neq f(y).
  • X has holomorphic charts, i.e. for every point x \in X, there are n holomorphic functions
f_1, \dots, f_n \in \mathcal O(X)
which form a coordinate system at x.

[edit] Properties and examples of Stein manifolds

  • The standard complex space \mathbb C^n is a Stein manifold.
  • It can be shown quite easily that every complex submanifold of a Stein manifold is a Stein manifold, too.
  • The embedding theorem for Stein manifolds states the following: Every Stein manifold X of complex dimension n can be embedded into \mathbb C^{2 n+1} by a biholomorphic proper map. (The proof of this theorem involves complex analysis).

These facts imply that a Stein manifold is a complex submanifold of complex space, whose complex structure is that of the ambient space (because the embedding is biholomorphic).

  • In one complex dimension the Stein condition can be simplified: a connected Riemann surface is a Stein manifold if and only if it is not compact.
  • Being a Stein manifold is equivalent to being a (complex) strongly pseudoconvex manifold. The latter means that it has a strongly pseudoconvex (or plurisubharmonic) exhaustive function, i.e. a smooth real function ψ on X (which can be assumed to be a morse function) with i \partial \bar \partial \psi >0, such that the subsets \{z \in X, \psi (z)\leq c \} are compact in X for every real number c. This is a solution to the so-called Levi problem. The function ψ invites a generalization of Stein manifold to the idea of a corresponding class of compact complex manifolds with boundary called Stein domains. A Stein domain is the preimage \{z|-\infty\leq\psi(z)\leq c\}. Some authors call such manifolds therefore strictly pseudoconvex manifolds.
  • Related to the previous item, another equivalent and more topological definition in complex dimension 2 is the following: a Stein surface is a complex surface X with a real-valued morse function f on X such that, away from the critical points of f, the field of complex tangencies to the preimage Xc = f-1(c) is a contact structure that induces an orientation on Xc agreeing with the usual orientation as the boundary of f-1(-∞,c). That is, f-1(-∞,c) is a Stein filling of Xc.

Numerous further characterizations of such manifolds exist, in particular capturing the property of their having "many" holomorphic functions taking values in the complex numbers. See for example Cartan's theorems A and B, relating to sheaf cohomology. The initial impetus was to have a description of the properties of the domain of definition of the (maximal) analytic continuation of an analytic function.

In the GAGA set of analogies, Stein manifolds correspond to affine varieties.

Stein manifolds are in some sense dual to the elliptic manifolds in complex analysis which admit "many" holomorphic functions from the complex numbers into themselves. It is known that a Stein manifold is elliptic if and only if it is fibrant in the sense of so-called "holomorphic homotopy theory".

[edit] Literature

  • Lars Hörmander, An Introduction to Complex Analysis in Several Variables, Van Nostrand (including a proof of the embedding theorem)
  • Robert Gompf, Handlebody Construction of Stein Surfaces, Ann. Math. 148, No. 2, p. 619-693 (definitions and constructions of Stein domains and manifolds in dimension 4)
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