nLab Stein manifold

Idea

A Stein manifold is a complex manifold satisfying some niceness conditions generalizing the concept of a domain of holomorphy: a Stein manifold is holomorphically convex and such that holomorphic functions separate points.

From the point of view of cohomology, Stein manifolds are to complex manifolds much as Cartesian spaces are to smooth manifolds:

Every complex manifold has a “good cover” by Stein manifolds and the positive-degree abelian sheaf cohomology with values in any analytic coherent sheaf on any Stein manifold vanishes (this is Cartan's theorem B, see below). This implies for instance that with respect to covers of complex manifolds by Stein manifolds usual Cech cohomology techniques work for analytic coherent sheaves (such as the structure sheaf of holomorphic functions).

Accordingly, Stein spaces are close to being the affine varieties over the complex numbers, but not quite, see below.

Definition

Stein manifold

The original definition (Stein 51) says that a Stein manifold $\Sigma$ is a complex manifold which is a “holomorphically convex” and “holomorphically separable” subset of a $\mathbb{C}^n$.

This is equivalent to (Reinhold Remmert, Narasimhan, Bishop) saying that $\Sigma$ admits a proper holomorphic immersion $\Sigma \hookrightarrow\mathbb{C}^n$.

Stein spaces

A complex analytic space is a Stein space precisely if its reduction is a Stein manifold.

Examples

Example

Every domain of holomorphy in some $\mathbb{C}^n$ is a Stein manifold.

Example

Every closed sub-complex manifold of a Stein manifold is itself Stein.

Example

(Stein surfaces are open Riemann surfaces)
A connected Riemann surface is a Stein manifold if and only if it is open (i.e. not compact).

(Behnke & Stein 1947, review in Wehler 2020, 15.3)

Properties

Basic properties

A Stein manifold is necessarily a non-compact topological space.

Good covers by Stein manifolds

Every complex manifold admits a good cover by Stein manifolds, in the sense that all finite non-empty intersections of the cover are Stein manifolds (e.g. Maddock 09, lemma 3.2.8), not in the sense that these intersections are contractible! Rather, all Dolbeault cohomology in positive degree vanishes.

Cohomology

The following central property of Stein manifolds is due to Henri Cartan.

Theorem

Cartan's theorem B

On a Stein manifold $\Sigma$ and for $A$ an analytic coherent sheaf on $\Sigma$ then all the positive-degree abelian sheaf cohomology groups of $\Sigma$ with coefficients in $A$ vanish:

$$H^{\bullet \geq 1}(\Sigma, A) = 0 \,.$$

(Serre 1954, Thm. B on p. 214, recalled for instance in Forstnerič 2011, theorem 2.4.1)

Theorem

Also all positive-degree Dolbeault cohomology groups vanish:

$$H^{\bullet, \bullet \geq 1}_{\bar \partial}(\Sigma) = 0 \,.$$

(recalled in Forstnerič 2011, theorem 2.4.6).

As an important special case, it follows that:

Proposition

The holomorphic de Rham cohomology of Stein manifolds coincides with ordinary de Rham cohomology (with complex coefficients).

(Serre 1954 §2.1, cf. also MO:q/273178)

Relation to affine varieties

The analytification of any affine variety over the complex numbers is a Stein space, but the converse is not quite true. (Zhang 06). There is also some theorem by Neeman to this effect.

See also at affine variety – Cohomology.

Oka-Grauert principle

The Oka-Grauert principle states that for any Stein manifold $X$ the holomorphic and the topological classification of complex vector bundles on $X$ coincide. The original reference is (Grauert 58).

Related concepts

• Stein site

• EFC-algebra

• complex analytic ∞-groupoid

References

The original article:

• Karl Stein, Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem, Math. Ann. (in German) 123: 201–222, (1951) (doi:10.1007/BF02054949, MR 0043219)

Further development:

• Hans Grauert, Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann. 135, 263–-273 (1958) (doi:10.1007/BF01351803)

Relation to Riemann surfaces:

• Heinrich Behnke, Karl Stein, Entwicklung analytischer Funktionen auf Riemannschen Flächen, Mathematische Annalen volume 120, pages 430–461 (1947) (doi:10.1007/BF01447838)

On how holomorphic de Rham cohomology on Stein manifolds coincides with ordinary de Rham cohomology:

• Jean-Pierre Serre: Quelques problemes globaux relatifs aux varietes de Stein, Colloque sur les fonctions de plusieurs variables (1953) [doi:10.1007/978-3-642-39816-2_23]

• Jean-Pierre Serre: §2.1 in: Cohomologie et fonctions de variables complexes, Séminaire Bourbaki 2 71 (1954) [numdam:SB_1951-1954__2__213_0]

Texbook accounts:

• Hans Grauert, Reinhold Remmert, Theory of Stein Spaces, Springer-Verlag, Berlin Heidelberg, 2004.

• Franc Forstnerič, Section 5.3 of: Stein manifolds and holomorphic mappings – The homotopy principle in complex analysis, Springer 2011 (doi:10.1007/978-3-642-22250-4)

  (in relation to the Oka principle)

Lecture notes:

• Joachim Wehler, Section 15 in: Riemann surfaces 2020 (pdf)

See also:

• Wikipedia, Stein manifold

• eom, Stein space

• Zachary Maddock, Dolbeault cohomology, notes 2009 (pdf)

Discussion of the relation to affine varieties includes

• Jing Zhang, Algebraic Stein Varieties (arXiv:math/0610886)

In relation to the homotopy-theoretic Oka principle:

• Franc Forstnerič, Stein manifolds and holomorphic mappings – The homotopy principle in complex analysis, Springer 2011 (doi:10.1007/978-3-642-22250-4)

As a site for higher complex analytic geometry the category of Stein manifolds appears in

• Jacob Lurie, section 4.4. of Structured Spaces