# Stein manifold

In the theory of several complex variables and complex manifolds in mathematics, a Stein manifold is a complex submanifold of the vector space of n complex dimensions. They were introduced by and named after Template:Harvs. A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry.

## Definition

A complex manifold ${\displaystyle X}$ of complex dimension ${\displaystyle n}$ is called a Stein manifold if the following conditions hold:

${\displaystyle {\bar {K}}=\{z\in X:|f(z)|\leq \sup _{K}|f|\ \forall f\in {\mathcal {O}}(X)\},}$
is again a compact subset of ${\displaystyle X}$. Here ${\displaystyle {\mathcal {O}}(X)}$ denotes the ring of holomorphic functions on ${\displaystyle X}$.
${\displaystyle f\in {\mathcal {O}}(X)}$
such that ${\displaystyle f(x)\neq f(y).}$

## Non-compact Riemann surfaces are Stein

Let X be a connected non-compact Riemann surface. A deep theorem of Behnke and Stein (1948) asserts that X is a Stein manifold.

Another result, attributed to Grauert and Röhrl (1956), states moreover that every holomorphic vector bundle on X is trivial.

In particular, every line bundle is trivial, so ${\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*})=0}$. The exponential sheaf sequence leads to the following exact sequence:

${\displaystyle H^{1}(X,{\mathcal {O}}_{X})\longrightarrow H^{1}(X,{\mathcal {O}}_{X}^{*})\longrightarrow H^{2}(X,\mathbb {Z} )\longrightarrow H^{2}(X,{\mathcal {O}}_{X})}$

This is related to the solution of the Cousin problems, and more precisely to the second Cousin problem.

## Properties and examples of Stein manifolds

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

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".

## Relation to smooth manifolds

Every compact smooth manifold of dimension 2n, which has only handles of index ≤ n, has a Stein structure provided n>2, and when n=2 the same holds provided the 2-handles are attached with certain framings (framing less than the Thurston-Bennequin framing).[2][3] Every closed smooth 4-manifold is a union of two Stein 4-manifolds glued along their common boundary.[4]

## Notes

1. PlanetMath: solution of the Levi problem
2. Y. Eliashberg, Topological characterization of Stein manifolds of dimension > 2, Int. J. of Math. vol. 1, no 1 (1990) 29-46.
3. R. Gompf, Handlebody construction of Stein surfaces, Ann. of Math. 148, (1998) 619-693.
4. S. Akbulut and R. Matveyev, A convex decomposition for four-manifolds, IMRN, no.7 (1998) 371-381.

## References

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