In mathematics, in the theory of several complex variables and complex manifolds, a **Stein manifold** is a complex submanifold of the vector space of *n* complex dimensions. They were introduced by and named after Karl Stein (1951). 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

Suppose is a complex manifold of complex dimension and let denote the ring of holomorphic functions on We call a **Stein manifold** if the following conditions hold:

- is holomorphically convex, i.e. for every compact subset , the so-called
*holomorphically convex hull*,

- is also a
*compact*subset of .

- is holomorphically separable, i.e. if are two points in , then there exists such that

## Non-compact Riemann surfaces are Stein manifolds

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

Another result, attributed to Hans Grauert and Helmut Röhrl (1956), states moreover that every holomorphic vector bundle on *X* is trivial. In particular, every line bundle is trivial, so . The exponential sheaf sequence leads to the following exact sequence:

Now Cartan's theorem B shows that , therefore .

This is related to the solution of the second Cousin problem.

## Properties and examples of Stein manifolds

- The standard complex space is a Stein manifold.

- Every domain of holomorphy in is a Stein manifold.

- It can be shown quite easily that every closed complex submanifold of a Stein manifold is a Stein manifold, too.

- The embedding theorem for Stein manifolds states the following: Every Stein manifold of complex dimension can be embedded into by a biholomorphic proper map.

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).

- Every Stein manifold of (complex) dimension
*n*has the homotopy type of an*n*-dimensional CW-complex.

- 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. This can be proved using a version of the Runge theorem for Riemann surfaces, due to Behnke and Stein.

- Every Stein manifold is holomorphically spreadable, i.e. for every point , there are holomorphic functions defined on all of which form a local coordinate system when restricted to some open neighborhood of .

- 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 on (which can be assumed to be a Morse function) with , such that the subsets are compact in for every real number . This is a solution to the so-called**Levi problem**,^{[1]}named after E. E. Levi (1911). The function 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 . 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 is a contact structure that induces an orientation on*X*agreeing with the usual orientation as the boundary of That is, is a Stein filling of_{c}*X*._{c}

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

**^**Onishchik, A.L. (2001) [1994], "Levi problem",*Encyclopedia of Mathematics*, EMS Press**^**Yakov Eliashberg, Topological characterization of Stein manifolds of dimension > 2, International Journal of Mathematics vol. 1, no 1 (1990) 29–46.**^**Robert Gompf, Handlebody construction of Stein surfaces, Annals of Mathematics 148, (1998) 619–693.**^**Selman Akbulut and Rostislav Matveyev, A convex decomposition for four-manifolds, International Mathematics Research Notices (1998), no.7, 371–381. MR1623402

## References

- Andrist, Rafael (2010). "Stein spaces characterized by their endomorphisms".
*Transactions of the American Mathematical Society*.**363**(5): 2341–2355. doi:10.1090/S0002-9947-2010-05104-9. S2CID 14903691. - Forster, Otto (1981),
*Lectures on Riemann surfaces*, Graduate Text in Mathematics, vol. 81, New-York: Springer Verlag, ISBN 0-387-90617-7 (including a proof of Behnke-Stein and Grauert–Röhrl theorems) - Hörmander, Lars (1990),
*An introduction to complex analysis in several variables*, North-Holland Mathematical Library, vol. 7, Amsterdam: North-Holland Publishing Co., ISBN 978-0-444-88446-6, MR 1045639 (including a proof of the embedding theorem) - Gompf, Robert E. (1998), "Handlebody construction of Stein surfaces",
*Annals of Mathematics*, Second Series, The Annals of Mathematics, Vol. 148, No. 2,**148**(2): 619–693, arXiv:math/9803019, doi:10.2307/121005, ISSN 0003-486X, JSTOR 121005, MR 1668563 (definitions and constructions of Stein domains and manifolds in dimension 4) - Grauert, Hans; Remmert, Reinhold (1979),
*Theory of Stein spaces*, Grundlehren der Mathematischen Wissenschaften, vol. 236, Berlin-New York: Springer-Verlag, ISBN 3-540-90388-7, MR 0580152 - Ornea, Liviu; Verbitsky, Misha (2010). "Locally conformal Kähler manifolds with potential".
*Mathematische Annalen*.**348**: 25–33. doi:10.1007/s00208-009-0463-0. S2CID 10734808. - Iss'Sa, Hej (1966). "On the Meromorphic Function Field of a Stein Variety".
*Annals of Mathematics*.**83**(1): 34–46. doi:10.2307/1970468. JSTOR 1970468. - Stein, Karl (1951), "Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem",
*Math. Ann.*(in German),**123**: 201–222, doi:10.1007/bf02054949, MR 0043219 - Zhang, Jing (2006). "Algebraic Stein Varieties". arXiv:math/0610886. Bibcode:2006math.....10886Z.
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