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| In the theory of [[several complex variables]] and [[complex manifold]]s in mathematics, a '''Stein manifold''' is a complex [[submanifold]] of the [[vector space]] of ''n'' [[complex number|complex]] dimensions. They were introduced by and named after {{harvs|txt|authorlink=Karl Stein (mathematician)|first=Karl |last=Stein|year=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.
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| == Definition ==
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| A [[complex manifold]] <math>X</math> of complex dimension <math>n</math> is called a '''Stein manifold''' if the following conditions hold:
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| * <math>X</math> is holomorphically convex, i.e. for every [[Compact space|compact]] subset <math>K \subset X</math>, the so-called ''holomorphic convex hull'',
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| ::<math>\bar K = \{z \in X: |f(z)| \leq \sup_K |f| \ \forall f \in \mathcal O(X) \},</math>
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| :is again a ''compact'' subset of <math>X</math>. Here <math>\mathcal O(X)</math> denotes the ring of [[holomorphic]] functions on <math>X</math>.
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| * <math>X</math> is holomorphically separable, i.e. if <math>x \neq y</math> are two points in <math>X</math>, then there is a holomorphic function
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| ::<math>f \in \mathcal O(X)</math>
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| :such that <math>f(x) \neq f(y).</math>
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| ==Non-compact Riemann surfaces are Stein==
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| Let ''X'' be a connected non-compact Riemann surface. A deep theorem of [[Heinrich Behnke|Behnke]] and Stein (1948) asserts that ''X'' is a Stein manifold.
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| Another result, attributed to [[Hans Grauert|Grauert]] and [[Helmut Röhrl|Röhrl]] (1956), states moreover that every holomorphic vector bundle on ''X'' is trivial.
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| In particular, every line bundle is trivial, so <math>H^1(X, \mathcal O_X^*) =0 </math>. The [[exponential sheaf sequence]] leads to the following exact sequence:
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| : <math>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) </math>
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| Now [[Cartan's theorems A and B|Cartan's theorem B]] shows that <math>H^1(X, \mathcal O_X)= H^2(X, \mathcal O_X)=0 </math>, therefore <math>H^2(X, \mathbb Z)=0</math>.
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| This is related to the solution of the [[Cousin problems]], and more precisely to the second Cousin problem.
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| == Properties and examples of Stein manifolds ==
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| * The standard complex space <math>\mathbb C^n</math> is a Stein manifold.
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| * Every [[domain of holomorphy]] in <math>\mathbb C^n</math> is a Stein manifold.
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| * It can be shown quite easily that every closed complex submanifold of a Stein manifold is a Stein manifold, too.
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| * The embedding theorem for Stein manifolds states the following: Every Stein manifold <math>X</math> of complex dimension <math>n</math> can be embedded into <math>\mathbb C^{2 n+1}</math> by a [[biholomorphic]] [[proper map]].
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| 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).
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| * 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.
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| * Every Stein manifold <math>X</math> is holomorphically spreadable, i.e. for every point <math>x \in X</math>, there are <math>n</math> holomorphic functions defined on all of <math>X</math> which form a local coordinate system when restricted to some open neighborhood of <math>x</math>.
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| * Being a Stein manifold is equivalent to being a (complex) ''strongly pseudoconvex manifold''. The latter means that it has a strongly pseudoconvex (or [[plurisubharmonic function|plurisubharmonic]]) exhaustive function, i.e. a smooth real function <math>\psi</math> on <math>X</math> (which can be assumed to be a [[Morse theory|Morse function]]) with <math>i \partial \bar \partial \psi >0</math>, such that the subsets <math>\{z \in X, \psi (z)\leq c \}</math> are compact in <math>X</math> for every real number <math>c</math>. This is a solution to the so-called '''Levi problem''',<ref>[http://planetmath.org/encyclopedia/LeviProblem.html PlanetMath: solution of the Levi problem]</ref> named after E. E. Levi (1911). The function <math>\psi</math> 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 <math>\{z|-\infty\leq\psi(z)\leq c\}</math>. Some authors call such manifolds therefore strictly pseudoconvex manifolds.
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| *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 ''X''<sub>''c''</sub> = ''f''<sup>−1</sup>(''c'') is a [[Contact geometry|contact structure]] that induces an orientation on ''X<sub>c</sub>'' agreeing with the usual orientation as the boundary of ''f''<sup>−1</sup>(−∞,''c''). That is, ''f''<sup>−1</sup>(−∞,''c'') is a Stein [[Symplectic filling|filling]] of ''X<sub>c</sub>''.
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| Numerous further characterizations of such manifolds exist, in particular capturing the property of their having "many" [[holomorphic function]]s 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]].
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| In the [[GAGA]] set of analogies, Stein manifolds correspond to [[affine variety|affine varieties]].
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| Stein manifolds are in some sense dual to the [[elliptic manifold]]s 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 object|fibrant]] in the sense of so-called "holomorphic homotopy theory".
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| == Relation to smooth manifolds ==
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| 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).<ref>[[Yakov Eliashberg|Y. Eliashberg]], Topological characterization of Stein manifolds of dimension > 2, Int. J. of Math. vol. 1, no 1 (1990) 29-46.</ref><ref>R. Gompf, Handlebody construction of Stein surfaces, Ann. of Math. 148, (1998) 619-693.</ref> Every closed smooth 4-manifold is a union of two Stein 4-manifolds glued along their common boundary.<ref>S. Akbulut and R. Matveyev, A convex decomposition for four-manifolds, IMRN, no.7 (1998) 371-381.</ref>
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| ==Notes==
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| <references/>
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| == References ==
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| * {{Citation | last1=Forster | first1=Otto | author1-link= Otto Forster | title=Lectures on Riemann surfaces | publisher=Springer Verlag | location=New-York | series=Graduate Text in Mathematics | isbn=0-387-90617-7 | year=1981 | volume=81}} (including a proof of Behnke-Stein and Grauert-Röhrl theorems)
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| * {{Citation | last1=Hörmander | first1=Lars | author1-link= Lars Hörmander | title=An introduction to complex analysis in several variables | publisher=North-Holland Publishing Co. | location=Amsterdam | series=North-Holland Mathematical Library | isbn=978-0-444-88446-6 | mr=1045639 | year=1990 | volume=7}} (including a proof of the embedding theorem)
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| * {{Citation | last1=Gompf | first1=Robert E. | title=Handlebody construction of Stein surfaces | mr=1668563 | year=1998 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=148 | issue=2 | pages=619–693 | doi=10.2307/121005 | jstor=121005 | publisher=The Annals of Mathematics, Vol. 148, No. 2}} (definitions and constructions of Stein domains and manifolds in dimension 4)
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| *{{citation|mr=0580152|last1=Grauert|first1= Hans|last2= Remmert|first2= Reinhold|title=Theory of Stein spaces|series=Grundlehren der Mathematischen Wissenschaften |volume=236|publisher= Springer-Verlag|place= Berlin-New York|year= 1979|isbn= 3-540-90388-7 }}
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| *{{citation|mr=0043219|last=Stein|first= Karl|title=Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem|language=German|journal=Math. Ann. |volume=123|year=1951|pages=201–222}}
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| [[Category:Complex manifolds]]
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