In mathematics, in the theory ofseveral complex variables andcomplex manifolds, aStein manifold is a complexsubmanifold of thevector space ofncomplex dimensions. They were introduced by and named afterKarl Stein (1951). AStein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues ofaffine varieties oraffine schemes in algebraic geometry.

Suppose${\displaystyle X}$ is acomplex manifold of complex dimension${\displaystyle n}$ and let${\displaystyle \mathcal{O}(X)}$ denote the ring ofholomorphic functions on${\displaystyle X.}$ We call${\displaystyle X}$ aStein manifold if the following conditions hold:

• ${\displaystyle X}$ is holomorphically convex, i.e. for everycompact subset${\displaystyle K\subset X}$, the so-calledholomorphically convex hull,

${\displaystyle \overline{K}=\left\{z\in X||f(z)|\le \underset{w\in K}{sup}|f(w)|\mathrm{\forall }f\in \mathcal{O}(X)\right\},}$

is also acompact subset of${\displaystyle X}$.

• ${\displaystyle X}$ is holomorphically separable, i.e. if${\displaystyle x\ne y}$ are two points in${\displaystyle X}$, then there exists${\displaystyle f\in \mathcal{O}(X)}$ such that${\displaystyle f(x)\ne f(y).}$

LetX be a connected, non-compactRiemann surface. A deeptheorem ofHeinrich Behnke and Stein (1948) asserts thatX is a Stein manifold.

Another result, attributed toHans Grauert andHelmut Röhrl (1956), states moreover that everyholomorphic vector bundle onX is trivial. In particular, every line bundle is trivial, so${\displaystyle H^1(X,{\mathcal{O}}_X^{\ast })=0}$. Theexponential sheaf sequence leads to the followingexact sequence:

${\displaystyle H^1(X,{\mathcal{O}}_X)⟶H^1(X,{\mathcal{O}}_X^{\ast })⟶H^2(X,\mathbb{Z})⟶H^2(X,{\mathcal{O}}_X)}$

NowCartan's theorem B shows that${\displaystyle H^1(X,{\mathcal{O}}_X)=H^2(X,{\mathcal{O}}_X)=0}$, therefore${\displaystyle H^2(X,\mathbb{Z})=0}$.

This is related to the solution of thesecond Cousin problem.

• The standard complex space${\displaystyle {\mathbb{C}}^n}$ is a Stein manifold.

• Everydomain of holomorphy in${\displaystyle {\mathbb{C}}^n}$ is a Stein manifold.
• EveryFatou–Bieberbach domain in${\displaystyle {\mathbb{C}}^n}$ is a Stein manifold.

• 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${\displaystyle X}$ of complex dimension${\displaystyle n}$ can be embedded into${\displaystyle {\mathbb{C}}^{2n+1}}$ by abiholomorphicproper map.

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

• Every Stein manifold of (complex) dimensionn has the homotopy type of ann-dimensional CW-complex.

• In one complex dimension the Stein condition can be simplified: a connectedRiemann surface is a Stein manifoldif and only if it is not compact. This can be proved using a version of theRunge theorem for Riemann surfaces, due to Behnke and Stein.

• Every Stein manifold${\displaystyle X}$ is holomorphically spreadable, i.e. for every point${\displaystyle x\in X}$, there are${\displaystyle n}$ holomorphic functions defined on all of${\displaystyle X}$ which form a local coordinate system when restricted to some open neighborhood of${\displaystyle x}$.

• Being a Stein manifold is equivalent to being a (complex)strongly pseudoconvex manifold. The latter means that it has a strongly pseudoconvex (orplurisubharmonic) exhaustive function, i.e. a smooth real function${\displaystyle \psi }$ on${\displaystyle X}$ (which can be assumed to be aMorse function) with${\displaystyle i\mathrm{\partial }\overline{\mathrm{\partial }}\psi >0}$, such that the subsets${\displaystyle \{z\in X\mid \psi (z)\le c\}}$ are compact in${\displaystyle X}$ for every real number${\displaystyle c}$. This is a solution to the so-calledLevi problem,^[1] named afterEugenio Levi (1911). The function${\displaystyle \psi }$ invites a generalization ofStein manifold to the idea of a corresponding class of compact complex manifolds with boundary calledStein domains. A Stein domain is the preimage${\displaystyle \{z\mid -\mathrm{\infty }\le \psi (z)\le c\}}$. 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 surfaceX with a real-valued Morse functionf onX such that, away from the critical points off, the field of complex tangencies to the preimage${\displaystyle X_c=f^{-1}(c)}$ is acontact structure that induces an orientation onX_c agreeing with the usual orientation as the boundary of${\displaystyle f^{-1}(-\mathrm{\infty },c).}$ That is,${\displaystyle f^{-1}(-\mathrm{\infty },c)}$ is a Steinfilling ofX_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 exampleCartan's theorems A and B, relating tosheaf cohomology. The initial impetus was to have a description of the properties of the domain of definition of the (maximal)analytic continuation of ananalytic function.

In theGAGA set of analogies, Stein manifolds correspond toaffine varieties.

Stein manifolds are in some sense dual to theelliptic 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 isfibrant in the sense of so-called "holomorphic homotopy theory".

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

• Andrist, Rafael (2010)."Stein spaces characterized by their endomorphisms".Transactions of the American Mathematical Society.363 (5):2341–2355.arXiv:0809.3919.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)
• Forstnerič, Franc (2011).Stein Manifolds and Holomorphic Mappings. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics. Vol. 56.doi:10.1007/978-3-642-22250-4.ISBN 978-3-642-22249-8.
• 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,148 (2), The Annals of Mathematics, Vol. 148, No. 2:619–693,arXiv:math/9803019,doi:10.2307/121005,ISSN 0003-486X,JSTOR 121005,MR 1668563,S2CID 17709531 (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,S2CID 122647212
• Zhang, Jing (2008). "Algebraic Stein varieties".Mathematical Research Letters.15 (4):801–814.arXiv:math/0610886.doi:10.4310/MRL.2008.v15.n4.a16.MR 2424914.

• Complex manifolds
• Several complex variables

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