several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...
and
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a ...
s, a Stein manifold is a complex
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
of the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of ''n'' complex dimensions. They were introduced by and named after . 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
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a ...
of complex dimension and let denote the ring of
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
s on We call a Stein manifold if the following conditions hold:
* is holomorphically convex, i.e. for every
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
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
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
. A deep
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
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 In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry.
Let ''M'' be a complex manifold, and write ''O'M'' for the sheaf of holomorphic functions on ''M''. Let ''O'M''* be th ...
leads to the following exact sequence:
:
Now
Cartan's theorem B
In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf on a Stein manifold . They are significant both as applied to several complex variables, and in the general development o ...
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 mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be extended to a bigger domain.
For ...
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 biholomorphicproper 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
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
is a Stein manifold
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
it is not compact. This can be proved using a version of the
Runge theorem
In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Runge who first proved it in the year 1885. It states the following:
Denoting by C the set of complex numbers, let '' ...
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, 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 ''Xc'' agreeing with the usual orientation as the boundary of That is, is a Stein
filling
Filling may refer to:
* a food mixture used for stuffing
* Frosting used between layers of a cake
* Dental restoration
* Symplectic filling, a kind of cobordism in mathematics
* Part of the leather crusting process
See also
* Fill (disambiguat ...
of ''Xc''.
Numerous further characterizations of such manifolds exist, in particular capturing the property of their having "many"
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
s taking values in the complex numbers. See for example
Cartan's theorems A and B
In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf on a Stein manifold . They are significant both as applied to several complex variables, and in the general development o ...
, relating to
sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally whe ...
. The initial impetus was to have a description of the properties of the domain of definition of the (maximal)
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...
of an
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
.
In the
GAGA
Gaga ( he, גע גע literally 'touch touch') (also: ga-ga, gaga ball, or ga-ga ball) is a variant of dodgeball that is played in a gaga "pit". The game combines dodging, striking, running, and jumping, with the objective of being the last perso ...
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 2''n'', 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). Every closed smooth 4-manifold is a union of two Stein 4-manifolds glued along their common boundary.Selman Akbulut and Rostislav Matveyev, A convex decomposition for four-manifolds,
International Mathematics Research Notices
The ''International Mathematics Research Notices'' is a peer-reviewed mathematics journal. Originally published by Duke University Press and Hindawi Publishing Corporation, it is now published by Oxford University Press.Complex manifolds