Several complex variables

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The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions in the space Cn of n-tuples of complex numbers.

As in complex analysis of functions of one variable, which is the case n = 1, the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables zi. Equivalently, they are locally uniform limits of polynomials; or local solutions to the n-dimensional Cauchy–Riemann equations. For one complex variable, the any domain was the domain of holomorphy, but for several complex variables, the any domain is not the domain of holomorphy, so the domain of holomorphy is one of the themes in this field. Patching the local data of meromorphic functions, i.e. the problem of creating a global meromorphic function from zeros and poles, is called the Cousin problem. Also, the interesting phenomena that occur in several complex variables are fundamentally important to the study of compact complex manifolds and projective complex varieties and has a different flavour to complex analytic geometry in or on Stein manifolds.

Historical perspective[edit]

Many examples of such functions were familiar in nineteenth-century mathematics: abelian functions, theta functions, and some hypergeometric series. Naturally also any function of one variable that depends on some complex parameter is a candidate. The theory, however, for many years didn't become a full-fledged area in mathematical analysis, since its characteristic phenomena weren't uncovered. The Weierstrass preparation theorem would now be classed as commutative algebra; it did justify the local picture, ramification, that addresses the generalization of the branch points of Riemann surface theory.

With work of Friedrich Hartogs, and of Kiyoshi Oka in the 1930s, a general theory began to emerge; others working in the area at the time were Heinrich Behnke, Peter Thullen and Karl Stein. Hartogs proved some basic results, such as every isolated singularity is removable, for any analytic function

whenever n > 1. Naturally the analogues of contour integrals will be harder to handle: when n = 2 an integral surrounding a point should be over a three-dimensional manifold (since we are in four real dimensions), while iterating contour (line) integrals over two separate complex variables should come to a double integral over a two-dimensional surface. This means that the residue calculus will have to take a very different character.

After 1945 important work in France, in the seminar of Henri Cartan, and Germany with Hans Grauert and Reinhold Remmert, quickly changed the picture of the theory. A number of issues were clarified, in particular that of analytic continuation. Here a major difference is evident from the one-variable theory: while for any open connected set D in C we can find a function that will nowhere continue analytically over the boundary, that cannot be said for n > 1. In fact the D of that kind are rather special in nature (satisfying a condition called pseudoconvexity). The natural domains of definition of functions, continued to the limit, are called Stein manifolds and their nature was to make sheaf cohomology groups vanish. In fact it was the need to put (in particular) the work of Oka on a clearer basis that led quickly to the consistent use of sheaves for the formulation of the theory (with major repercussions for algebraic geometry, in particular from Grauert's work).

From this point onwards there was a foundational theory, which could be applied to analytic geometry, [note 1] automorphic forms of several variables, and partial differential equations. The deformation theory of complex structures and complex manifolds was described in general terms by Kunihiko Kodaira and D. C. Spencer. The celebrated paper GAGA of Serre[ref 1] pinned down the crossover point from géometrie analytique to géometrie algébrique.

C. L. Siegel was heard to complain that the new theory of functions of several complex variables had few functions in it, meaning that the special function side of the theory was subordinated to sheaves. The interest for number theory, certainly, is in specific generalizations of modular forms. The classical candidates are the Hilbert modular forms and Siegel modular forms. These days these are associated to algebraic groups (respectively the Weil restriction from a totally real number field of GL(2), and the symplectic group), for which it happens that automorphic representations can be derived from analytic functions. In a sense this doesn't contradict Siegel; the modern theory has its own, different directions.

Subsequent developments included the hyperfunction theory, and the edge-of-the-wedge theorem, both of which had some inspiration from quantum field theory. There are a number of other fields, such as Banach algebra theory, that draw on several complex variables.

The Cn space[edit]

is defined as the Cartesian product of n copies of , and when is a domain of holomorphy, can be regarded as a Stein manifold. It can be considered as an n-dimensional vector space over complex numbers, which gives its dimension 2n over R.[note 2] Hence, as a set, and as topological space, Cn is identical to R2n and its topological dimension is 2n.

In coordinate-free language, any vector space over complex numbers may be thought of as a real vector space of twice as many dimensions, where a complex structure is specified by a linear operator J (such that J 2 = I) which defines multiplication by the imaginary unit i.

Any such space, as a real space, is oriented. On the complex plane thought of as a Cartesian plane, multiplication by a complex number w = u + iv may be represented by the real matrix

with determinant

Likewise, if one expresses any finite-dimensional complex linear operator as a real matrix (which will be composed from 2 × 2 blocks of the aforementioned form), then its determinant equals to the square of absolute value of the corresponding complex determinant. It is a non-negative number, which implies that the (real) orientation of the space is never reversed by a complex operator. The same applies to Jacobians of holomorphic functions from Cn to Cn.

Connected space[edit]

Every product of a family of connected (resp. path-connected) spaces is connected (resp. path-connected).

Compact[edit]

From Tychonoff's theorem, the space mapped by the cartesian product consisting of any combination of compact spaces is a compact space.

Holomorphic functions[edit]

A function defined on a domain is called holomorphic if satisfies the following two conditions.[note 3][ref 2]

  1. is continuous[note 4] on D[note 5]
  2. For each variable , is holomorphic, namely,

     

     

     

     

    (1)

    which is a generalization of the Cauchy–Riemann equations (using a partial Wirtinger derivative), and has the origin of Riemann's differential equation methods.

Cauchy–Riemann equations[edit]

For each index λ let

and generalize the usual Cauchy–Riemann equation for one variable for each index λ, then we obtain

 

 

 

 

 

(2)

Let

through

the above equations (1) and (2) turn to be equivalent.

Cauchy's integral formula[edit]

f meets the conditions of being continuous and separately homorphic on domain D. Each disk has a rectifiable curve , is piecewise smoothness, class Jordan closed curve. () Let be the domain surrounded by each . Cartesian product closure is . Also, take the polydisc so that it becomes . ( and let be the center of each disk.) Using Cauchy's integral formula of one variable repeatedly,

Because is a rectifiable Jordanian closed curve[note 6] and f is continuous, so the order of products and sums can be exchanged so the iterated integral can be calculated as a multiple integral. Therefore,

 

 

 

 

(3)

While in the one-variable case Cauchy's integral formula is an integral over the circumference of a disc with some radius r, in several variables case over the surface of a polydisc with radii 's as in (3).

Cauchy's evaluation formula[edit]

Because the order of products and sums is interchangeable, from (3) we get

 

 

 

 

(4)

f is differentiable any number of times and the derivative is continuous.

From (4), if is holomorphic, on polydisc and , the following evaluation equation is obtained.

Therefore, Liouville's theorem hold.

Power series expansion of holomorphic functions[edit]

If is holomorphic, on polydisc , from Cauchy's integral formula, we can see that it can be uniquely expanded to the next power series.

 

 

 

 

(5)

In addition, that satisfies the following conditions is called an analytic function.

For each point , is expressed as a power series expansion that is convergent on D :

We have already explained that holomorphic functions are analytic. Also, from the theorem derived by Weierstrass , we can see that the analytic function (convergent power series) is holomorphic.

If a sequence of functions which converges uniformly on compacta inside a domain D, the limit function f of also uniformly on compacta inside a domain D. Also, respective partial derivative of also compactly converges on domain D to the corresponding derivative of f.
[ref 3]
Radius of convergence of power series[edit]

It is possible to define a combination of positive real numbers such that the power series converges uniformly at and does not converge uniformly at .

In this way it is possible to have a similar, combination of radius of convergence[note 7] for a one complex variable. This combination is generally not unique and there are an infinite number of combinations.

Identity theorem[edit]

When the function f,g is holomorphic in the concatenated domain D,[note 8] even for several complex variables, the identity theorem[note 9] holds on the domain D, because it has a power series expansion the neighbourhood of holomorphic point. Therefore, the maximal principle hold. Also, the inverse function theorem and implicit function theorem hold.

Analytic continuation[edit]

Let U, V be open subsets in , and . Assume that and is a connected component of . If then f is said to be connected to V, and g is said to be analytic continuation of f. From the identity theorem, if g exists, for each way of choosing w it is unique. Whether or not the definition of this analytic continuation is well-defined should be considered whether the domains U,V and W can be defined well. Several complex variables have restrictions on this domain, and depending on the shape of the domain , all holomorphic functions f belonging to U are connected to V, and there may be not exist function f with as the natural boundary. In other words, U cannot be defined. There is called the Hartogs's phenomenon. Therefore, researching when domain boundaries become natural boundaries has become one of the main research themes of Several complex variables. Also, in the general dimension, there may be multiple intersections between U and V. That is, f is not connected as a monovalent holomorphic function, but as an multivalued holomorphic function. This means that W is not unique and has different properties in the neighborhood of the branch point than in the case of one variable.

Reinhardt domain[edit]

Power series expansion of several complex variables it is possible to define the combination of radius of convergence similar to that of one complex variable, but each variable cannot independently define a unique radius of convergence. The Reinhardt domain is considered in order to investigate the characteristics of the convergence domain of the power series, but when considering the Reinhardt domain, it can be seen that the convergence domain of the power series satisfies the convexity called Logarithmically-convex. There are various convexity for the convergence domain of Several complex variables.

A domain D in the complex space , , with centre at a point , with the following property: Together with any point , the domain also contains the set

A Reinhardt domain D with is invariant under the transformations , , . The Reinhardt domains constitute a subclass of the Hartogs domains [ref 4] and a subclass of the circular domains, which are defined by the following condition: Together with any , the domain contains the set

i.e. all points of the circle with center and radius that lie on the complex line through and .

A Reinhardt domain D is called a complete Reinhardt domain if together with any point it also contains the polydisc

A complete Reinhardt domain is star-like with respect to its centre a. Therefore, the complete Reinhardt domain is simply connected, also when the complete Reinhardt domain is the boundary line, there is a way to prove Cauchy's integral theorem without using the Jordan curve theorem.

Logarithmically-convex[edit]

A Reinhardt domain D is called logarithmically convex if the image of the set

under the mapping

is a convex set in the real space . An important property of logarithmically-convex Reinhardt domains is the following: Every such domain in is the interior of the set of points of absolute convergence (i.e. the domain of convergence) of some power series in , and conversely: The domain of convergence of any power series in is a logarithmically-convex Reinhardt domain with centre . [note 10]

Some results[edit]

Thullen's classic results[edit]

Thullen's[ref 5] classical result says that a 2-dimensional bounded Reinhard domain containing the origin is biholomorphic to one of the following domains provided that the orbit of the origin by the automorphism group has positive dimension:

  1. (polydisc);
  2. (unit ball);
  3. (Thullen domain).

Hartogs's phenomenon[edit]

Let's look at the example on the Hartogs's extension theorem page in terms of the Reinhardt domain.

On the polydisk consisting of two disks when .

Internal domain of

Theorem Hartogs (1906)[ref 6] Let f be a holomorphic function on a set G \ K, where G is an open subset of Cn (n ≥ 2) and K is a compact subset of G. If the complement G \ K is connected, then f can be extended to a unique holomorphic function on G.

From Hartogs's extension theorem the convergence domain extends from to . Looking at this from the perspective of the Reinhardt domain, is the Reinhardt domain containing the center z = 0, and the convergence domain of has been extended to the smallest complete Reinhardt domain containing .[ref 7]

Sunada's results[edit]

Toshikazu Sunada (1978)[ref 8] established a generalization of Thullen's result:

Two n-dimensional bounded Reinhardt domains and are mutually biholomorphic if and only if there exists a transformation given by , being a permutation of the indices), such that .

Domain of holomorphy[edit]

The sets in the definition. Note: On this page, replace in the figure with D

When moving from the theory of one complex variable to the theory of several complex variables, depending on the range of the domain, it may not be possible to define a holomorphic function such that the boundary of the domain becomes a natural boundary. Considering the domain where the boundaries of the domain are natural boundaries (i.e. domain of holomorphy), the first result in the domain of holomorphy was the holomorphic convexity of H. Cartan and Thullen. Levi's problem shows that the pseudoconvex domain was a domain of holomorphy.[ref 9][ref 10][ref 11][ref 12] Also Kiyoshi Oka's idéal de domaines indéterminés[ref 13] is interpreted by Cartan.[note 11] In sheaf[ref 14] theory, the domain of holomorphy has come to be interpreted as the theory of Stein manifolds.[ref 15]

Definition[edit]

When a function f is holomorpic on the domain and cannot directly connect to the domain outside D, including the point of the domain boundary , the domain D is called the domain of holomorphy of f and the boundary is called the natural boundary of f. In other words, the domain of holomorphy D is the supremum of the domain where the holomorphic function f is holomorphic, and the domain D, which is holomorphic, cannot be extended any more. For Several complex variables, i.e. domain , the boundaries may not be natural boundaries. Hartogs' extension theorem gives an example of a domain where boundaries are not natural boundaries.

Formally, an open set D in the n-dimensional complex space is called a domain of holomorphy if there do not exist non-empty open sets and where V is connected, and such that for every holomorphic function f on D there exists a holomorphic function g on V with on U.

In the case, every open set is a domain of holomorphy: we can define a holomorphic function with zeros accumulating everywhere on the boundary of the domain, which must then be a natural boundary for a domain of definition of its reciprocal.

Holomorphically convex hull[edit]

The first result on the properties of the domain of holomorphy is the holomorphic convexity of Henri Cartan and Peter Thullen (1932).[ref 16]

The holomorphically convex hull of a given compact set in the n-dimensional complex space is defined as follows.

Let be a domain (an open set and connected set), or alternatively for a more general definition, let be an dimensional complex analytic manifold. Further let stand for the set of holomorphic functions on G. For a compact set , the holomorphically convex hull of K is

One obtains a narrower concept of polynomially convex hull by taking instead to be the set of complex-valued polynomial functions on G. The polynomially convex hull contains the holomorphically convex hull.

The domain is called holomorphically convex if for every compact subset is also compact in G. Sometimes this is just abbreviated as holomorph-convex.

When , any domain is holomorphically convex since then is the union of with the relatively compact components of .

If satisfies the above holomorphically convexity it has the following properties. The radius of the polydisc satisfies condition also the compact set satisfies and is the domain. In the time that, any holomorphic function on the domain can be direct analytic continuated up to .

Levi convex (Approximate from the inside on the analytic polyhedron domain)[edit]

is union of ascending sequence of analytic compact surfaces with paracompact and Holomorphically convex properties such that . i.e. Approximate from the inside by analytic polyhedron. [note 12]

Pseudoconvex[edit]

Pseudoconvex Hartogs showed that is subharmonic for the radius of convergence in the Hartogs series when the Hartogs series is a one-variable . If such a relationship holds in the domain of holomorphy of Several complex variables, it looks like a more manageable condition than a holomorphically convex. The subharmonic function looks like a kind of convex function, so it was named by Levi as a pseudoconvex domain. Pseudoconvex domain are important, as they allow for classification of domains of holomorphy.

Definition of plurisubharmonic function[edit]

A function
with domain

is called plurisubharmonic if it is upper semi-continuous, and for every complex line

with
the function is a subharmonic function on the set
In full generality, the notion can be defined on an arbitrary complex manifold or even a Complex analytic space as follows. An upper semi-continuous function
is said to be plurisubharmonic if and only if for any holomorphic map

the function

is subharmonic, where denotes the unit disk.

Strictly plurisubharmonic function[edit]

Necessary and sufficient condition that the real-valued function u(z), that can be second-order differentiable with respect to z of one-variable complex function is subharmonic is . When the Hermitian matrix of u is positive-definite and class , we call u a strict plural subharmonic function.

(Weakly) pseudoconvex (p-pseudoconvex)[edit]

Weak pseudoconvex[ref 17] is defined as : Let be a domain, that is, an open connected subset. One says that X is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function on X such that the set is a relatively compact subset of X for all real numbers x. [note 13] i.e there exists a smooth plurisubharmonic exhaustion function .

Strongly pseudoconvex[edit]

Strongly pseudoconvex if there exists a smooth strictly plurisubharmonic exhaustion function ,i.e. is positive definite at every point. The strongly pseudoconvex domain is the pseudoconvex domain.[ref 17]

Levi–Krzoska pseudoconvexity[edit]

If boundary (i.e. When D is a strongly pseudoconvex domain.), it can be shown that D has a defining function; i.e., that there exists which is so that , and . Now, D is pseudoconvex iff for every and in the complex tangent space at p, that is,

, we have

If D does not have a boundary, the following approximation result can be useful.

Proposition 1 If D is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains with (smooth) boundary which are relatively compact in D, such that

This is because once we have a as in the definition we can actually find a C exhaustion function.

Levi Strongly Pseudoconvex (Levi total Pseudoconvex)[edit]

If for every boundary point of D, there exists an analytic variety passing which lies entirely outside D in some neighborhood around , except the point itself. Domain D that satisfies these conditions is called Levi Strongly Pseudoconvex or Levi total Pseudoconvex.[ref 18]

Oka pseudoconvex[edit]

Family of Oka's disk[edit]

Let n-functions be continuous on , holomorphic in when the parameter t is fixed in [0, 1], and assume that are not all zero at any point on . Then the set is called an analytic disc de-pending on a parameter t, and is called its shell. If and , Q(t) is called Family of Oka's disk.[ref 18]

Definition[edit]

When holds on any Family of Oka's disk, D is called Oka pseudoconvex.[ref 18] Oka's proof of Levi's problem was proved by the fact that each boundary point of the domain of holomorphy is an Oka pseudoconvex.[ref 10]

Cartan pseudoconvex (Local Levi property)[edit]

For every point there exist a neighbourhood U of x and f holomorphic on such that f cannot be extended to any neighbourhood of x. Such a property is called local Levi property, and the domain that satisfies this property is called the Cartan pseudoconvex domain. The Cartan pseudoconvex domain is itself a pseudoconvex domain and is a domain of holomorphy.[ref 18]

Equivalent conditions (In connection with Levi problem)[edit]

For a domain the following conditions are equivalent.[note 14]:

  1. D is a domain of holomorphy.
  2. D is holomorphically convex.
  3. D is Levi convex.
  4. D is pseudoconvex.
  5. D is Cartan pseudoconvex.

The implications ,[note 15] ,[note 16] and are standard results. Proving , i.e. constructing a global holomorphic function which admits no extension from non-extendable functions defined only locally. This is called the Levi problem (after E. E. Levi) and was first solved by Kiyoshi Oka,[note 17] and then by Lars Hörmander using methods from functional analysis and partial differential equations (a consequence of -problem).

Properties of the domain of holomorphy[edit]

  • If are domains of holomorphy, then their intersection is also a domain of holomorphy.
  • If is an ascending sequence of domains of holomorphy, then their union is also a domain of holomorphy (see Behnke–Stein theorem).
  • If and are domains of holomorphy, then is a domain of holomorphy.
  • The first Cousin problem is always solvable in a domain of holomorphy; this is also true, with additional topological assumptions, for the second Cousin problem.

Sheaf[edit]

Coherent sheaf[edit]

Definition[edit]

The definition of the coherent sheaf is as follows.[ref 24]

A coherent sheaf on a ringed space is a sheaf satisfying the following two properties:

  1. is of finite type over , that is, every point in has an open neighborhood in such that there is a surjective morphism for some natural number ;
  2. for any open set , any natural number , and any morphism of -modules, the kernel of is of finite type.

Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of -modules.

Also, Jean-Pierre Serre (1955)[ref 24] proves that

If in an exact sequence of sheaves of -modules two of the three sheaves are coherent, then the third is coherent as well.

A quasi-coherent sheaf on a ringed space is a sheaf of -modules which has a local presentation, that is, every point in has an open neighborhood in which there is an exact sequence

for some (possibly infinite) sets and .

Oka's coherent theorem for sheaf of holomorphic function germ[edit]

Kiyoshi Oka (1950)[ref 13][ref 25] proved the following

Sheaf of holomorphic function germ on the complex manifold is the coherent sheaf. Therefore, from the above Serre(1955) theorem, is also a coherent sheaf. This theorem is also used to prove Cartan's theorems A and B.

Ideal sheaf[edit]

If is a closed subscheme of a locally Noetherian scheme , the sheaf of all regular functions vanishing on is coherent. Likewise, if is a closed analytic subspace of a complex analytic space , the ideal sheaf is coherent.

Cousin problem[edit]

In the case of one variable complex functions, Mittag-Leffler's theorem was able to create a global meromorphic function from a given pole, and Weierstrass factorization theorem was able to create a global meromorphic function from a given zero. The theory of Riemann's surface suggests that in multivariate complex functions, the similar theorem that holds for one-variable complex functions does not hold unless Several restrictions are added to the open Complex manifold. This problem is called the Cousin problem and is formulated in Sheaf cohomology terms. They were introduced in special cases by Pierre Cousin in 1895. It was Kiyoshi Oka who gave the complete answer to this question.[ref 26][ref 27][ref 28]

First Cousin problem[edit]

Definition without Sheaf words[edit]

Each difference is a holomorphic function, where it is defined. It asks for a meromorphic function f on M such that is holomorphic on Ui; in other words, that f shares the singular behaviour of the given local function.

Definition using Sheaf words[edit]

Let K be the sheaf of meromorphic functions and O the sheaf of holomorphic functions on M. If the next map is surjective, Cousin first problem can be solved.

By the long exact cohomology sequence,

is exact, and so the first Cousin problem is always solvable provided that the first cohomology group H1(M,O) vanishes. In particular, by Cartan's theorem B, the Cousin problem is always solvable if M is a Stein manifold.

Second Cousin problem[edit]

Definition without Sheaf words[edit]

Each ratio is a non-vanishing holomorphic function, where it is defined. It asks for a meromorphic function f on M such that is holomorphic and non-vanishing.

Definition using Sheaf words[edit]

let be the sheaf of holomorphic functions that vanish nowhere, and the sheaf of meromorphic functions that are not identically zero. These are both then sheaves of abelian groups, and the quotient sheaf is well-defined. If the next map is surjective, then Second Cousin problem can be solved.

The long exact sheaf cohomology sequence associated to the quotient is

so the second Cousin problem is solvable in all cases provided that

The cohomology group for the multiplicative structure on can be compared with the cohomology group with its additive structure by taking a logarithm. That is, there is an exact sequence of sheaves

where the leftmost sheaf is the locally constant sheaf with fiber . The obstruction to defining a logarithm at the level of H1 is in , from the long exact cohomology sequence

When M is a Stein manifold, the middle arrow is an isomorphism because for so that a necessary and sufficient condition in that case for the second Cousin problem to be always solvable is that

Manifolds considered with Several complex variables[edit]

Stein manifold[edit]

Since the open Riemann surface always has a non-constant monovalent holomorphic function and satisfies the second axiom of countability, the Riemann surface was considered for embedding the one-dimensional complex plane into a complex manifolds. In fact, taking one point at infinity on the one-dimensional complex plane extended it to the Riemann sphere. The Whitney embedding theorem tells us that every smooth n-dimensional manifold can be embedded as a smooth submanifold of , whereas it is "rare" for a complex manifold to have a holomorphic embedding into . Consider for example any compact connected complex manifold X: any holomorphic function on it is constant by Liouville's theorem. Now that we know that for Several complex variables, complex manifolds do not always have holomorphic functions that are not constants, consider the conditions that have holomorphic functions. Now if we had a holomorphic embedding of X into , then the coordinate functions of would restrict to nonconstant holomorphic functions on X, contradicting compactness, except in the case that X is just a point. Complex manifolds that can be embedded in Cn are called Stein manifolds. Also Stein manifolds satisfy the second axiom of countability.

Stein manifold is a complex submanifold of the vector space of n complex dimensions. They were introduced by and named after Karl Stein (1951).[ref 29] 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. If the univalent domain on is connection to a manifold, can be regarded as a complex manifold and satisfies the separation condition described later, the condition for becoming a Stein manifold is to satisfy the holomorphic convexity. Therefore, the Stein manifold is the properties of the domain of definition of the (maximal) analytic continuation of an analytic function.

Definition[edit]

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

  • X is holomorphically convex, i.e. for every compact subset , the so-called holomorphically convex hull,
is also a compact subset of X.
  • X is holomorphically separable, i.e. if are two points in X, then there exists such that
  • The open neighborhood of any point on the manifold has a holomorphic Chart to the .

Non-compact Riemann surfaces are Stein[edit]

Let X be a connected, non-compact Riemann surface. A deep theorem of Heinrich Behnke and Stein (1948)[ref 30] 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 (multiplicative) Cousin problem.

Levi problem[edit]

Cartan extended Levi's problem to Stein manifolds.[ref 31]

If the relative compact open subset of the Stein manifold X is a Cartan pseudoconvex, then D is a Stein manifold, and conversely, if D is a Cartan pseudoconvex, then X is a Stein manifold. i.e. Then X is a Stein manifold if and only if D is locally the Stein manifold.[ref 32]

This was proved by Bremermann[ref 33] by embedding it in a sufficiently high dimensional , and reducing it to the result of Oka.[ref 10]

Also, Grauert proved for arbitrary complex manifolds.[ref 34][ref 12]

If the relative compact subset of a arbitrary complex manifold X is a strongly pseudoconvex on X, then X is a holomorphically convex (i.e. Stein manifold). Also, D is itself a Stein manifold.

Properties and examples of Stein manifolds[edit]

  • The standard[note 18] 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 X 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 x.
  • The first Cousin problem can always be solved on a Stein manifold.
  • 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 X for every real number . This is a solution to the so-called Levi problem,[ref 35] 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.[ref 34]
  • 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 of Xc.

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.

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".

See also[edit]

Annotation[edit]

  1. ^ a name adopted, confusingly, for the geometry of zeroes of analytic functions: this is not the analytic geometry learned at school
  2. ^ The field of complex numbers is a 2-dimensional vector space over real numbers.
  3. ^ This may seem nontrivial, but it's known as Osgood's lemma. Osgood's lemma can be proved from the establishment of Cauchy's integral formula, also Cauchy's integral formula can be proved by assuming separate holomorphicity and continuity, so it is appropriate to define it in this way.
  4. ^ It is not separate continuous.
  5. ^ Using Hartogs's theorem on separate holomorphicity, If condition (ii) is met, it will be derived to be continuous. But, there is no theorem similar to several real variables, and there is no theorem that indicates the continuity of the function, assuming differentiability.
  6. ^ According to the Jordan curve theorem, domain D is bounded closed set.
  7. ^ But there is a point where it converges outside the circle of convergence. For example if one of the variables is 0, then some terms, represented by the product of this variable, will be 0 regardless of the values taken by the other variables. Therefore, even if you take a variable that diverges when a variable is other than 0, it may converge.
  8. ^ For several variables, the boundary of any domain is not always the natural boundary, so depending on how the domain is taken, there may not be a holomorphic function that makes that domain the natural boundary. See domain of holomorphy for an example of a condition where the boundary of a domain is a natural boundary.
  9. ^ Note that from Hartogs' extension theorem, the zeros of holomorphic functions of several variables are not isolated points. Therefore, for several variables it is not enough that is satisfied at the accumulation point.
  10. ^ The final paragraph reduces to: A Reinhardt domain is a domain of holomorphy if and only if it is logarithmically convex.
  11. ^ The idea of the sheaf itself is by Jean Leray.
  12. ^ cannot be "touched from inside" by a sequence of analytic surfaces
  13. ^ This is a hullomorphically convex hull condition expressed by a plurisubharmonic function. For this reason, it is also called p-pseudoconvex or simply p-convex.
  14. ^ In algebraic geometry, there is a problem whether it is possible to remove the singular point of the complex analytic space by performing an operation called modification[ref 19][ref 20] on the complex analytic space (when n = 2, the result by Hirzebruch,[ref 21] when n = 3 the result by Zariski[ref 22] for algebraic varietie.), but, Grauert and Remmert has reported an example of a domain that is neither pseudoconvex nor holomorphic convex, even though it is a domain of holomorphy. [ref 23]
  15. ^ The Cartan–Thullen theorem
  16. ^ See Oka's lemma
  17. ^ Oka's proof uses Oka pseudoconvex instead of Cartan pseudoconvex.
  18. ^ ( is a projective complex varieties) does not become a Stein manifold, even if it satisfies the holomorphic convexity.

References[edit]

  1. ^ Serre, Jean-Pierre (1956), "Géométrie algébrique et géométrie analytique", Annales de l'Institut Fourier, 6: 1–42, doi:10.5802/aif.59, ISSN 0373-0956, MR 0082175
  2. ^ Osgood, William F. (1899), "Note über analytische Functionen mehrerer Veränderlichen", Mathematische Annalen, Springer Berlin / Heidelberg, 52: 462–464, doi:10.1007/BF01476172, ISSN 0025-5831, JFM 30.0380.02, S2CID 121407666 CS1 maint: discouraged parameter (link)
  3. ^ Solomentsev, E.D. (2001) [1994], "Weierstrass theorem", Encyclopedia of Mathematics, EMS Press
  4. ^ Chirka, E.M. (2001) [1994], "Hartogs domain", Encyclopedia of Mathematics, EMS Press
  5. ^ Peter Thullen, Zu den Abbildungen durch analytische Funktionen mehrerer komplexer Veraenderlichen Die Invarianz des Mittelpunktes von Kreiskoerpern, Matt. Ann. 104 (1931), 244–259
  6. ^ Hartogs, Fritz (1906), "Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veränderlichen.", Sitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften zu München, Mathematisch-Physikalische Klasse (in German), 36: 223–242, JFM 37.0443.01
  7. ^ Cartan, Henri (1931). "Les fonctions de deux variables complexes et le problème de la représentation analytique". Journal de Mathématiques Pures et Appliquées. 10: 1–116. Zbl 0001.28501.
  8. ^ Tosikazu Sunada, Holomorphic equivalence problem for bounded Reinhaldt domains, Math. Ann. 235 (1978), 111–128
  9. ^ In the case of Oka, Kiyoshi (1942), "Sur les fonctions analytiques de plusieurs variables. VI. Domaines pseudoconvexes", Tohoku Mathematical Journal, First Series, 49: 15–52, ISSN 0040-8735, Zbl 0060.24006
  10. ^ a b c Extension to Oka, Kiyoshi (1953), "Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique intérieur", Japanese Journal of Mathematics :transactions and Abstracts, 23: 97–155, doi:10.4099/jjm1924.23.0_97, ISSN 0075-3432
  11. ^ Extension to Hans J. Bremermann (1954), "Über die Äquivalenz der pseudokonvexen Gebiete und der Holomorphiegebiete im Raum vonn komplexen Veränderlichen", Mathematische Annalen, 106: 63–91, doi:10.1007/BF01360125, S2CID 119837287
  12. ^ a b Huckleberry, Alan (2013). "Hans Grauert (1930–2011)". Jahresbericht der Deutschen Mathematiker-Vereinigung. 115: 21–45. arXiv:1303.6933. doi:10.1365/s13291-013-0061-7.
  13. ^ a b Oka, Kiyoshi (1950), "Sur les fonctions analytiques de plusieurs variables. VII. Sur quelques notions arithmétiques", Bulletin de la Société Mathématique de France, 78: 1–27, doi:10.24033/bsmf.1408, ISSN 0037-9484, MR 0035831
  14. ^ Numdam.org, Cartan, H., Eilenberg, Samuel., Serre, J-P., Séminaire Henri Cartan Tome 3 (SHC_1950-1951)
  15. ^ Numdam.org, Cartan, H., Bruhat, F., Cerf, Jean., Dolbeault, P., Frenkel, Jean., Hervé, Michel., Malatian., Serre, J-P., Séminaire Henri Cartan Tome 4 (1951-1952)
  16. ^ Henri Cartan&Peter Thullen (1932), "Zur Theorie der Singularitäten der Funktionen mehrerer komplexen Veränderlichen Regularitäts-und Konvergenzbereiche", Mathematische Annalen, 106: 617–647, doi:10.1007/BF01455905
  17. ^ a b Complex Analytic and Differential Geometry p.49
  18. ^ a b c d Sin Hitomatsu (1958), "On some conjectures concerning pseudo-convex domains", Journal of the Mathematical Society of Japan, 6 (No.2) (2): 177–195, doi:10.2969/jmsj/00620177, Zbl 0057.31503
  19. ^ Heinrich Behnke & Karl Stein (1951), "Modifikationen komplexer Mannigfaltigkeiten und Riernannscher Gebiete", Mathematische Annalen, 124: 1–16, doi:10.1007/BF01343548, S2CID 120455177, Zbl 0043.30301
  20. ^ Onishchik, A.L. (2001) [1994], "Modification", Encyclopedia of Mathematics, EMS Press
  21. ^ Friedrich Hirzebruch (1953), "Über vierdimensionaleRIEMANNsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen", Mathematische Annalen, 126: 1–22, doi:10.1007/BF01343146, hdl:21.11116/0000-0004-3A47-C, S2CID 122862268
  22. ^ Oscar Zariski (1944), "Reduction of the Singularities of Algebraic Three Dimensional Varieties", Annals of Mathematics, Second Series, 45 (3): 472–542, doi:10.2307/1969189, JSTOR 1969189
  23. ^ Hans Grauert & Reinhold Remmert (1956), "Konvexität in der komplexen Analysis. Nicht-holomorph-konvexe Holomorphiegebiete und Anwendungen auf die Abbildungstheorie.", Commentarii Mathematici Helvetici Volume, 31: 152–183, doi:10.1007/BF02564357, S2CID 117913713, Zbl 0073.30301
  24. ^ a b Serre, Jean-Pierre (1955), "Faisceaux algébriques cohérents", Annals of Mathematics, 61 (2): 197–278, doi:10.2307/1969915, JSTOR 1969915, MR 0068874
  25. ^ Noguchi, Junjiro (2019), "A Weak Coherence Theorem and Remarks to the Oka Theory" (PDF), Kodai Math. J., 42 (3): 566–586, arXiv:1704.07726, doi:10.2996/kmj/1572487232, S2CID 119697608
  26. ^ Oka, Kiyoshi (1936). "Sur les fonctions analytiques de plusieurs variables. I. Domaines convexes par rapport aux fonctions rationnelles". Journal of Science of the Hiroshima University. 6: 245–255. doi:10.32917/hmj/1558749869.
  27. ^ Oka, Kiyoshi (1937). "Sur les fonctions analytiques de plusieurs variables. II–Domaines d'holomorphie". Journal of Science of the Hiroshima University. 7: 115–130. doi:10.32917/hmj/1558576819.
  28. ^ Oka, Kiyoshi (1939). "Sur les fonctions analytiques de plusieurs variables. III–Deuxième problème de Cousin". Journal of Science of the Hiroshima University. 9: 7–19. doi:10.32917/hmj/1558490525.
  29. ^ 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
  30. ^ In the case of Heinrich Behnke & Karl Stein (1948), "Entwicklung analytischer Funktionen auf Riemannschen Flächen", Mathematische Annalen, 120: 430–461, doi:10.1007/BF01447838, S2CID 122535410, Zbl 0038.23502
  31. ^ Cartan, Henri (1957). "Variétés analytiques réelles et variétés analytiques complexes". Bulletin de la Société Mathématique de France. 85: 77–99. doi:10.24033/bsmf.1481.
  32. ^ Barth, Theodore J. (1968). "Families of nonnegative divisors". Trans. Amer. Math. Soc. 131: 223–245. doi:10.1090/S0002-9947-1968-0219751-3.
  33. ^ Bremermann, Hans J. (1957). "On Oka's theorem for Stein manifolds". Seminars on Analytic Functions. Institute for Advanced Study (Princeton, N.J.). 1: 29–35.
  34. ^ a b Hans Grauert (1958), "On Levi's Problem and the Imbedding of Real-Analytic Manifolds", Annals of Mathematics, Second Series, 68 (2): 460–472, doi:10.2307/1970257, JSTOR 1970257, Zbl 0108.07804
  35. ^ Onishchik, A.L. (2001) [1994], "Levi problem", Encyclopedia of Mathematics, EMS Press

Textbooks[edit]

Encyclopedia of Mathematics[edit]

PlanetMath[edit]

This article incorporates material from Reinhardt domain on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. This article incorporates material from Holomorphically convex on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. This article incorporates material from Domain of holomorphy on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Further reading[edit]

  • Krantz, Steven G. (1987), "What is Several Complex Variables?", The American Mathematical Monthly, 94 (3): 236–256, doi:10.2307/2323391, JSTOR 2323391
  • Oka, Kiyoshi; R., Remmert(Ed.) (1984), Collected Papers, Springer-Verlag Berlin Heidelberg, p. XIV, 226, ISBN 978-3-662-43412-3CS1 maint: extra text: authors list (link)

External links[edit]