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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. One of the reasons why this field has come to be studied is that if you increase several complicated variables that were one, the boundary of all domains may not be the natural boundary. Therefore, in the neighborhood of a branch point, it is not possible to discuss analytic continuation in the same way as in the case of a single complex variable. We consider the domain of holomorphy so that the domain that becomes holomorphic inside becomes the natural domain, but 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 1][ref 2][ref 3] Also Kiyoshi Oka's idéal de domaines indéterminés[ref 4] is interpreted by Cartan. In sheaf[ref 5] theory, the domain of holomorphy has come to be interpreted as the theory of Stein manifolds.[ref 6] 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.
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 (a name adopted, confusingly, for the geometry of zeroes of analytic functions: this is not the analytic geometry learned at school), 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 7] 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
is defined as the cartesian product of n complex planes , 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 1] 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.
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.
Every product of a family of connected (resp. path-connected) spaces is connected (resp. path-connected).
From Tychonoff's theorem, the space mapped by the cartesian product consisting of any combination of compact spaces is a compact space.
A function defined on a domain is called holomorphic if satisfies one of the following two conditions.
- (i) If is continuous on D[note 2]
- (ii) For each variable , is holomorphic, namely,
- which is a generalization of the Cauchy–Riemann equations (using a partial Wirtinger derivative), and has the origin of Riemann's differential equation methods.
For each index λ let
and generalize the usual Cauchy–Riemann equation for one variable for each index λ, then we obtain
the above equations (1) and (2) turn to be equivalent.
Cauchy's integral formula
f is meets condition 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 3] 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,
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
Because the order of products and sums is interchangeable, from (3) we get
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
If is holomorphic, on polydisc , from Cauchy's integral formula, we can see that it can be uniquely expanded to the next power series.
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.
Radius of convergence of power series
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 .
When the function f,g is holomorphic in the concatenated domain D,[note 6] even for several complex variables, the identity theorem[note 7] 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.
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 and V 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.
Power series expansion of Several complex variables have some points of convergence outside the circle of convergence, but it is possible to define a radius of convergence similar to that of one complex variable. Therefore, in order to investigate the characteristics of the domain of convergence of power series expansion of several complex variables, we define the domain of convergence of the invariant region by rotation and examine this characteristic. In other words, the convergent characteristics of the Reinhardt domain apply to the convergent characteristics of power series expansion 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 (cf. Hartogs domain) 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 centre 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.
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 8]
Thullen's classic results
Thullen's[ref 9] 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:
(2) (unit ball);
(3) (Thullen domain).
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 10] any holomorphic functions f on are analytically continued to . Namely, there is a holomorphic function F on such that on .
The convergence domain extends from to . i.e. The convergent domain of is extended to the smallest complete Reinhardt domain that can cover .
- 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
Function f is holomorpic on the domain , When f 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
The first result on the properties of the domain of holomorphy is the regular convexity of Henri Cartan and Peter Thullen (1932).[ref 12]
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 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 .
Pseudoconvex Hartogs showed that is subharmonic for the radius of convergence 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 sets are important, as they allow for classification of domains of holomorphy.
Definition of plurisubharmonic function
- A function
- with domain
is called plurisubharmonic if it is upper semi-continuous, and for every complex line
- 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
is subharmonic, where denotes the unit disk.
Strictly plurisubharmonic function
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.
Weak pseudoconvex[ref 13] 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 i.e there exists a smooth plurisubharmonic exhaustion function .
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 13]
If boundary, it can be shown that has a defining function; i.e., that there exists which is so that , and . Now, is pseudoconvex iff for every and in the complex tangent space at p, that is,
- , we have
If does not have a boundary, the following approximation result can be useful.
This is because once we have a as in the definition we can actually find a C∞ exhaustion function.
For domain D, always holds for any analytical disk that satisfies and .
Cartan pseudoconvex (local Levi property)
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
For a domain D the following conditions are equivalent:
- D is a domain of holomorphy.
- D is holomorphically convex.
- D is pseudoconvex.
- D is Levi convex - for every sequence of analytic compact surfaces such that for some set we have ( cannot be "touched from inside" by a sequence of analytic surfaces)
- D is Cartan pseudoconvex.
Implications [note 9] are standard results (for , see Oka's lemma). 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, and then by Lars Hörmander using methods from functional analysis and partial differential equations (a consequence of -problem).
Properties of the domain of holomorphy
- 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.
The definition of the coherent sheaf is as follows.[ref 14]
A coherent sheaf on a ringed space is a sheaf satisfying the following two properties:
- 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 ;
- 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.
- If in an exact sequence of sheaves of -modules two of the three sheaves are coherent, then the third is coherent as well.
for some (possibly infinite) sets and .
Oka's coherent theorem for sheaf of holomorphic functions
Kiyoshi Oka (1950)[ref 4] proved the following
- Sheaf of holomorphic function germ on the analytic variety is the coherent sheaf. Therefore, is also a coherent sheaf. This theorem is also used to prove Cartan's theorems A and B.
Manifolds considered with Several complex variables
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 analytic manifold. 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. So let's consider a manifold with enough 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.
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 15] 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.
- 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
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.
Properties and examples of Stein manifolds
- The standard[note 10] 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 17] 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 18]
- 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.
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".
- Complex geometry
- Complex projective space
- Several real variables
- Harmonic maps
- Harmonic morphisms
- Infinite-dimensional holomorphy
- The field of complex numbers is a 2-dimensional vector space over real numbers.
- Using Hartogs' theorem on separate holomorphicity, If condition (ii) is met, it will be derived to be continuous.
- According to the Jordan curve theorem, domain D is bounded closed set.
- This combination is generally not unique.
- 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.
- 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.
- 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.
- The final paragraph reduces to: A Reinhardt domain is a domain of holomorphy if and only if it is logarithmically convex.
- The Cartan–Thullen theorem
- ( is a projective complex varieties) does not become a Stein manifold, even if it satisfies the holomorphic convexity.
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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.
- Steven G. Krantz, What is Several Complex Variables? The American Mathematical Monthly Vol. 94, No. 3 (Mar., 1987), pp. 236-256 (21 pages) Published By: Taylor & Francis, Ltd. https://doi.org/10.2307/2323391