Multivariable Complex Analysis

[Channing Arther]

Thetheory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space C n \displaystyle \mathbb C ^n , that is, n-tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space), which the Mathematics Subject Classification has as a top-level heading.

Many examples of such functions were familiar in nineteenth-century mathematics; abelian functions, theta functions, and some hypergeometric series, and also, as an example of an inverse problem; the Jacobi inversion problem.[7] Naturally also same 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 field 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.

From this point onwards there was a foundational theory, which could be applied to analytic geometry, [note 2] 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[8] pinned down the crossover point from gometrie analytique to gometrie algbrique.

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 complex coordinate space C n \displaystyle \mathbb C ^n is the Cartesian product of n copies of C \displaystyle \mathbb C , and when C n \displaystyle \mathbb C ^n is a domain of holomorphy, C n \displaystyle \mathbb C ^n can be regarded as a Stein manifold, and more generalized Stein space. C n \displaystyle \mathbb C ^n is also considered to be a complex projective variety, a Khler manifold,[9] etc. It is also an n-dimensional vector space over the complex numbers, which gives its dimension 2n over R \displaystyle \mathbb R .[note 3] Hence, as a set and as a topological space, C n \displaystyle \mathbb C ^n may be identified to the real coordinate space R 2 n \displaystyle \mathbb R ^2n and its topological dimension is thus 2n.

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 C n \displaystyle \mathbb C ^n to C n \displaystyle \mathbb C ^n .

are holomorphic as functions of one complex variable : we say that f is holomorphic in each variable separately. Conversely, if f is holomorphic in each variable separately, then f is in fact holomorphic : this is known as Hartog's theorem, or as Osgood's lemma under the additional hypothesis that f is continuous.

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

In this way it is possible to have a similar, combination of radius of convergence[note 6] for a one complex variable. This combination is generally not unique and there are an infinite number of combinations.

The maximal principle, inverse function theorem, and implicit function theorems also hold. For a generalized version of the implicit function theorem to complex variables, see the Weierstrass preparation theorem.

When n > 1 \displaystyle n>1 , open balls and open polydiscs are not biholomorphically equivalent, that is, there is no biholomorphic mapping between the two.[12] This was proven by Poincar in 1907 by showing that their automorphism groups have different dimensions as Lie groups.[5][13] However, even in the case of several complex variables, there are some results similar to the results of the theory of uniformization in one complex variable.[14]

In polydisks, the Cauchy's integral formula holds and the power series expansion of holomorphic functions is defined, but polydisks and open unit balls are not biholomorphic mapping because the Riemann mapping theorem does not hold, and also, polydisks was possible to separation of variables, but it doesn't always hold for any domain. Therefore, in order to study of the domain of convergence of the power series, it was necessary to make additional restriction on the domain, this was the Reinhardt domain. Early knowledge into the properties of field of study of several complex variables, such as Logarithmically-convex, Hartogs's extension theorem, etc. , were given in the Reinhardt domain.

A complete Reinhardt domain D is star-like with regard 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 the Cauchy's integral theorem without using the Jordan curve theorem.

When examining the domain of convergence on the Reinhardt domain, Hartogs found the Hartogs's phenomenon in which holomorphic functions in some domain on the C n \displaystyle \mathbb C ^n were all connected to larger domain.[19]

From Hartogs's extension theorem the domain of convergence extends from H ε \displaystyle H_\varepsilon to Δ 2 \displaystyle \Delta ^2 . Looking at this from the perspective of the Reinhardt domain, H ε \displaystyle H_\varepsilon is the Reinhardt domain containing the center z = 0, and the domain of convergence of H ε \displaystyle H_\varepsilon has been extended to the smallest complete Reinhardt domain Δ 2 \displaystyle \Delta ^2 containing H ε \displaystyle H_\varepsilon .[25]

Thullen's[26] 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:

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 (In the complex coordinate space C n \displaystyle \mathbb C ^n call the domain of holomorphy), the first result of the domain of holomorphy was the holomorphic convexity of H. Cartan and Thullen.[28] Levi's problem shows that the pseudoconvex domain was a domain of holomorphy. (First for C 2 \displaystyle \mathbb C ^2 ,[29] later extended to C n \displaystyle \mathbb C ^n .[30][31])[32] Kiyoshi Oka's[35][36] notion of idal de domaines indtermins is interpreted theory of sheaf cohomology by H. Cartan and more development Serre.[note 10][37][38][39][40][41][42][6] In sheaf cohomology, the domain of holomorphy has come to be interpreted as the theory of Stein manifolds.[43] The notion of the domain of holomorphy is also considered in other complex manifolds, furthermore also the complex analytic space which is its generalization.[4]

The domain G \displaystyle G is called holomorphically convex if for every compact subset K , K ^ G \displaystyle K,\hat K_G is also compact in G. Sometimes this is just abbreviated as holomorph-convex.

If such a relations holds in the domain of holomorphy of several complex variables, it looks like a more manageable condition than a holomorphically convex.[note 11] The subharmonic function looks like a kind of convex function, so it was named by Levi as a pseudoconvex domain (Hartogs's pseudoconvexity). Pseudoconvex domain (boundary of pseudoconvexity) are important, as they allow for classification of domains of holomorphy. A domain of holomorphy is a global property, by contrast, pseudoconvexity is that local analytic or local geometric property of the boundary of a domain.[47]

The origin of indeterminate domains comes from the fact that domains change depending on the pair ( f , δ ) \displaystyle (f,\delta ) . Cartan[37][38] translated this notion into the notion of the coherent (sheaf) (Especially, coherent analytic sheaf) in sheaf cohomology.[68][69] This name comes from H. Cartan.[70] Also, Serre (1955) introduced the notion of the coherent sheaf into algebraic geometry, that is, the notion of the coherent algebraic sheaf.[71] The notion of coherent (coherent sheaf cohomology) helped solve the problems in several complex variables.[40]

In the case of one variable complex functions, Mittag-Leffler's theorem was able to create a global meromorphic function from a given and principal parts (Cousin I problem), and Weierstrass factorization theorem was able to create a global meromorphic function from a given zeroes or zero-locus (Cousin II problem). However, these theorems do not hold in several complex variables because the singularities of analytic function in several complex variables are not isolated points; these problems are called the Cousin problems and are formulated in terms of sheaf cohomology. They were first introduced in special cases by Pierre Cousin in 1895.[80] It was Oka who showed the conditions for solving first Cousin problem for the domain of holomorphy[note 18] on the complex coordinate space,[83][84][81][note 19] also solving the second Cousin problem with additional topological assumptions. The Cousin problem is a problem related to the analytical properties of complex manifolds, but the only obstructions to solving problems of a complex analytic property are pure topological;[81][40][32] Serre called this the Oka principle.[85] They are now posed, and solved, for arbitrary complex manifold M, in terms of conditions on M. M, which satisfies these conditions, is one way to define a Stein manifold. The study of the cousin's problem made us realize that in the study of several complex variables, it is possible to study of global properties from the patching of local data,[37] that is it has developed the theory of sheaf cohomology. (e.g.Cartan seminar.[43])[40]

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