# Function of several complex variables

Many examples of such functions were familiar in nineteenth-century mathematics; abelian functions, theta functions, and some hypergeometric series. 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.

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 every 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.

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[ref 3] 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.

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

Every product of a family of an 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.

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

Because the order of products and sums is interchangeable, from (1) we get

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

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 7] for a one complex variable. This combination is generally not unique and there are an infinite number of combinations.

The Cauchy integral formula holds only for polydiscs, and in the domain of several complex variables, polydiscs are only one of many domain, so we introduce Bochner–Martinelli formula.

When the function f,g is analytic in the 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 point of analytic. Therefore, the maximal principle hold. Also, the inverse function theorem and implicit function theorem hold.

From the establishment of the inverse function theorem, the following mapping can be defined.

In polydisks and the Reinhardt domain, Cauchy's integral formula holds and the power series expansion of holomorphic functions is defined, but the unique radius of convergence is not defined for each variable. Also, since the Riemann mapping theorem does not hold, polydisks and open unit balls are not biholomorphic mapping. Therefore, the domain of convergence of the power series is not as simple as the case of one variable, but it satisfies the property called Logarithmically-convex. There are various convexity for the domain of convergence of several complex variables.

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.

Thullen's[ref 15] 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:

Toshikazu Sunada (1978)[ref 16] established a generalization of Thullen's result:

Levi convex (approximate from the inside on the analytic polyhedron domain)

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

For arbitrary complex manifold, Levi (–Krzoska) pseudoconvexity does not always have an plurisubharmonic exhaustion function, i.e. it does not necessarily have a (p-)pseudoconvex domain.[ref 31]

When the Levi (–Krzoska) form is positive-definite, it is called Strongly Levi (–Krzoska) pseudoconvex or often called simply Strongly pseudoconvex.[ref 2]

Locally pseudoconvex (locally Stein, Cartan pseudoconvex, local Levi property)

Oka's coherent theorem[ref 22] says that each sheaf that meets the following conditions is a coherent.[ref 45]

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. However, these theorems do not hold because the singularities of analytic function in several complex variables is not isolated points, 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.[ref 46] It was Kiyoshi Oka who showed[note 20] the conditions for solving first Cousin problem for the domain of holomorphy on the complex coordinate space, and also solving with additional topological assumptions, for the second Cousin problem, the Cousin problem is a problem related to the analytical properties of complex manifolds, but the condition for solving second Cousin problem is pure a topological property,[ref 27] and J.P., Serre[ref 47] called this the Oka principle.[ref 48][ref 49][ref 50] 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 Cousin problems has developed the theory of sheaf cohomology. (e.g.Cartan seminar.[ref 28])[ref 27]

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.

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.

The long exact sheaf cohomology sequence associated to the quotient is

Let X be a connected, non-compact (open) Riemann surface. A deep theorem (1939)[ref 56] of Heinrich Behnke and Stein (1948)[ref 52] asserts that X is a Stein manifold.

This is related to the solution of the second (multiplicative) Cousin problem.

Also, Grauert proved for arbitrary complex manifolds M.[note 22][ref 62][ref 21][ref 60]

And Narasimhan[ref 63][ref 64] extended Levi's problem to Complex analytic space, a generalized in the singular case of complex manifolds.

This means that Behnke–Stein theorem, which holds for Stein manifolds, has not found a conditions to be established in Stein space. [ref 67]

Grauert introduced the concept of K-complete in the proof of Levi's problem.

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).

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".