Function of several complex variables

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.

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, Pierre Cousin [fr], E. E. Levi, 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, Karl Stein, Wilhelm Wirtinger and Francesco Severi.

Hartogs proved some basic results, such as every isolated singularity is removable, for every analytic function

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.

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, on the other hand, the Grauert–Riemenschneider vanishing theorem is known as a similar result for compact complex manifolds, and the Grauert–Riemenschneider conjecture is a special case of the conjecture of Narasimhan.

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.

It is a non-negative number, which implies that the (real) orientation of the space is never reversed by a complex operator.

Let f meets the conditions of being continuous and separately homorphic on domain D. Each disk has a rectifiable curve

is expressed as a power series expansion that is convergent on D : 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.

Contrary to the one variable case, it is possible that two different holomorphic functions coincide on a set which has an accumulation point, for instance the maps

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.

[29][30])[31] Kiyoshi Oka's[34][35] notion of idéal de domaines indéterminés is interpreted theory of sheaf cohomology by H. Cartan and more development Serre.

[note 10][36][37][38][39][40][41][6] In sheaf cohomology, the domain of holomorphy has come to be interpreted as the theory of Stein manifolds.

is a subharmonic function on D.[4]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.

by Kiyoshi Oka,[note 17] but for ramified Riemann domains, pseudoconvexity does not characterize holomorphically convexity,[66] and then by Lars Hörmander using methods from functional analysis and partial differential equations (a consequence of

[1][43][3][67] The introduction of sheaves into several complex variables allowed the reformulation of and solution to several important problems in the field.

[79] It was Oka who showed the conditions for solving first Cousin problem for the domain of holomorphy[note 18] on the complex coordinate space,[82][83][80][note 19] also solving the second Cousin problem with additional topological assumptions.

Now, let K be the sheaf of meromorphic functions and O the sheaf of holomorphic functions on M. The first Cousin problem can always be solved if the following map is surjective: 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.

, from the long exact cohomology sequence When M is a Stein manifold, the middle arrow is an isomorphism because

Since a non-compact (open) Riemann surface[85] always has a non-constant single-valued holomorphic function,[86] and satisfies the second axiom of countability, the open Riemann surface is in fact a 1-dimensional complex manifold possessing a holomorphic mapping into the complex plane

(In fact, Gunning and Narasimhan have shown (1967)[87] that every non-compact Riemann surface actually has a holomorphic immersion into the complex plane.

Another result, attributed to Hans Grauert and Helmut Röhrl (1956), states moreover that every holomorphic vector bundle on X is trivial.

Numerous further characterizations of such manifolds exist, in particular capturing the property of their having "many" holomorphic functions taking values in the complex numbers.

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".

[115] In fact, Hopf found a class of compact complex manifolds without nonconstant meromorphic functions.

As an application of this theorem, the Kodaira embedding theorem[127] says that a compact Kähler manifold M, with a Hodge metric, there is a complex-analytic embedding of M into complex projective space of enough high-dimension N. In addition the Chow's theorem[128] shows that the complex analytic subspace (subvariety) of a closed complex projective space to be an algebraic that is, so it is the common zero of some homogeneous polynomials, such a relationship is one example of what is called Serre's GAGA principle.

Then combined with Kodaira's result, a compact Kähler manifold M embeds as an algebraic variety.

The combination of analytic and algebraic methods for complex projective varieties lead to areas such as Hodge theory.