Several complex variables

The theory of functions of several complex variables is the branch of mathematics dealing with functions

f(z₁, z₂, ... , z_n)

on the space Cⁿ of n-tuples of complex numbers. As in complex analysis, which is the case n = 1 but of a distinct character, these are not just any functions: they are supposed to be analytic, so that locally speaking they are power series in the variables z_i.

Equivalently, as it turns out, they are limits of polynomials, uniformly on compact sets; or locally square-integrable solutions to the n-dimensional Cauchy-Riemann equations.

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 fully-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 generalisation of the branch points of Riemann surface theory.

With work of Hartogs, and of Kiyoshi Oka in the 1930s, a general theory began to emerge. Hartogs proved some basic results, including showing that there can be no isolated singularity in the theory when 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 Grauert and 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 (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 PDEs. 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 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 generalisations 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.