Function theory of several variables
The apprentice in the theory of functions of a single complex variable is inclined to explore on a naïve comparison basis several-variables analogues of the basic principles. The surprise is big, and fully justified, to discover a completely new landscape: the domain of convergence of a power series, analytic continuation, growth at infinity, integral representations, potential theory, orthogonal function expansions, classification of domains, groups of automorphisms are each revealing quite different phenomena. To learn and enter into this universe takes time, dedication, curiosity and especially a solid, authoritative guidance. The present book by Jörg Eschmeier offers to the German readership such a flawless, gentle, yet comprehensive, introduction to function theory of several complex variables. Compared to the several centuries old field of functions of one complex variable, the study of several complex variables is modern. It can be traced back to the pioneering works of Weierstrass and Poincaré, the foundational contributions of Hartogs, Oka, Weil, Siegel and the golden era, around mid XX-th century, marked by Cartan, Behnke, Stein, Serre, to name only a few prominent figures. The basic theory is nowadays well settled and should constitute, in harmony with the author's book clear intention, a part of the general mathematical curriculum of a Master Program in Mathematical Analysis. The research frontier in several complex variables remains active today in more sophisticated chapters contingent to algebraic geometry, mathematical physics, differential geometry of Cauchy-Riemann structures and global analysis. Each direction of current investigation would fill a separate volume and demand a rather long time of acquaintanceship. The book starts with the definition of holomorphy in several variables, elementary properties of analytic functions and a discussion of the possible convergence domains of a power series. Then emphasis is put on the structure of null sets of systems of analytic functions, developing minimally on this purpose the necessary techniques of local algebra. The central role of Weierstrass preparation lemma is well highlighted on this ground. The notion of domain of holomorphy then takes shape, in the light of analytic continuation phenomena specific to several variables. At this stage enter techniques of partial differential equations, as agents of solvability of the overdetermined Cauchy-Riemann equations at the level of differential forms. With this minimal and masterly organized material at hand, one attacks the main questions of function theory: interpolation, division and approximation of analytic functions. Examples abound and will delight the patient reader. Next a full chapter is devoted to the growth of analytic functions at the boundary of their domain of definition, in the specific and natural study of plurisubharmonic functions and their role in the elementary theory of several complex variables. A final chapter, illustrating the effectiveness of the new mathematical tools, touches deep theorems referring to commutative Banach algebras. The presentation is light, with complete details, avoiding long technical preparations. The many accompanying problems are selected with rigor and good taste. This elegant textbook is renewing a precious trend set by two notable introductions to the subject: [\textit{B. Malgrange}, Lectures on the theory of functions of several complex variables. Bombay: Tata Institute of Fundamental Research (1958); \textit{R. Narasimhan}, Several complex variables. Chicago Lectures in Mathematics. Chicago-London: The University of Chicago Press (1971; Zbl 0223.32001)]. Among the many textbooks and monographs exploring deeper aspects of function theory of several complex variables, I would mention and suggest as a continuation the second volume by \textit{B. Shabat} [Introduction to complex analysis. II: Functions of several variables. Providence, RI: American Mathematical Society (1992). The accessible and informative textbook by the author will educate and prepare many generations of mathematicians to come. Highly recommended.
function theory of several complex variables
scientific article; zbMATH DE number 6840360