Several Complex Variables (Chicago Lectures in Mathematics) Review

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Several Complex Variables (Chicago Lectures in Mathematics) ReviewThis book is a short overview of the theory of several complex variables that emphasized the elementary aspects of the subject, such as Hartogs' theory, and domains of holomorphy. The price quoted here for the book is surely a mistake, and even though it is out of print, if obtained it could still be of use for self-study. Readers should have a background of course in the theory of one complex variable, and when reading it will clearly see the differences in the theory of complex variables when raising the dimension.
In chapter 1, the author defines the basic concepts needed for a study of this subject, such as the polydisc and the definition of real analyticity. As expected, the Cauchy integral formula holds in several dimensions, with the proof easily generalized from the case of one dimension. In fact, most of the results in one dimension, such as Montel's and Weierstrass theorems, are shown by the author to hold in several dimensions. After defining what is means for a function of several complex variables to be holomorphic, he shows that such a function is not dependent on the complex conjugate of its dependent variables, just like in the one variable case. He proves the converse, namely that the vanishing of the partial derivative with respect to the conjugate variables implies the function is holomorphic, in chapter three.
In chapter two, the author immediately shows where the things begin to change in several variables, the first result being Hartogs' theorem. This theorem states that one can always find an analytic continuation of a holomorphic function defined on an open disc to the boundary of the disk. One can clearly see the origins of sheaf theory in this chapter, when the author discusses the construction of domains of existence for one or more holomorphic functions. The author proves that two holomorphic functions on a domain that agree on a nonempty open subset of this domain are identical.
Chapter three is an introduction to the study of subharmonic functions in several complex variables. The author proves the theorem of Hartogs, that shows the converse of the holomorphicity result in chapter 1. The reader familiar with elementary harmonic analysis will see it here in the context of several complex variables.
In chapter four, the author introduces analytic sets, these being essentially subsets of an open set in complex n-space on which a finite collection of holomorphic functions vanish. He shows to what extent the singularities of a holomorphic function are an analytic set.
The author studies the collection of automorphisms on bounded domains in complex n-space in chapter five. An automorphism from an open connected set in complex n-space to itself is a holomorphic map if there exists another holomorphic map whose composition with the automorphism is the identity. After putting a topology on the group of automorphisms, he characterizes all automorphisms of a polydisc. In addition, he again illustrates the differences between complex 1-space and complex n-space for n greater than 2. In complex 2-space for example, he shows that there is no analytic isomorphism from the open square onto the open disk. This is not the case in complex 1-space, where all simply connected domains are analytically equivalent. He also shows that there are no proper holomorphic maps of domains in complex n-space into any ball, and that there are "flat" directions for holomorphic maps from domains in complex n-space to products of bounded domains in complex 1-space. The latter in particular is a radical departure from the behavior in complex 1-space.
The author considers families of holomorphic functions in chapter six, generalizing the results in chapter two. Sheaf theory again makes its presence known, and the key concept introduced is that of envelopes of holomorphy. This is essentially the maximal domain such that each function holomorphic on a domain can be analytically continued to the envelope of holomorphy. Envelopes of holomorphy are shown to exist and to be unique. And, this is one of the few places in the book where the author gives examples, i.e. he details examples of envelopes of holomorphy that are not in complex n-space. The theory in this chapter is of considerable importance in some formulations of axiomatic quantum field theory.
Holomorphic convexity and domains of holomorphy are the subject of chapter seven. A domain of holomorphy is essentially the "natural" domain for at least one function. If a given domain is not a domain of holomorphy, then any function holomorphic in this domain is holomorphic in some larger domain. Any domain in complex 1-space is a domain of holomorphy, but in complex n-space this need not be the case. One can construct domains in complex n-space so that every holomorphic function on this domain extends holomorphically to a strictly larger domain containing the original. The author shows the role of holomorphic convexity in the consideration of functions that are bounded by their supremum norm. Every convex open set in complex n-space is a domain of holomorphy.
Hadamard's three domains theorem and one part of Oka's theorem are proved in chapter eight. The former is essentially an inequality (in the sup norm) for a function defined on a connected domain that has two others as successive proper subsets. Oka's theorem as stated gives that a domain of holomorphicially is holomorphically convex. The author does not prove the converse, explaining that such a proof would take an excursion into global ideal theory.
The last chapter of the book deals with automorphisms of bounded domains, wherein the author details the proof of Henri Cartan that the collection of analytic automorphisms of a bounded domain is a Lie group.Several Complex Variables (Chicago Lectures in Mathematics) Overview

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