Elementary Local Properties of Holomorphic Functions

Abstract

In §1 and §2 of this chapter we present the standard local properties of holomorphic functions and maps which are obtained by combining basic one complex variable theory with the calculus of several (real) variables. The reader should go through this material rapidly, with the goal of familiarizing himself with the results, notation, and terminology, and return to the appropriate sections later on, as needed. The inclusion at this stage of holomorphic maps and of complex submanifolds, i.e., the level sets of nonsingular holomorphic maps, is quite natural in several variables. In particular, it allows us to present elementary proofs of two results which distinguish complex analysis from real analysis, namely: (i) the only compact complex submanifolds of ℂ n are finite sets, and (ii) the Jacobian determinant of an injective holomorphic map from an open set in ℂ n into ℂ n is nowhere zero. Section 3, which gives an introduction to analytic sets, may be omitted without loss of continuity. We have included it mainly to familiarize the reader with a topic which is fundamental for many aspects of the general theory of several complex variables, and in order to show, by means of the Weierstrass Preparation Theorem, how algebraic methods become indispensable for a thorough understanding of the deeper local properties of holomorphic functions and their zero sets.