Complex Analysis: A Brief Tour into Higher Dimensions

Abstract

1. INTRODUCTION. Why is it that most graduate students of mathematics (and many undergraduates as well) are exposed to complex analysis in one variable, yet only a small minority of students or, for that matter, professional mathematicians ever learn even the most basic corresponding theory in several variables? Think about it another way: Could anyone seriously argue that it might be sufficient to train a mathematics major in calculus of functions of one real variable without expecting him or her to learn at least something about partial derivatives, multiple integrals, and some higher dimensional version of the Fundamental Theorem of Calculus? Of course not, the real world is not one-dimensional! But neither is the complex world: witness classical applications of complex analysis to quantum field theory, more recent uses in twistor and gravitation theory, and the latest developments in string theory! Multidimensional complex analysis is an indispensable tool in modern theoretical physics. (See, for example, Green, Schwarz, and Witten [6], Manin [12], Henkin and Novikov [9].) Aside from questions of applicability, shouldn’t the pure mathematician’s mind wonder about the restriction to functions of only one complex variable? It should not surprise anyone that there is a natural extension of complex analysis to the multivariable setting. What is surprising is the many new and intriguing phenomena that appear when one considers more than one variable. Indeed, these phenomena presented major challenges to any straightforward generalization of familiar theorems. Amazing progress was made in the 1940s and 1950s, when the area was enriched by new and powerful tools such as coherent analytic sheaves and their cohomology theory. Yet for decades these apparently quite abstract techniques constituted a formidable barrier that deterred analysts from exploring this territory unless they were committed to a research career in multidimensional complex analysis. Fortunately, many of the technical hardships of the pioneering days can now be overcome with much less effort by approaching the subject along different routes. The purpose of this article is to lead you, as painlessly as possible, on a tour through some of the foundations of complex analysis in several variables, and to take you to some vantage points from which we can enjoy some of the fascinating sights that remain hidden from us as long as we restrict ourselves to the complex plane. Along the way, we shall encounter a few of the unexpected higher dimensional phenomena and explore fundamental new concepts, always trying to gain an understanding of the underlying difficulties. Good walking shoes (i.e., an elementary introduction to basic complex analysis in one variable, and standard multivariable real calculus) are all the equipment that is needed, no technical gear or skills will be required. I will be rewarded if, at the end of our hike, many of you will recommend this tour to friends and colleagues. Perhaps a few of you will be sufficiently intrigued and challenged that you will pick up one of the excellent books that are available to guide you through more advanced terrain in order to reach the “high peaks” of the subject.