Function Theory of Several Complex Variables

The idea for this book came when I was an assistant at the Department of Mat- matics and Computer Science at the Philipps-University Marburg, Germany. S- eraltimesIfacedthetaskofsupportinglecturesandseminarsoncomplexanalysis of several variables and found out that there are very few books on the subject, compared to the vast amount of literature on function theory of one variable, let alone on real variables or basic algebra. Even fewer books, to my understanding, were written primarily with the student in mind. So it was quite hard to ?nd s- porting examples and exercises that helped the student to become familiar with the fascinating theory of several complex variables. Of course, there are notable exceptions, like the books of R.M. Range [9] or B. and L. Kaup [6], however, even those excellent books have a drawback: they are quite thick and thus quite expensive for a student’s budget. So an additional motivation to write this book was to give a comprehensive introduction to the theory of several complex variables, illustrate it with as many examples as I could ?nd and help the student to get deeper insight by giving lots of exercises, reaching from almost trivial to rather challenging. There are not many illustrations in this book, in fact, there is exactly one, becauseinthetheoryofseveralcomplexvariablesI?ndmostofthemeithertrivial or misleading. The readers are of course free to have a di?erent opinion on these matters.

The idea for this book came when I was an assistant at the Department of Mat- matics and Computer Science at the Philipps-University Marburg, Germany. S- eraltimesIfacedthetaskofsupportinglecturesandseminarsoncomplexanalysis of several variables and found out that there are very few books on the subject, compared to the vast amount of literature on function theory of one variable, let alone on real variables or basic algebra. Even fewer books, to my understanding, were written primarily with the student in mind. So it was quite hard to ?nd s- porting examples and exercises that helped the student to become familiar with the fascinating theory of several complex variables. Of course, there are notable exceptions, like the books of R.M. Range [9] or B. and L. Kaup [6], however, even those excellent books have a drawback: they are quite thick and thus quite expensive for a student’s budget. So an additional motivation to write this book was to give a comprehensive introduction to the theory of several complex variables, illustrate it with as many examples as I could ?nd and help the student to get deeper insight by giving lots of exercises, reaching from almost trivial to rather challenging. There are not many illustrations in this book, in fact, there is exactly one, becauseinthetheoryofseveralcomplexvariablesI?ndmostofthemeithertrivial or misleading. The readers are of course free to have a di?erent opinion on these matters.

Around 1970, an abrupt change occurred in the study of holomorphic functions of several complex variables. Sheaves vanished into the back­ ground, and attention was focused on integral formulas and on the "hard analysis" problems that could be attacked with them: boundary behavior, complex-tangential phenomena, solutions of the J-problem with control over growth and smoothness, quantitative theorems about zero-varieties, and so on. The present book describes some of these developments in the simple setting of the unit ball of en. There are several reasons for choosing the ball for our principal stage. The ball is the prototype of two important classes of regions that have been studied in depth, namely the strictly pseudoconvex domains and the bounded symmetric ones. The presence of the second structure (i.e., the existence of a transitive group of automorphisms) makes it possible to develop the basic machinery with a minimum of fuss and bother. The principal ideas can be presented quite concretely and explicitly in the ball, and one can quickly arrive at specific theorems of obvious interest. Once one has seen these in this simple context, it should be much easier to learn the more complicated machinery (developed largely by Henkin and his co-workers) that extends them to arbitrary strictly pseudoconvex domains. In some parts of the book (for instance, in Chapters 14-16) it would, however, have been unnatural to confine our attention exclusively to the ball, and no significant simplifications would have resulted from such a restriction.

Around 1970, an abrupt change occurred in the study of holomorphic functions of several complex variables. Sheaves vanished into the back­ ground, and attention was focused on integral formulas and on the "hard analysis" problems that could be attacked with them: boundary behavior, complex-tangential phenomena, solutions of the J-problem with control over growth and smoothness, quantitative theorems about zero-varieties, and so on. The present book describes some of these developments in the simple setting of the unit ball of en. There are several reasons for choosing the ball for our principal stage. The ball is the prototype of two important classes of regions that have been studied in depth, namely the strictly pseudoconvex domains and the bounded symmetric ones. The presence of the second structure (i.e., the existence of a transitive group of automorphisms) makes it possible to develop the basic machinery with a minimum of fuss and bother. The principal ideas can be presented quite concretely and explicitly in the ball, and one can quickly arrive at specific theorems of obvious interest. Once one has seen these in this simple context, it should be much easier to learn the more complicated machinery (developed largely by Henkin and his co-workers) that extends them to arbitrary strictly pseudoconvex domains. In some parts of the book (for instance, in Chapters 14-16) it would, however, have been unnatural to confine our attention exclusively to the ball, and no significant simplifications would have resulted from such a restriction.

This carefully written textbook is an introduction to the beautiful concepts and results of complex analysis. It is intended for international bachelor and master programmes in Germany and throughout Europe; in the Anglo-American system of university education the content corresponds to a beginning graduate course.