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Complex Analysis 2: Riemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular Functions

Eberhard Freitag
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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The English edition of Freitag and Busam’s Funktionentheorie I, which I reviewed here some years ago, was just called Complex Analysis, without any suggestion that it was to be the first of two volumes. Nevertheless, here is Complex Analysis 2, this time by Freitag alone. No translator is listed, so my guess is Freitag himself has rendered his Funktionentheorie II into English.

This volume has two parts that differ in content and in the level of detail. The first four chapters give a thorough treatment of the theory of Riemann surfaces and uniformization. The last three are a sort of introduction to several complex variables. The link between the two is provided by the theory of abelian functions, which arises naturally from the study of Riemann surfaces of genus bigger than one. That theory, in turn, leads to the theory of Siegel modular forms with which the book concludes.

As is inevitable, the material in this book is much more geometric than that covered in the first volume. The author provides a (very brief) introduction to fundamental notions of topology, but develops fully the theory of surfaces and covering spaces he needs. In particular, the book includes a proof of the classification of compact orientable surfaces by their genus.

The fact that sheaves only appear once, and in an exercise at that, signals that the treatment of several complex variables is intended to be introductory. As the author notes in the introduction, it is pretty much restricted to classical material.

While the first volume could conceivably be used with (very good) undergraduates, this one is definitely a graduate text. It is well written but terse. Some exercises are included; in contrast with the first volume, solutions are not. There are some typos, such as “if we where to abandon” on page 202, that might have been caught with careful copyediting but cannot be detected by a spelling and grammar checker.

There is a lot of mathematics in this book, presented efficiently and well. The author has good taste in selecting topics (or maybe his taste just coincides with mine). It is a book I am glad to have, and that I will certainly refer to in the future.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. He is the editor of MAA Reviews.

Chapter I. Riemann Surfaces
Chapter II. Harmonic Functions on Riemann Surfaces
Chapter III. Uniformization
Chapter IV. Compact Riemann Surfaces
Appendices to Chapter IV
Chapter V. Analytic Functions of Several Complex Variables
Chapter V. Analytic Functions of Several Complex Variable
Chapter VI. Abelian Functions
Chapter VII. Modular Forms of Several Variables
Chapter VIII. Appendix: Algebraic Tools