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Analytic functions of several complex variables. Reprint of the 1965 original. (English) Zbl 1204.01045

Providence, RI: AMS Chelsea Publishing (ISBN 978-0-8218-2165-7/hbk). xii, 317 p. (2009).
The book under review is the unaltered reprint of one of the great classics in the textbook literature on the modern theory of analytic functions of several complex variables. The first edition of this excellent introduction to multivariate complex analysis was published in 1965 by Prentice-Hall [Series in Modern Analysis. Englewood Cliffs, N.J.: Prentice-Hall, Inc. (1965; Zbl 0141.08601)], and it was to become a leading standard text on this subject for several decades thereafter. Originally grown out of the authors’ joint graduate courses on functions of several complex variables at Princeton University in the early 1960’s, the book provided the first modern account, in comprehensive textbook form, of a subject that was amidst a very active and transitional period of development. The fundamental new ideas and contributions by H. Cartan, K. Oka, H. Grauert, R. Remmert, K. Stein, and others were far beyond what had been covered in the classical books by W. F. Osgood [Lehrbuch der Funktionentheorie. Leipzig: B. G. Teubner. (1924; JFM 50.0209.04); H. Behnke und P. Thullen [Theorie der Funktionen mehrerer komplexer Veränderlichen (1935; Zbl 0008.36504)] or S. Bochner and W. T. Martin [Several complex variables, Princeton, Princeton University Press (1948; Zbl 0041.05205)], which led the authors of the present book to provide an extensive introduction to these new achievements, especially to the Oka-Cartan approach and some of its applications, and to the rapidly developing general theory of analytic spaces. After the appearance of the once modern primer of multivariate complex analysis by R. Gunning and H. Rossi in the early 1960’s, that is, in the course of the last half-century, the subject has undergone a vast expansion, in particular with a view toward its topological and algebro-geometric aspects, and today there are a great many excellent textbooks covering the variety of new results and methods from various viewpoints. Nevertheless, the book by R. Gunning and H. Rossi was the historically first one of its kind and, moreover, a true milestone in the development of modern complex analysis and its allied areas in contemporary mathematics. In fact, this book reflects, in a unique manner, both the mathematical spirit of the time of its writing and the state of the subject a half-century ago, when the then new methods of sheaf theory, sheaf cohomology, local algebra, and algebraic geometry just had revolutionized the understanding of complex analysis, and had helped solving various long-standing problems in the theory of analytic functions of several complex variables. Apart from its historical value, this venerable textbook still represents a highly useful introduction to modern multivariate complex analysis, which profoundly prepares the reader for studying the vast topical, more advanced, and more specialized literature in the field.
As the authors point out, they deliberately have not attempted to rework their classic text, even not in order to correct the surprising number of detected minor errors and misprints. This is perhaps a little drawback of the highly welcome reprint of the authors’ evergreen classic, as a list of errata could have improved the book belatedly, thereby preventing a new generation of unexperienced readers from getting stuck somewhere in the course of the text. On the other hand, the occurring errors and misprints are not that terribly severe or confusing, but just rather numerous, and the alert reader may take them as an additional, perhaps particularly educating challenge. Thus this reprint remains what it was intended to be: a unique historical document of the development of multivariate complex analysis in the period after World War II, being of enduring significance, also in the current textbook literature and due to the authors’ outstanding expository mastery.
Recall that the contents of the book are organized in nine chapters and two appendices. As for detailed comments, we may refer to N. Knoche’s extensive review of the original [Zbl 0141.08601 (loc.cit.)] from 1965; likewise as a historical document. However, it seems appropriate to mention again the main topics of the book as indicated by the headlines of the single chapters, which are as follows.
I. Holomorphic functions: elementary properties, holomorphic mappings and complex manifolds, removable singularities, differential forms, the Cousin Theorem, polynomial approximations, envelopes of holomorphy, and some applications to uniform algebras.
II. Local rings of holomorphic functions: the Weierstrass theorems, modules over local rings, the Extended Weierstrass Division Theorem, and germs of varieties.
III. Varieties: the Nullstellensatz for prime ideals, local parametrizations, analytic covers, and analytic dimension.
IV. Analytic sheaves: elementary properties of sheaves, sheaves of modules, analytic sheaves on subdomains and on subvarieties of \(\mathbb{C}^n\).
V. Analytic spaces: definitions and examples, holomorphic functions on analytic spaces, the Proper Mapping Theorem, and nowhere degenerate maps.
VI. Cohomology theory: soft sheaves and fine sheaves, the axioms of sheaf cohomology, the Theorem of Dolbeault on cohomology, Leray’s Theorem on cohomology, Cartan’s Lemma ,and amalgamation of syzygies.
VII. Stein spaces, geometric theory: approximation theorems, special analytic polyhedra, the imbedding theorem, and uses of special analytic polyhedra.
VIII. Stein spaces, sheaf theory: Fréchet sheaves, meromorphic functions, and locally free sheaves.
IX. Pseudoconvexity: the complex Hessian, Grauert’s solution of Levi’s problem, plurisubharmonic functions, Oka’s Pseudoconvexity Theorem, Kodaira’s Embedding Theorem.
Appendix A: Partitions of Unity.
Appendix B: The Theorem of Schwartz on Fréchet spaces.
It remains to emphasize again that the present reprint of the famous classic textbook “Analytic Functions of Several Complex Variables” by R. Gunning and H. Rossi, finally issued after forty-five years since the appearance of the original edition, must be seen as an invaluable, highly welcome enhancement of the current textbook literature in the field. It had become a very rare book of still greatest importance, and it had become quite difficult to find copies of it. Now it has been made available again, very much so to the benefit of further generations of students, teachers, and researchers in the field. No doubt, this primer will maintain its enduring place in the standard textbook literature in modern complex analysis.

MSC:

01A75 Collected or selected works; reprintings or translations of classics
32-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces
32Axx Holomorphic functions of several complex variables
32Bxx Local analytic geometry
32Cxx Analytic spaces
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