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Several Complex Variables with Connections to Algebraic Geometry and Lie Groups
Joseph L. Taylor, University of Utah, Salt Lake City, UT
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2002; 507 pp; hardcover
Volume: 46
ISBN-10: 0-8218-3178-X
ISBN-13: 978-0-8218-3178-6
List Price: US$89 Member Price: US$71.20
Order Code: GSM/46

Analytic Functions of Several Complex Variables - Robert C Gunning and Hugo Rossi

This text presents an integrated development of the theory of several complex variables and complex algebraic geometry, leading to proofs of Serre's celebrated GAGA theorems relating the two subjects, and including applications to the representation theory of complex semisimple Lie groups. It includes a thorough treatment of the local theory using the tools of commutative algebra, an extensive development of sheaf theory and the theory of coherent analytic and algebraic sheaves, proofs of the main vanishing theorems for these categories of sheaves, and a complete proof of the finite dimensionality of the cohomology of coherent sheaves on compact varieties. The vanishing theorems have a wide variety of applications and these are covered in detail.

Of particular interest are the last three chapters, which are devoted to applications of the preceding material to the study of the structure and representations of complex semisimple Lie groups. Included are introductions to harmonic analysis, the Peter-Weyl theorem, Lie theory and the structure of Lie algebras, semisimple Lie algebras and their representations, algebraic groups and the structure of complex semisimple Lie groups. All of this culminates in Miličić's proof of the Borel-Weil-Bott theorem, which makes extensive use of the material developed earlier in the text.

There are numerous examples and exercises in each chapter. This modern treatment of a classic point of view would be an excellent text for a graduate course on several complex variables, as well as a useful reference for the expert.

Graduate students and research mathematicians interested in ODEs, algebraic geometry, group theory, generalizations, and abstract harmonic analysis.

Reviews

"The book can serve as an excellent text for a graduate course on modern methods of complex analysis, as well as a useful reference for those working in analysis."

-- Zentralblatt MATH

"The author succeeded in making the text as self-contained as possible by giving results and proofs of many results from this background material ... This very well-written book is a pleasant text for any graduate student and a base for any lecture course on several complex variables."

-- Mathematical Reviews