Analytic Functions of Several Complex Variables (AMS Chelsea Publishing) by Robert C. Gunning (PDF)

Description

The theory of analytic functions of several complex variables enjoyed a period of remarkable development in the middle part of the twentieth century. After initial successes by Poincare and others in the late 19th and early 20th centuries, the theory encountered obstacles that prevented it from growing quickly into an analogue of the theory for functions of one complex variable. Beginning in the 1930s, initially through the work of Oka, then H. Cartan, and continuing with the work of Grauert, Remmert, and others, new tools were introduced into the theory of several complex variables that resolved many of the open problems and fundamentally changed the landscape of the subject. These tools included a central role for sheaf theory and increased uses of topology and algebra. The book by Gunning and Rossi was the first of the modern era of the theory of several complex variables, which is distinguished by the use of these methods. The intention of Gunning and Rossi’s book is to provide an extensive introduction to the Oka-Cartan theory and some of its applications, and to the general theory of analytic spaces. Fundamental concepts and techniques are discussed as early as possible. The first chapter covers material suitable for a one-semester graduate course, presenting many of the central problems and techniques, often in special cases. The later chapters give more detailed expositions of sheaf theory for analytic functions and the theory of complex analytic spaces. Since its original publication, this book has become a classic resource for the modern approach to functions of several complex variables and the theory of analytic spaces.

User’s Reviews

Editorial Reviews: Review This book is an excellent survey of the present state of the modern theory of several complex variables. Because of the style of presentation, the wide scope and precise treatment of the material, it is destined to become a classic. –Mathematical Reviews…it is a pure pleasure to read: the prose is crystal clear and anything but prolix. … They are happy to get into relatively elementary material in some detail, and sophisticated stuff. And the i’s are dotted and the t’s are crossed. It’s a wonderful book! –MAA Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a classic book and I do recommend it to the readers

⭐A classic book should not go unreviewed, even if a more worthy reviewer would be better. This book was already a standard in the 1980’s, when I began studying the area. It is very carefully written and insightful. It would be wonderful if our science and mathematics were to progress so quickly that such a book would become obsolete instantly or even quickly. Sadly, that is not the case. This is still an important book for every new student to read. Michael Range has a newer book that appeals more to me. In my own view, the theory expressed in this book (and still) is limited by a less reflective view of what “holomorphic” really means. If you ask a classical mathematician why in a single variable you have a beautiful and powerful theory like analytic continuation, and you can preserve nothing more than a Hartog’s phenomenon in even two variables, you are likely to get quite a hostile response (I certainly did). There is a reason for it. The existing theory is missing the idea that holomorphicity in higher dimensions is a naturally graded phenomenon. But that is my own prejudice. The book is old, but still indispensable reading.

⭐Cette réédition d’un livre paru en 1965 reste très pertinent.Le domaine de fonctions analytiques de plusieurs variables complexes est riche et difficile. Il fait appel à des résultats d’algèbre commutative (notamment les modules sur les anneaux locaux), à la théorie des faisceaux et à leur cohomologie, et bien sûr aux notions qui lui sont propres. Ces éléments sont bien motivés dans le premier chapitre, qui sert d’introduction, et sont ensuite traités de façon très claire, tout particulièrement ceux relatifs à la cohomologie des faisceaux, avec les résultats de Dolbeault et de Leray. Il s’agit d’une très bonne introduction à la théorie des espaces analytiques, qui a été approfondie par les travaux de Grauert et Remmert (en particulier dans leur livre “Theory of Stein Spaces”, paru en 1979 chez Springer, et qui est un bon complément du livre discuté ici).Ce livre, actualisé en 1990 par Gunning, est devenu sous sa plume un traité en trois volumes. Ce n’est pas le même investissement, ni en temps, ni au plan financier!Pour une introduction à ce vaste domaine, je trouve donc le Gunning et Rossi tout à fait excellent. Il mériterait 5 étoiles… si le livre de Taylor “Several Complex Variables with Connections to Algebraic Geometry and Lie Groups” n’existait pas. Contrairement au livre de Taylor, le Gunning et Rossi ne propose pas d’exercices, en revanche il comporte d’intéressantes notes historiques à la fin de chaque chapitre.

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