Several Complex Variables Iii
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Several Complex Variables Iii
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Author : G.M. Khenkin
language : en
Publisher: Springer Science & Business Media
Release Date : 1989-02-23
Several Complex Variables Iii written by G.M. Khenkin and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1989-02-23 with Mathematics categories.
We consider the basic problems, notions and facts in the theory of entire functions of several variables, i. e. functions J(z) holomorphic in the entire n space en (i. e. JEH( 1 variables, as in the case n = 1, a central theme deals with questions of growth of functions and the distribu tion of their zeros. However, there are significant differences between the cases of one and several variables. In the first place there is the fact that for n 1 the zero set of an entire function is not discrete and therefore one has no analogue of a tool such as the canonical Weierstrass product, which is fundamental in the case n = 1. Second, for n> 1 there exist several different natural ways of exhausting the space
Analytic Functions Of Several Complex Variables
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Author : Robert C. Gunning
language : en
Publisher: American Mathematical Society
Release Date : 2022-08-25
Analytic Functions Of Several Complex Variables written by Robert C. Gunning and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-08-25 with Mathematics categories.
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 Poincaré 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. Further information about this book, including updates, can be found at the following URL: www.ams.org/publications/authors/books/postpub/chel-368.
Several Complex Variables Iii
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Author : G.M. Khenkin
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Several Complex Variables Iii written by G.M. Khenkin and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
We consider the basic problems, notions and facts in the theory of entire functions of several variables, i. e. functions J(z) holomorphic in the entire n space 1 the zero set of an entire function is not discrete and therefore one has no analogue of a tool such as the canonical Weierstrass product, which is fundamental in the case n = 1. Second, for n> 1 there exist several different natural ways of exhausting the space
Several Complex Variables Vii
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Author : H. Grauert
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09
Several Complex Variables Vii written by H. Grauert and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Mathematics categories.
Of making many books there is no end; and much study is a weariness of the flesh. Eccl. 12.12. 1. In the beginning Riemann created the surfaces. The periods of integrals of abelian differentials on a compact surface of genus 9 immediately attach a g dimensional complex torus to X. If 9 ~ 2, the moduli space of X depends on 3g - 3 complex parameters. Thus problems in one complex variable lead, from the very beginning, to studies in several complex variables. Complex tori and moduli spaces are complex manifolds, i.e. Hausdorff spaces with local complex coordinates Z 1, ... , Zn; holomorphic functions are, locally, those functions which are holomorphic in these coordinates. th In the second half of the 19 century, classical algebraic geometry was born in Italy. The objects are sets of common zeros of polynomials. Such sets are of finite dimension, but may have singularities forming a closed subset of lower dimension; outside of the singular locus these zero sets are complex manifolds.
Several Complex Variables And Complex Manifolds I
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Author : Mike Field
language : en
Publisher: Cambridge University Press
Release Date : 1982-04-01
Several Complex Variables And Complex Manifolds I written by Mike Field and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1982-04-01 with Mathematics categories.
This self-contained and relatively elementary introduction to functions of several complex variables and complex (especially compact) manifolds was first published in 1982. It was intended be a synthesis of those topics and a broad introduction to the field. The work as a whole will be useful to professional mathematicians or mathematical physicists who wish to acquire a further knowledge of this area of mathematics. Many exercises have been included and indeed they form an integral part of the text. The prerequisites for understanding Part I would be met by any mathematics student with a first degree and together the two parts were designed to provide an introduction to the more advanced works in the subject.
Several Complex Variables
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Author : H. Grauert
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Several Complex Variables written by H. Grauert and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
The present book grew out of introductory lectures on the theory offunctions of several variables. Its intent is to make the reader familiar, by the discussion of examples and special cases, with the most important branches and methods of this theory, among them, e.g., the problems of holomorphic continuation, the algebraic treatment of power series, sheaf and cohomology theory, and the real methods which stem from elliptic partial differential equations. In the first chapter we begin with the definition of holomorphic functions of several variables, their representation by the Cauchy integral, and their power series expansion on Reinhardt domains. It turns out that, in l:ontrast ~ 2 there exist domains G, G c en to the theory of a single variable, for n with G c G and G "# G such that each function holomorphic in G has a continuation on G. Domains G for which such a G does not exist are called domains of holomorphy. In Chapter 2 we give several characterizations of these domains of holomorphy (theorem of Cartan-Thullen, Levi's problem). We finally construct the holomorphic hull H(G} for each domain G, that is the largest (not necessarily schlicht) domain over en into which each function holomorphic on G can be continued.
Banach Algebras And Several Complex Variables
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Author : John Wermer
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29
Banach Algebras And Several Complex Variables written by John Wermer and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-29 with Mathematics categories.
During the past twenty years many connections have been found between the theory of analytic functions of one or more complex variables and the study of commutative Banach algebras. On the one hand, function theory has been used to answer algebraic questions such as the question of the existence of idempotents in a Banach algebra. On the other hand, concepts arising from the study of Banach algebras such as the maximal ideal space, the Silov boundary, Gleason parts, etc. have led to new questions and to new methods of proof in function theory. Roughly one third of this book isconcerned with developing some of the principal applications of function theory in several complex variables to Banach algebras. We presuppose no knowledge of severalcomplex variables on the part of the reader but develop the necessary material from scratch. The remainder of the book deals with problems of uniform approximation on compact subsets of the space of n complex variables. For n > I no complete theory exists but many important particular problems have been solved. Throughout, our aim has been to make the exposition elementary and self-contained. We have cheerfully sacrificed generality and completeness all along the way in order to make it easier to understand the main ideas.
Introduction To Complex Analysis In Several Variables
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Author : Volker Scheidemann
language : en
Publisher: Springer Nature
Release Date : 2023-04-26
Introduction To Complex Analysis In Several Variables written by Volker Scheidemann and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-04-26 with Mathematics categories.
This book gives a comprehensive introduction to complex analysis in several variables. While it focusses on a number of topics in complex analysis rather than trying to cover as much material as possible, references to other parts of mathematics such as functional analysis or algebras are made to help broaden the view and the understanding of the chosen topics. A major focus are extension phenomena alien to the one-dimensional theory, which are expressed in the famous Hartog's Kugelsatz, the theorem of Cartan-Thullen, and Bochner's theorem. The book aims primarily at students starting to work in the field of complex analysis in several variables and instructors preparing a course. To that end, a lot of examples and supporting exercises are provided throughout the text. This second edition includes hints and suggestions for the solution of the provided exercises, with various degrees of support.
Nevanlinna Theory In Several Complex Variables And Diophantine Approximation
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Author : Junjiro Noguchi
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-12-09
Nevanlinna Theory In Several Complex Variables And Diophantine Approximation written by Junjiro Noguchi and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-12-09 with Mathematics categories.
The aim of this book is to provide a comprehensive account of higher dimensional Nevanlinna theory and its relations with Diophantine approximation theory for graduate students and interested researchers. This book with nine chapters systematically describes Nevanlinna theory of meromorphic maps between algebraic varieties or complex spaces, building up from the classical theory of meromorphic functions on the complex plane with full proofs in Chap. 1 to the current state of research. Chapter 2 presents the First Main Theorem for coherent ideal sheaves in a very general form. With the preparation of plurisubharmonic functions, how the theory to be generalized in a higher dimension is described. In Chap. 3 the Second Main Theorem for differentiably non-degenerate meromorphic maps by Griffiths and others is proved as a prototype of higher dimensional Nevanlinna theory. Establishing such a Second Main Theorem for entire curves in general complex algebraic varieties is a wide-open problem. In Chap. 4, the Cartan-Nochka Second Main Theorem in the linear projective case and the Logarithmic Bloch-Ochiai Theorem in the case of general algebraic varieties are proved. Then the theory of entire curves in semi-abelian varieties, including the Second Main Theorem of Noguchi-Winkelmann-Yamanoi, is dealt with in full details in Chap. 6. For that purpose Chap. 5 is devoted to the notion of semi-abelian varieties. The result leads to a number of applications. With these results, the Kobayashi hyperbolicity problems are discussed in Chap. 7. In the last two chapters Diophantine approximation theory is dealt with from the viewpoint of higher dimensional Nevanlinna theory, and the Lang-Vojta conjecture is confirmed in some cases. In Chap. 8 the theory over function fields is discussed. Finally, in Chap. 9, the theorems of Roth, Schmidt, Faltings, and Vojta over number fields are presented and formulated in view of Nevanlinna theory with results motivated by those in Chaps. 4, 6, and 7.
Functions Of Several Complex Variables And Their Singularities
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Author : Wolfgang Ebeling
language : en
Publisher: American Mathematical Soc.
Release Date : 2007
Functions Of Several Complex Variables And Their Singularities written by Wolfgang Ebeling and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with Mathematics categories.
The book provides an introduction to the theory of functions of several complex variables and their singularities, with special emphasis on topological aspects. The topics include Riemann surfaces, holomorphic functions of several variables, classification and deformation of singularities, fundamentals of differential topology, and the topology of singularities. The aim of the book is to guide the reader from the fundamentals to more advanced topics of recent research. All the necessary prerequisites are specified and carefully explained. The general theory is illustrated by various examples and applications.