从全纯函数到复流形

出版时间：2009-6  出版社：世界图书出版公司  作者：弗里切  页数：392  
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前言

　　The aim of this book is to give an understandable introduction to the the-ory of complex manifolds. With very few exceptions we give complete proofs.Many examples and figures along with quite a few exercises are included.Our intent is to familiarize the reader with the most important branches andmethods in complex analysis of several variables and to do this as simply aspossible. Therefore, the abstract concepts involved with sheaves, coherence,and higher-dimensional cohomology are avoided. Only elementary methodssuch as power series, holomorphic vector bundles, and one-dlmensional co-cycles are used. Nevertheless, deep results can be proved, for example theRemmert-Stein theorem for analytic sets, finiteness theorems for spaces ofcross sections in holomorphic vector bundles, and the solution of the Leviproblem.　　The first chapter deals with holomorphic functions defined in open sub-sets of the space Cn. Many of the well-known properties of holomorphicfunctions of one variable, such as the Canchy integral formula or the maxi-mum principle, can be applied directly to obtain corresponding properties ofholomorphic functions of several variables. Furthermore, certain properties of differentiable functions of several variables, such as the implicit and inversefunction theorems, extend easily to holomorphic functions.　　In Chapter II the following phenomenon is considered: For n＞2, there're pairs of open subsets H∈ P ∈ Cn such that every function holomorphicin H extends to a holomorpbic function in P. Special emphasis is put on domains G ∈ Cn for which there is no such extension to a bigger domain.They are called domains of holomorphy and have a number of interesting convexity properties. These are described using plurisubharmonie functions.If G is not a domain of holomorphy, one asks for a maximal set E to which allholomorpbic functions in G extend. Such an "envelope of holomorphy" existsin the category of Riemann domains, i.e., unbranched domains over Cn.　　The common zero locus of a system of holomorphie functions is calledan analytic set. In Chapter III we use Weierstrass's division theorem forpower series to investigate the local and global structure of analytic sets.Two of the main results are the decomposition of analytic sets into irreduciblecomponents and the extension theorem of Remmert and Stein. This is theonly place in the book where singularities play an essential role.　　Chapter IV establishes the theory of complex manifolds and holomorphicfiber bundles. Numerous examples are given, in particular branched and un-branched coverings of Cn quotient manifolds such as tore and Hoof manifolds, projective spaces and Grassmannians, algebraic manifolds, modifications, andtoric varieties. We do not present the abstract theory of complex spaces, but do provide an elementary introduction to complex algebraic geometry. For example.

内容概要

　　本书是一部介绍复流形理论的入门书籍。作者用尽可能简单的方法使读者熟悉多变量复分析中的重要分支和方法，所以避免出现比较抽象的概念，如，层、凝聚和高维上同调等，仅运用了基本方法幂级数、正则向量丛和一维上闭链。然而，解析集Remmert-Stein定理，正则向量丛中的截面空间有限定理以及Levi问题解这些深层次的都得到了完整的证明。每章的结束都有大量的例子和练习。具备实分析、代数、拓扑以及单变量理论知识就可以完全读懂这本书。本书可以作为学习多变量的入门教程，也是一本很好的参考书。　　读者对象：本书适用于数学专业的广大师生。

书籍目录

PrefaceI　Holomorphic Functions　1.Complex Geometry　　Real and Complex Structures　　Hermitian Forms and Inner Products　　Balls and Polydisks　　Connectedness　　Reinhardt Domains　2.Power Series　　Polynomials　　Convergence　　Power Series　3.Complex Differentiable Functions　　The Complex Gradient　　Weakly Holomorphic Functions　　Holomorphic Functions　4.The Cauchy Integral　　The Integral Formula　　Holomorphy of the Derivatives　　The Identity Theorem　5.The Hartogs Figure　　Expansion in Reinhardt Domains　　Hartogs Figures　6.The Cauchy-Riemann Equations　　Real Differentiable Functions　　Wirtinger's Calculus　　The Cauchy-Riemann Equations　7.Holomorphic Maps　　The Jacobian　　Chain Rules　　Tangent Vectors　　The Inverse Mapping　8.Analytic Sets　　Analytic Subsets　　Bounded Holomorphic Functions　　Regular Points　　Injective Holomorphic MappingsII  Domains of Holomorphy　1.The Continuity Theorem　　General Hartogs Figures　　Removable Singularities　　The Continuity Principle　　Hartogs Convexity　　Domains of Holomorphy　2.Plurisubharmonic Functions　　Subharmonic Functions　　The Maximum Principle　　Differentiable Subharmonic Functions　　Plurisubharmonic Functions　　The Levi Form　　Exhaustion Functions　3.Pseudoconvexity　　Pseudoconvexity　　The Boundary Distance　　Properties of Pseudoconvex Domains　4.Levi Convex Boundaries　　Boundary Functions　　The Levi Condition　　Affine Convexity　　A Theorem of Levi　5.Holomorphic Convexity　　Affine Convexity　　Holomorphic Convexity　　The Cartan-Thullen Theorem　6.Singular Functions　　Normal Exhaustions　　Unbounded Holomorphic Functions　　Sequences　7.Examples and Applications　　Domains of Holomorphy　　Complete Reinhardt Domains　　Analytic Polyhedra　8.Riemann Domains over Cn　　Riemann Domains　　Union of Riemann Domains　9.The Envelope of Holomorphy　　Holomorphy on Riemann Domains　　Envelopes of Holomorphy　　Pseudoconvexity　　Boundary Points　　Analytic DisksIII　Analytic SetsIV　Complex MannifoldsV  Stein TheoryVI  Kahler ManifoldsVII  Boundary BehaviorReferencesIndex of NotationIndex

用户评论 (总计2条)

•   是一本讲解复流形的好书，值得一读。复流形是现代数学中既重要又难理解的内容之一。该书能从全纯函数引入基本概念，继而讲解复流形，使之容易理解。
•   不错 影印版都这样吧