From Holomorphic Functions to Complex Manifolds

This book is an introduction to the theory of complex manifolds. The authors's intent is to familiarize the reader with the most important branches and methods in complex analysis of several variables and to do this as simply as possible. Therefore, the abstract concepts involving sheaves, coherence, and higher-dimensional cohomology have been completely avoided. Each chapter is complemented by a variety of examples and exercises. The only prerequisite needed to read this book is a knowledge of real analysis and some basic facts from algebra, topology, and the theory of one complex variable.

Includes supplementary material: sn.pub/extras

Autorentext
Hans Grauert studierte in Münster und Zürich, wo er 1958 promovierte. Seit dem 1. Oktober 1959 war er bis zu seiner Emeritierung ordentlicher Professor in Göttingen. Er hatte Gastprofessuren u.a. in Princeton und Paris. Er gilt als einer der bedeutendsten deutschen Mathematiker der Nachkriegszeit. Sein Spezialgebiet ist die Funktionentheorie mehrerer 'Veränderlicher'. Prof. Dr. Klaus Fritzsche lehrt Mathematik an der Universität Wuppertal. Er hat bereits mehrfach den Brückenkurs "Mathematik für Mathematiker" gehalten.

Klappentext
This book is an introduction to the theory of complex manifolds. The author's intent is to familiarize the reader with the most important branches and methods in complex analysis of several variables and to do this as simply as possible. Therefore, the abstract concepts involving sheaves, coherence, and higher-dimensional cohomology have been completely avoided. Only elementary methods such as power series, holomorphic vector bundles, and one-dimensional cocycles are used. Nevertheless, deep results can be proved, for example the Remmert-Stein theorem for analytic sets, finiteness theorems for spaces of cross sections in holomorphic vector bundles, and the solution of the Levi problem. Each chapter is complemented by a variety of examples and exercises. The only prerequisite needed to read this book is a knowledge of real analysis and some basic facts from algebra, topology, and the theory of one complex variable. The book can be used as a first introduction to several complex variables as well as a reference for the expert.
Klaus Fritzsche received his PhD from the University of Göttingen in 1975, under the direction of Professor Hans Grauert. Since 1984, he has been Professor of Mathematics at the University of Wuppertal, where he has been investigating convexity problems on complex spaces and teaching undergraduate and graduate courses on Real and Complex Analysis. Hans Grauert studied physics and mathematics in Mainz, Münster and Zürich. He received his PhD in mathematics from the University of Münster and in 1959 he became a full professor at the University of Göttingen. Professor Grauert is responsible for many important developments in mathematics in the Twentieth Century. Along with Reinhold Remmert, Karl Stein and Henri Cartan, he founded the theory of Several Complex Variables in its modern form. He also proved various important theorems, including Levi's Problem and the coherence of higher direct image sheaves. Professor Grauert is the author of 10 books and his Selected Papers was published by Springer in 1994.

Zusammenfassung
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 and methods in complex analysis of several variables and to do this as simply as possible. Therefore, the abstract concepts involved with sheaves, coherence, and higher-dimensional cohomology are avoided. Only elementary methods such as power series, holomorphic vector bundles, and one-dimensional co­ cycles are used. Nevertheless, deep results can be proved, for example the Remmert-Stein theorem for analytic sets, finiteness theorems for spaces of cross sections in holomorphic vector bundles, and the solution of the Levi problem. The first chapter deals with holomorphic functions defined in open sub­ sets of the space en. Many of the well-known properties of holomorphic functions of one variable, such as the Cauchy integral formula or the maxi­ mum principle, can be applied directly to obtain corresponding properties of holomorphic functions of several variables. Furthermore, certain properties of differentiable functions of several variables, such as the implicit and inverse function theorems, extend easily to holomorphic functions.

Inhalt
I Holomorphic Functions.- 1. Complex Geometry.- 2. Power Series.- 3. Complex Differentiable Functions.- 4. The Cauchy Integral.- 5. The Hartogs Figure.- 6. The Cauchy-Riemann Equations.- 7. Holomorphic Maps.- 8. Analytic Sets.- II Domains of Holomorphy.- 1. The Continuity Theorem.- 2. Plurisubharmonic Functions.- 3. Pseudoconvexity.- 4. Levi Convex Boundaries.- 5. Holomorphic Convexity.- 7. Examples and Applications.- 8. Riemann Domains over Cn.- 9. The Envelope of Holomorphy.- III Analytic Sets.- 1. The Algebra of Power Series.- 2. The Preparation Theorem.- 3. Prime Factorization.- 4. Branched Coverings.- 5. Irreducible Components.- 6. Regular and Singular Points.- IV Complex Manifolds.- 1. The Complex Structure.- 2. Complex Fiber Bundles.- 3. Cohomology.- 4. Meromorphie Functions and Divisors.- 5. Quotients and Submanifolds.- 6. Branched Riemann Domains.- 7. Modifications and Toric Closures.- V Stein Theory.- 1. Stein Manifolds.- 2. The Levi Form.- 3. Pseudoconvexity.- 4. Cuboids.- 5. Special Coverings.- 6. The Levi Problem.- VI Kahler Manifolds.- 1. Differential Forms.- 2. Dolbeault Theory.- 3. Kähler Metrics.- 4. The Inner Product.- 5. Hodge Decomposition.- 6. Hodge Manifolds.- 7. Applications.- VII Boundary Behavior.- 1. Strongly Pseudoconvex Manifolds.- 2. Subelliptic Estimates.- 3. Nebenhüllen.- 4. Boundary Behavior of Biholomorphic Maps.- References.- Index of Notation.

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