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From Holomorphic Functions to Complex Manifolds

Klaus Fritzsche, Hans Grauert (Autoren)

Buch | Hardcover

397 Seiten

2002
Springer-Verlag New York Inc.
978-0-387-95395-3 (ISBN)

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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.

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.

Reihe/Serie	Graduate Texts in Mathematics ; 213
Zusatzinfo	XV, 397 p.
Verlagsort	New York, NY
Sprache	englisch
Maße	156 x 234 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
	Mathematik / Informatik ► Mathematik ► Analysis
	Naturwissenschaften ► Physik / Astronomie
ISBN-10	0-387-95395-7 / 0387953957
ISBN-13	978-0-387-95395-3 / 9780387953953
Zustand	Neuware