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From Holomorphic Functions to Complex Manifolds


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Table of Contents

I 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 Mappings.- II 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.- Differentiate 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 Disks.- III Analytic Sets.- 1. The Algebra of Power Series.- The Banach Algebra Bt.- Expansion with Respect to z1.- Convergent Series in Banach Algebras.- Convergent Power Series.- Distinguished Directions.- 2. The Preparation Theorem.- Division with Remainder in Bt.- The Weierstrass Condition.- Weierstrass Polynomials.- Weierstrass Preparation Theorem.- 3. Prime Factorization.- Unique Factorization.- Gauss's Lemma.- Factorization in Hn.- Hensel's Lemma.- The Noetherian Property.- 4. Branched Coverings.- Germs.- Pseudopolynomials.- Euclidean Domains.- The Algebraic Derivative.- Symmetric Polynomials.- The Discriminant.- Hypersurfaces.- The Unbranched Part.- Decompositions.- Projections.- 5. Irreducible Components.- Embedded-Analytic Sets.- Images of Embedded-Analytic Sets.- Local Decomposition.- Analyticity.- The Zariski Topology.- Global Decompositions.- 6. Regular and Singular Points.- Compact Analytic Sets.- Embedding of Analytic Sets.- Regular Points of an Analytic Set.- The Singular Locus.- Extending Analytic Sets.- The Local Dimension.- IV Complex Manifolds.- 1. The Complex Structure.- Complex Coordinates.- Holomorphic Functions.- Riemann Surfaces.- Holomorphic Mappings.- Cartesian Products.- Analytic Subsets.- Differentiable Functions.- Tangent Vectors.- The Complex Structure on the Space of Derivations.- The Induced Mapping.- Immersions and Submersions.- Gluing.- 2. Complex Fiber Bundles.- Lie Groups and Transformation Groups.- Fiber Bundles.- Equivalence.- Complex Vector Bundles.- Standard Constructions.- Lifting of Bundles.- Subbundles and Quotients.- 3. Cohomology.- Cohomology Groups.- Refinements.- Acyclic Coverings.- Generalizations.- The Singular Cohomology.- 4. Meromorphie Functions and Divisors.- The Ring of Germs.- Analytic Hypersurfaces.- Meromorphic Functions.- Divisors.- Associated Line Bundles.- Meromorphic Sections.- 5. Quotients and Submanifolds.- Topological Quotients.- Analytic Decompositions.- Properly Discontinuously Acting Groups.- Complex Tori.- Hopf Manifolds.- The Complex Projective Space.- Meromorphie Functions.- Grassmannian Manifolds.- Submanifolds and Normal Bundles.- Projective Algebraic Manifolds.- Projective Hypersurfaces.- The Euler Sequence.- Rational Functions.- 6. Branched Riemann Domains.- Branched Analytic Coverings.- Branched Domains.- Torsion Points.- Concrete Riemann Surfaces.- Hyperelliptic Riemann Surfaces.- 7. Modifications and Toric Closures.- Proper Modifications.- Blowing Up.- The Tautological Bundle.- Quadratic Transformations.- Monoidal Transformations.- Meromorphic Maps.- Toric Closures.- V Stein Theory.- 1. Stein Manifolds.- Fundamental Theorems.- Cousin-I Distributions.- Cousin-II Distributions.- Chern Class and Exponential Sequence.- Extension from Submanifolds.- Unbranched Domains of Holomorphy.- The Embedding Theorem.- The Serre Problem.- 2. The Levi Form.- Covariant Tangent Vectors.- Hermitian Forms.- Coordinate Transformations.- Plurisubharmonic Functions.- The Maximum Principle.- 3. Pseudoconvexity.- Pseudoconvex Complex Manifolds.- Examples.- Analytic Tangents.- 4. Cuboids.- Distinguished Cuboids.- Vanishing of Cohomology.- Vanishing on the Embedded Manifolds.- Cuboids in a Complex Manifold.- Enlarging U?.- Approximation.- 5. Special Coverings.- Cuboid Coverings.- The Bubble Method.- Frechet Spaces.- Finiteness of Cohomology.- Holomorphic Convexity.- Negative Line Bundles.- Bundles over Stein Manifolds.- 6. The Levi Problem.- Enlarging: The Idea of the Proof.- Enlarging: The First Step.- Enlarging: The Whole Process.- Solution of the Levi Problem.- The Compact Case.- VI Kahler Manifolds.- 1. Differential Forms.- The Exterior Algebra.- Forms of Type (p, q).- Bundles of Differential Forms.- 2. Dolbeault Theory.- Integration of Differential Forms.- The Inhomogeneous Cauchy Formula.- The ??-Equation in One Variable.- A Theorem of Hartogs.- Dolbeault's Lemma.- Dolbeault Groups.- 3. Kahler Metrics.- Hermitian metrics.- The Fundamental Form.- Geodesic Coordinates.- Local Potentials.- Pluriharmonic Functions.- The Fubini Metric.- Deformations.- 4. The Inner Product.- The Volume Element.- The Star Operator.- The Effect on (p, q)-Forms.- The Global Inner Product.- Currents.- 5. Hodge Decomposition.- Adjoint Operators.- The Kahlerian Case.- Bracket Relations.- The Laplacian.- Harmonic Forms.- Consequences.- 6. Hodge Manifolds.- Negative Line Bundles.- Special Holomorphic Cross Sections.- Projective Embeddings.- Hodge Metrics.- 7. Applications.- Period Relations.- The Siegel Upper Halfplane.- Semipositive Line Bundles.- Moishezon Manifolds.- VII Boundary Behavior.- 1. Strongly Pseudoconvex Manifolds.- The Hilbert Space.- Operators.- Boundary Conditions.- 2. Subelliptic Estimates.- Sobolev Spaces.- The Neumann Operator.- Real-Analytic Boundaries.- Examples.- 3. Nebenhullen.- General Domains.- A Domain with Nontrivial Nebenhulle.- Bounded Domains.- Domains in C2.- 4. Boundary Behavior of Biholomorphic Maps.- The One-Dimensional Case.- The Theory of Henkin and Vormoor.- Real-Analytic Boundaries.- Fefferman's Result.- Mappings.- The Bergman Metric.- References.- Index of Notation.

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From the reviews:


"This new book is a valuable addition to the literature."

K. Fritzsche and H. Grauert

From Holomorphic Functions to Complex Manifolds

"A valuable addition to the literature."-MATHEMATICAL REVIEW

"The book is a nice introduction to the theory of complex manifolds. The authors' intention is to introduce the reader in a simple way to the most important branches and methods in the theory of several complex variables. ... The book is written in a very readable way; it is a nice introduction into the topic." (EMS, March 2004)

"About 25 years ago, the same couple of authors published the forerunner of this work with the title Several Complex Variables ... . The experience of forty years of active teaching besides the well-known research career resulted in an admirably well readable simple clean and polished style. ... I find this book of extraordinary importance and I recommend it to all students, teachers and researchers in mathematics and even in physics as well." (Laszlo L. Stacho, Acta Scientarum Mathematicarum, Vol. 69, 2003)

"This book gives an easily understandable introduction to the theory of complex manifolds. It is self-contained ... and leads to deep results such as the Remmert-Stein theorem for analytic sets, finiteness theorems for spaces of cross sections in holomorphic vector bundles, and the solution if the Levi problem, using only elementary methods such as power series, holomorphic vector bundles, and one-dimensional cocycles." (F. Haslinger, Monatshefte fur Mathematik, Vol. 142 (3), 2004)

"The book is an essentially extended and modified version of the classical monograph 'Several complex variables' by the same authors. ... The monograph is strongly recommended to everybody interested in modern complex analysis, both for students and researchers." (Marek Jarnicki, Zentralblatt MATH, Vol. 1005, 2003)

"The authors state that this book 'grew out of' their earlier graduate textbook [Several complex variables, Translated from the German, Springer, New York, 1976; MR 54 # 3004]. The book should not, however, be thought of as merely a second edition. ... Where the two books do overlap in content, the exposition in the new volume has been largely rewritten. This new book is a valuable addition to the literature." (Harold P. Boas, Mathematical Reviews, 2003 g)

"This book is an introduction to the theory of complex manifolds. The authors' 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. ... The book can be used as a first introduction to several complex variables as well as a reference for the expert." (L'ENSEIGNEMENT MATHEMATHIQUE, Vol. 48 (3-4), 2002)

"Due to its interior unity and its many-sided applicability, Complex Analysis became an absolutely essential part of today's Mathematics. ... It is a merit of the authors that their book is an introduction into holomorphic functions of several complex variables which is easily understandable. ... K. Fritzsche's and H. Grauert's book will give a fresh impetus not only to mathematicians who are interested in holomorphic functions in several complex variables but also to those who deal with generalized multi-regular functions." (W. Tutschke, ZAA, Vol. 22 (1), 2003)

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