From holomorphic functions to complex manifolds
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書誌事項
From holomorphic functions to complex manifolds
（Graduate texts in mathematics, 213）
Springer, c2002
- : pbk
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注記
Bibliography: p. [375]-379
Includes index
Pbk.: c2010
内容説明・目次
内容説明
This introduction to the theory of complex manifolds covers the most important branches and methods in complex analysis of several variables while completely avoiding abstract concepts involving sheaves, coherence, and higher-dimensional cohomology. Only elementary methods such as power series, holomorphic vector bundles, and one-dimensional cocycles are used. Each chapter contains a variety of examples and exercises.
目次
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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