从全纯函数到复流形
目 录内容简介
Preface
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
查看完整
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
查看完整
目 录内容简介
《从全纯函数到复流形》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《从全纯函数到复流形》。
目 录内容简介
Preface
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
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 Disks
Ⅲ Analytic Sets
1.The Algebra of Power Series
The Banach Algebra Bt
Expansion with Respect to zt
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 Grot ps
Fiber Bt ndles
Equivalence
Complex Vector Bundles
Standard Constructions
Lifting of Bundles
Subbundles and Quotients
3. Cehomolcgy
Cohomology Groups
Refinements
Acyclic Coverings
Generalizations
The Singular Cohomology
4. Meromorphic 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
Meromorphic Functions
Grassmannian Manifolds
Ⅴ Stein Theory
Ⅵ Kahler Manifolds
Ⅶ Boundary Behavior
References
Index of Notation
Index
^ 收 起
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
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 Disks
Ⅲ Analytic Sets
1.The Algebra of Power Series
The Banach Algebra Bt
Expansion with Respect to zt
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 Grot ps
Fiber Bt ndles
Equivalence
Complex Vector Bundles
Standard Constructions
Lifting of Bundles
Subbundles and Quotients
3. Cehomolcgy
Cohomology Groups
Refinements
Acyclic Coverings
Generalizations
The Singular Cohomology
4. Meromorphic 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
Meromorphic Functions
Grassmannian Manifolds
Ⅴ Stein Theory
Ⅵ Kahler Manifolds
Ⅶ Boundary Behavior
References
Index of Notation
Index
^ 收 起
目 录内容简介
《从全纯函数到复流形》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《从全纯函数到复流形》。
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