代数几何原理
CHAPTER 0 FOUNDATIONAL MATERIAL
1. Rudiments of Several Complex Variables
Cauchy's Formula and Applications
Several Variables
Weierstrass Theorems and Corollaries
Analytic Varieties
2. Complex Manifolds
Complex Manifolds
Submanifolds and Subvarieties
De Rham and Dolbeault Cohomology
查看完整
1. Rudiments of Several Complex Variables
Cauchy's Formula and Applications
Several Variables
Weierstrass Theorems and Corollaries
Analytic Varieties
2. Complex Manifolds
Complex Manifolds
Submanifolds and Subvarieties
De Rham and Dolbeault Cohomology
查看完整
代数几何是数学中古老和发展比较快的学科之一，它与投影几何、复分析、拓扑学、数论以及数学领域的其它分支有着紧密的联系。然而近些年代数几何不论是风格还是语言都发生了巨大的变化，本书展示了相关理论的主要研究结果和计算工具的发展。本书有如下特点：（1）本书以研究具体几何问题和特殊类代数簇为中心来展开。（2）注重实例的复杂性与通常模式的对称性这两者之间的均衡，在选择的论题和叙述顺序中，书中尽量体现这种关系。（3）尤其对于涉及到的“复杂”结果，都有充分完整的证明。目次：多复变初步；复代数簇；Liemann曲面和代数曲线；深入技巧；曲面；留数；二次线丛。
CHAPTER 0 FOUNDATIONAL MATERIAL
1. Rudiments of Several Complex Variables
Cauchy's Formula and Applications
Several Variables
Weierstrass Theorems and Corollaries
Analytic Varieties
2. Complex Manifolds
Complex Manifolds
Submanifolds and Subvarieties
De Rham and Dolbeault Cohomology
Calculus on Complex Manifolds
3. Sheaves and Cohomology
Origins: The Mittag-Leffler Problem
Sheaves
Cohomology of Sheaves
The de Rham Theorem
The Dolbeault Theorem
4. Topology of Manifolds
Intersection of Cycles
Poincare Duality
Intersection of Analytic Cycles
5. Vector Bundles, Connections, and Curvature
Complex and Holomorphic Vector Bundles
Metrics, Connections, and Curvature
6. Harmonic Theory on Compact Complex Manifolds
The Hodge Theorem
Proof of the Hodge Theorem I: Local Theory
Proof of the Hodge Theorem II: Global Theory
Applications of the Hodge Theorem
7. Kahler Manifolds
The Kahler Condition
The Hodge Identities and the Hodge Decomposition
The Lefschetz Decomposition
CHAPTER 1 COMPLEX ALGEBRAIC VARIETIES
1. Divisors and Line Bundles
Divisors
Line Bundles
Chern Classes of Line Bundles
2. Some Vanishing Theorems and Corollaries
The Kodaira Vanishing Theorem
The Lefschetz Theorem on Hyperplane Sections
Theorem B
The Lefschetz Theorem on (1, l)-classes
3. Algebraic Varieties
Analytic and Algebraic Varieties
Degree of a Variety
Tangent Spaces to Algebraic Varieties
4. The Kodaira Embedding Theorem
Line Bundles and Maps to Projective Space
Blowing Up
Proof of the Kodaira Theorem
5. Grassmannians
Definitions
The Cell Decomposition
The Schubert Calculus
Universal Bundles
The Pliicker Embedding
CHAPTER 2 RIEMANN SURFACES AND ALGEBRAIC
CURVES
Preliminaries
Embedding Riemann Surfaces
The Riemann-Hurwitz Formula
The Genus Formula
Cases g=0, 1
2. Abel's Theorem
Abel's Theorem——First Version
The First Reciprocity Law and Corollaries
Abel's Theorem——Second Version
Jacobi Inversion
3. Linear Systems on Curves
Reciprocity Law II
The Riemann-Roch Formula
Canonical Curves
Special Linear Systems I
Hyperelliptic Curves and Riemann's Count
Special Linear Systems II
4. Plucker Formulas
Associated Curves
Ramification
The General Plucker Formulas I
The General Plucker Formulas II
Weierstrass Points
Plucker Formulas for Plane Curves
5. Correspondences
Definitions and Formulas
Geometry of Space Curves
Special Linear Systems III
6. Complex Tori and Abelian Varieties
The Riemann Conditions
Line Bundles on Complex Tori
Theta-Functions
The Group Structure on an Abelian Variety
Intrinsic Formulations
7. Curves and Their Jacobians
Preliminaries
Riemann's Theorem
Riemann's Singularity Theorem
Special Linear Systems IV
Torelli's Theorem
CHAPTER 3 FURTHER TECHNIQUES
1. Distributions and Currents
Definitions; Residue Formulas
Smoothing and Regularity
Cohomology of Currents
2. Applications of Currents to Complex Analysis
Currents Associated to Analytic Varieties
Intersection Numbers of Analytic Varieties
The Levi Extension and Proper Mapping Theorems
3. Chern Classes
Definitions
The Gauss Bonnet Formulas
Some Remarks——Not Indispensable——Concerning
Chern Classes of Holomorphic Vector Bundles
4. Fixed-Point and Residue Formulas
The Lefschetz Fixed-Point Formula
The Holomorphic Lefschetz Fixed-Point Formula
The Bott Residue Formula
The General Hirzebruch-Riemann-Roch Formula
5. Spectral Sequences and Applications
Spectral Sequences of Filtered and Bigraded Complexes
Hypercohomology
Differentials of the Second Kind
The Leray Spectral Sequence
CHAPTER 4 SURFACES
1. Preliminaries
Intersection Numbers, the Adjunction Formula,
and Riemann-Roch
Blowing Up and Down
The Quadric Surface
The Cubic Surface
2. Rational Maps
Rational and Birationai Maps
Curves on an Algebraic Surface
The Structure of Birational Maps Between Surfaces
3. Rational Surfaces I
Noether's Lemma
Rational Ruled Surfaces
The General Rational Surface
Surfaces of Minimal Degree
Curves of Maximal Genus
Steiner Constructions
The Enriques-Petri Theorem
4. Rational Surfaces II
The Castelnuovo-Enriques Theorem
The Enriques Surface
Cubic Surfaces Revisited
The Intersection of Two Quadrics in p4
5. Some Irrational Surfaces
The Albanese Map
Irrational Ruled Surfaces
A Brief Introduction to Elliptic Surfaces
Kodaira Number and the Classification Theorem I
The Classification Theorem II
K-3 Surfaces
Enriques Surfaces
6. Noether's Formula
Noether's Formula for Smooth Hypersurfaces
Blowing Up Submanifolds
Ordinary Singularities of Surfaces
Noether's Formula for General Surfaces
Some Examples
Isolated Singularities of Surfaces
CHAPTER 5 RESIDUES
1. Elementary Properties of Residues
Definition and Cohomological Interpretation
The Global Residue Theorem
The Transformation Law and Local Duality
2. Applications of Residues
Intersection Numbers
Finite Holomorphic Mappings
Applications to Plane Projective Geometry
3. Rudiments of Commutative and Homological Algebra
with Applications
Commutative Algebra
Homological Algebra
The Koszul Complex and Applications
A Brief Tour Through Coherent Sheaves
4. Global Duality
Global Ext
Explanation of the General Global Duality Theorem
Global Ext and Vector Fields with Isolated Zeros
Global Duality and Superabundance of
Points on a Surface
Extensions of Modules
Points on a Surface and Rank-Two Vector Bundles
Residues and Vector Bundles
CHAPTER 6 THE QUADRIC LINE COMPLEX
1. Preliminaries: Quadrics
Rank of a Quadric
Linear Spaces on Quadrics
Linear Systems of Quadrics
Lines on Linear Systems of Quadrics
The Problem of Five Conics
2. The Quadric Line Complex: Introduction
Geometry of the Grassmannian G(2,4)
Line Complexes
The Quadric Line Complex and Associated Kummer Surface I
Singular Lines of the Quadric Line Complex
Two Configurations
3. Lines on the Quadric Line Complex
The Variety of Lines on the Quadric Line Complex
Curves on the Variety of Lines
Two Configurations Revisited
The Group Law
4. The Quadric Line Complex: Reprise
The Quadric Line Complex and Associated Kummer Surface II
Rationality of the Quadric Line Complex
INDEX
^ 收 起
1. Rudiments of Several Complex Variables
Cauchy's Formula and Applications
Several Variables
Weierstrass Theorems and Corollaries
Analytic Varieties
2. Complex Manifolds
Complex Manifolds
Submanifolds and Subvarieties
De Rham and Dolbeault Cohomology
Calculus on Complex Manifolds
3. Sheaves and Cohomology
Origins: The Mittag-Leffler Problem
Sheaves
Cohomology of Sheaves
The de Rham Theorem
The Dolbeault Theorem
4. Topology of Manifolds
Intersection of Cycles
Poincare Duality
Intersection of Analytic Cycles
5. Vector Bundles, Connections, and Curvature
Complex and Holomorphic Vector Bundles
Metrics, Connections, and Curvature
6. Harmonic Theory on Compact Complex Manifolds
The Hodge Theorem
Proof of the Hodge Theorem I: Local Theory
Proof of the Hodge Theorem II: Global Theory
Applications of the Hodge Theorem
7. Kahler Manifolds
The Kahler Condition
The Hodge Identities and the Hodge Decomposition
The Lefschetz Decomposition
CHAPTER 1 COMPLEX ALGEBRAIC VARIETIES
1. Divisors and Line Bundles
Divisors
Line Bundles
Chern Classes of Line Bundles
2. Some Vanishing Theorems and Corollaries
The Kodaira Vanishing Theorem
The Lefschetz Theorem on Hyperplane Sections
Theorem B
The Lefschetz Theorem on (1, l)-classes
3. Algebraic Varieties
Analytic and Algebraic Varieties
Degree of a Variety
Tangent Spaces to Algebraic Varieties
4. The Kodaira Embedding Theorem
Line Bundles and Maps to Projective Space
Blowing Up
Proof of the Kodaira Theorem
5. Grassmannians
Definitions
The Cell Decomposition
The Schubert Calculus
Universal Bundles
The Pliicker Embedding
CHAPTER 2 RIEMANN SURFACES AND ALGEBRAIC
CURVES
Preliminaries
Embedding Riemann Surfaces
The Riemann-Hurwitz Formula
The Genus Formula
Cases g=0, 1
2. Abel's Theorem
Abel's Theorem——First Version
The First Reciprocity Law and Corollaries
Abel's Theorem——Second Version
Jacobi Inversion
3. Linear Systems on Curves
Reciprocity Law II
The Riemann-Roch Formula
Canonical Curves
Special Linear Systems I
Hyperelliptic Curves and Riemann's Count
Special Linear Systems II
4. Plucker Formulas
Associated Curves
Ramification
The General Plucker Formulas I
The General Plucker Formulas II
Weierstrass Points
Plucker Formulas for Plane Curves
5. Correspondences
Definitions and Formulas
Geometry of Space Curves
Special Linear Systems III
6. Complex Tori and Abelian Varieties
The Riemann Conditions
Line Bundles on Complex Tori
Theta-Functions
The Group Structure on an Abelian Variety
Intrinsic Formulations
7. Curves and Their Jacobians
Preliminaries
Riemann's Theorem
Riemann's Singularity Theorem
Special Linear Systems IV
Torelli's Theorem
CHAPTER 3 FURTHER TECHNIQUES
1. Distributions and Currents
Definitions; Residue Formulas
Smoothing and Regularity
Cohomology of Currents
2. Applications of Currents to Complex Analysis
Currents Associated to Analytic Varieties
Intersection Numbers of Analytic Varieties
The Levi Extension and Proper Mapping Theorems
3. Chern Classes
Definitions
The Gauss Bonnet Formulas
Some Remarks——Not Indispensable——Concerning
Chern Classes of Holomorphic Vector Bundles
4. Fixed-Point and Residue Formulas
The Lefschetz Fixed-Point Formula
The Holomorphic Lefschetz Fixed-Point Formula
The Bott Residue Formula
The General Hirzebruch-Riemann-Roch Formula
5. Spectral Sequences and Applications
Spectral Sequences of Filtered and Bigraded Complexes
Hypercohomology
Differentials of the Second Kind
The Leray Spectral Sequence
CHAPTER 4 SURFACES
1. Preliminaries
Intersection Numbers, the Adjunction Formula,
and Riemann-Roch
Blowing Up and Down
The Quadric Surface
The Cubic Surface
2. Rational Maps
Rational and Birationai Maps
Curves on an Algebraic Surface
The Structure of Birational Maps Between Surfaces
3. Rational Surfaces I
Noether's Lemma
Rational Ruled Surfaces
The General Rational Surface
Surfaces of Minimal Degree
Curves of Maximal Genus
Steiner Constructions
The Enriques-Petri Theorem
4. Rational Surfaces II
The Castelnuovo-Enriques Theorem
The Enriques Surface
Cubic Surfaces Revisited
The Intersection of Two Quadrics in p4
5. Some Irrational Surfaces
The Albanese Map
Irrational Ruled Surfaces
A Brief Introduction to Elliptic Surfaces
Kodaira Number and the Classification Theorem I
The Classification Theorem II
K-3 Surfaces
Enriques Surfaces
6. Noether's Formula
Noether's Formula for Smooth Hypersurfaces
Blowing Up Submanifolds
Ordinary Singularities of Surfaces
Noether's Formula for General Surfaces
Some Examples
Isolated Singularities of Surfaces
CHAPTER 5 RESIDUES
1. Elementary Properties of Residues
Definition and Cohomological Interpretation
The Global Residue Theorem
The Transformation Law and Local Duality
2. Applications of Residues
Intersection Numbers
Finite Holomorphic Mappings
Applications to Plane Projective Geometry
3. Rudiments of Commutative and Homological Algebra
with Applications
Commutative Algebra
Homological Algebra
The Koszul Complex and Applications
A Brief Tour Through Coherent Sheaves
4. Global Duality
Global Ext
Explanation of the General Global Duality Theorem
Global Ext and Vector Fields with Isolated Zeros
Global Duality and Superabundance of
Points on a Surface
Extensions of Modules
Points on a Surface and Rank-Two Vector Bundles
Residues and Vector Bundles
CHAPTER 6 THE QUADRIC LINE COMPLEX
1. Preliminaries: Quadrics
Rank of a Quadric
Linear Spaces on Quadrics
Linear Systems of Quadrics
Lines on Linear Systems of Quadrics
The Problem of Five Conics
2. The Quadric Line Complex: Introduction
Geometry of the Grassmannian G(2,4)
Line Complexes
The Quadric Line Complex and Associated Kummer Surface I
Singular Lines of the Quadric Line Complex
Two Configurations
3. Lines on the Quadric Line Complex
The Variety of Lines on the Quadric Line Complex
Curves on the Variety of Lines
Two Configurations Revisited
The Group Law
4. The Quadric Line Complex: Reprise
The Quadric Line Complex and Associated Kummer Surface II
Rationality of the Quadric Line Complex
INDEX
^ 收 起
代数几何是数学中古老和发展比较快的学科之一，它与投影几何、复分析、拓扑学、数论以及数学领域的其它分支有着紧密的联系。然而近些年代数几何不论是风格还是语言都发生了巨大的变化，本书展示了相关理论的主要研究结果和计算工具的发展。本书有如下特点：（1）本书以研究具体几何问题和特殊类代数簇为中心来展开。（2）注重实例的复杂性与通常模式的对称性这两者之间的均衡，在选择的论题和叙述顺序中，书中尽量体现这种关系。（3）尤其对于涉及到的“复杂”结果，都有充分完整的证明。目次：多复变初步；复代数簇；Liemann曲面和代数曲线；深入技巧；曲面；留数；二次线丛。
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