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本書是一部全面介紹代數曲線、代數流形的教程（全英文版）。主體內容有兩部分組成：一部分以V. V. Shokurov所寫的學術著作為藍本，主要講述黎曼面和代數曲面理論，深刻地揭示了黎曼面和其模型——復射影面中的復代數曲面的相互關係；另外一部分以V. I. Danilov的學術論文為藍本主要討論了代數變量及其概型。 本書結構框架清晰，敘述簡明扼要，可以幫助讀者在很短的時間內了解并掌握代數幾何的精華。
讀者對象:數學專業的高年級本科生、研究生以及相關的科研人員。
讀者對象:數學專業的高年級本科生、研究生以及相關的科研人員。
目次
Introduction by I. R. Shafarevich
Chapter 1. Riemann Surfaces
1. Basic Notions
1.1. Complex Chart; Complex Coordinates
1.2. Complex Analytic Atlas
1,3. Complex Analytic Manifolds
1.4. Mappings of Complex Manifolds
1.5. Dimension of a Complex Manifold
1.6. Riemann Surfaces
1.7. Differentiable Manifolds
2. Mappings of Riemann Surfaces
2.1. Nonconstant Mappings of Riemann Surfaces are Discrete
2.2. Meromorphic Functions on a Riemann Surface
2.3. Meromorphic Functions with Prescribed Behaviour at Poles
2.4. Multiplicity of a Mapping; Order of a Function
2.5. Topological Properties of Mappings of Riemann Surfaces
2.6. Divisors on Riemann Surfaces
2.7. Finite Mappings of Riemann Surfaces
2.8. Unramified Coverings of Riemann Surfaces
2.9. The Universal Covering
2.10. Continuation of Mappings
2.11. The Riemann Surface of an Algebraic Function
3. Topology of Riemann Surfaces
3.1. Orientability
3.2. Triangulability
3.3. Development; Topological Genus
3.4. Structure of the Fundamental Group
3.5. The Euler Characteristic
3.6. The Hurwitz Formulae
3.7. Homology and Cohomology; Betti Numbers
3.8. Intersection Product; Poincare Duality
4. Calculus on Riemann Surfaces
4.1. Tangent Vectors; Differentiations
4.2. Differential Forms
4.3. Exterior Differentiations; de Rham Cohomology
4.4. Kahler and Riemann Metrics
4.5. Integration of Exterior Differentials; Greens Formula
4.6. Periods; de Rham Isomorphism
4.7. Holomorphic Differentials; Geometric GenuS;Riemanns Bilinear Relations
4.8. Meromorphic Differentials; Canonical Divisors
4.9. Meromorphic Differentials with Prescribed Behaviour at Poles; Residues
4.10. Periods of Meromorphic Differentials
4.11. Harmonic Differentials
4.12. Hilbert Space of Differentials; Harmonic Projection
4.13. Hodge Decomposition
4.14. Existence of Meromorphic Differentials and Functions
4.15. Dirichlets Principle
5. Classification of Riemann Surfaces
5.1. Canonical Regions
5.2. Uniformization
5.3. Types of Riemann Surfaces
5.4. Automorphisms of Canonical Regions
5.5. Riemann Surfaces of Elliptic Type
5.6. Riemann Surfaces of Parabolic Type
5.7. Riemann Surfaces of Hyperbolic Type
5.8. Automorphic Forms; Poincare Series
5.9. Quotient Riemann Surfaces; the Absolute Invariant
5.10. Moduli of Riemann Surfaces
6. Algebraic Nature of Compact Riemann Surfaces
6.1. Function Spaces and Mappings Associated with Divisors
6.2. Riemann-Roch Formula; Reciprocity Law for Differentials of the First and Second Kind
6.3. Applications of the Riemann-Roch Formula to Problems of Existence of Meromorphic Functions and Differentials
6.4. Compact Riemann Surfaces are Projective
……
Chapter 2. Algebraic Curves
Chapter 3. Jacobians and Abelian Varieties
References
Chapter 1. Riemann Surfaces
1. Basic Notions
1.1. Complex Chart; Complex Coordinates
1.2. Complex Analytic Atlas
1,3. Complex Analytic Manifolds
1.4. Mappings of Complex Manifolds
1.5. Dimension of a Complex Manifold
1.6. Riemann Surfaces
1.7. Differentiable Manifolds
2. Mappings of Riemann Surfaces
2.1. Nonconstant Mappings of Riemann Surfaces are Discrete
2.2. Meromorphic Functions on a Riemann Surface
2.3. Meromorphic Functions with Prescribed Behaviour at Poles
2.4. Multiplicity of a Mapping; Order of a Function
2.5. Topological Properties of Mappings of Riemann Surfaces
2.6. Divisors on Riemann Surfaces
2.7. Finite Mappings of Riemann Surfaces
2.8. Unramified Coverings of Riemann Surfaces
2.9. The Universal Covering
2.10. Continuation of Mappings
2.11. The Riemann Surface of an Algebraic Function
3. Topology of Riemann Surfaces
3.1. Orientability
3.2. Triangulability
3.3. Development; Topological Genus
3.4. Structure of the Fundamental Group
3.5. The Euler Characteristic
3.6. The Hurwitz Formulae
3.7. Homology and Cohomology; Betti Numbers
3.8. Intersection Product; Poincare Duality
4. Calculus on Riemann Surfaces
4.1. Tangent Vectors; Differentiations
4.2. Differential Forms
4.3. Exterior Differentiations; de Rham Cohomology
4.4. Kahler and Riemann Metrics
4.5. Integration of Exterior Differentials; Greens Formula
4.6. Periods; de Rham Isomorphism
4.7. Holomorphic Differentials; Geometric GenuS;Riemanns Bilinear Relations
4.8. Meromorphic Differentials; Canonical Divisors
4.9. Meromorphic Differentials with Prescribed Behaviour at Poles; Residues
4.10. Periods of Meromorphic Differentials
4.11. Harmonic Differentials
4.12. Hilbert Space of Differentials; Harmonic Projection
4.13. Hodge Decomposition
4.14. Existence of Meromorphic Differentials and Functions
4.15. Dirichlets Principle
5. Classification of Riemann Surfaces
5.1. Canonical Regions
5.2. Uniformization
5.3. Types of Riemann Surfaces
5.4. Automorphisms of Canonical Regions
5.5. Riemann Surfaces of Elliptic Type
5.6. Riemann Surfaces of Parabolic Type
5.7. Riemann Surfaces of Hyperbolic Type
5.8. Automorphic Forms; Poincare Series
5.9. Quotient Riemann Surfaces; the Absolute Invariant
5.10. Moduli of Riemann Surfaces
6. Algebraic Nature of Compact Riemann Surfaces
6.1. Function Spaces and Mappings Associated with Divisors
6.2. Riemann-Roch Formula; Reciprocity Law for Differentials of the First and Second Kind
6.3. Applications of the Riemann-Roch Formula to Problems of Existence of Meromorphic Functions and Differentials
6.4. Compact Riemann Surfaces are Projective
……
Chapter 2. Algebraic Curves
Chapter 3. Jacobians and Abelian Varieties
References
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