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; Green's Formula
4.6. Periods; de Rham Isomorphism
4.7. Holomorphic Differentials; Geometric GenuS;Riemann's 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. Dirichlet's 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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