《代数曲线和黎曼面（影印版）》作者认为复数域是与代数曲线酋次邂逅的好地方，在那里，读者对于曲面、积分和其他概念的经典直觉可以发挥作用。因此，第一章列举了代数曲线的许多例子。如此一来，该书便以复坐标图表和亚纯函数为中心舞台，开启了一场对黎曼面的启蒙教程。但是，该书主要的例子来自射影曲线，从而内容逐步而坚定地转向了代数范畴。Riemann-Roch定理和Setre对偶定理的证明都以一种代数的方式给出，它是一种修改了的阿代尔证明（adelic proof），借助于解Mittag-Leffler问题来完全表达。作为后面几章的统一构架引进了层和上同调，它们的用处和自然性直接可见。 该书要求读者有一学期的复变函数和一年的抽象代数的学习背景，从而很适合作为第二学期的复变函数课或一年期的代数几何课的参考书。
Preface
Chapter Ⅰ． Riemann Surfaces： Basic Definitions
1．Complex Charts and Complex Structures
Complex Charts
Complex Atlases
The Definition of a Piemann Surface
Real 2-Manifolds
The Genus of a Compact Riemann Surface
Complex Manifolds
Problems 1．1
2．First Examples of Riemann Surfaces
A Remark on Defining Riemann Surfaces
The Projective Line
Complex Tori
Graphs of Holomorphic Functions
Smooth Affine Plane Curves
Problems 1．2
3．Projective Curves
The Projective Plane P2
Smooth Projective Plane Curves
Higher-Dimensional Projective Spaces
Complete Intersections
Local Complete Intersections
Problems 1．3
Further Reading
Chapter Ⅱ． Functions and Maps
1．Functions on Riemann Surfaces
Holomorphic Functions
Singularities of Functions； Meromorphic Functions
Lanrent Series
The Order of a Meromorphic Function at a Point
C∞o Functions
Harmonic Functions
Theorems Inherited from One Complex Variable
Problems Ⅱ．1
2．Examples of Meromorphic Functions
Meromorphic Functions on the Riemann Sphere
Meromorphic Functions on the Projective Line
Meromorphic Functions on a Complex Torus
Meromorphic Functions on Smooth Plane Curves
Smooth Projective Curves
Problems Ⅱ．2
3．Holomorphic Maps Between Riemann Surfaces
The Definition of a Holomorphic Map
Isomorphisms and Automorphisms
Easy Theorems about Holomorphic Maps
Meromorphic Functions and Holomorphic Maps to the Riemann Sphere
Meromorphic Functions on a Complex Torus， Again
Problems Ⅱ．3
4．Global Properties of Holomorphic Maps
Local Normal Form and Multiplicity
The Degree of a Holomorphic Map between Compact Riemann Sur-faces
The Sum of the Orders of a Meromorphic Function
Meromorphic Functions on a Complex Torus， Yet Again
The Euler Number of a Compact Surface
Hurwitz's Formula
Problems Ⅱ．4
Further Reading
……
Chapter Ⅳ． Integration on Riemann Surfaces
Chapter Ⅴ． Divisors and Meromorphic Functions
Chapter Ⅵ． Algebraic Curves and the Riemann-Roch Theorem
Chapter Ⅶ． Applications of Riemann-Roch
Chapter Ⅷ． Abel's Theorem
Chapter Ⅸ． Sheaves and Cech Cohomology
Chapter Ⅹ． Algebraic Sheaves
Chapter Ⅺ． Invertible Sheaves， Line Bundles， and H1
Further Reading
References
Index of Notation
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