美国数学会经典影印系列 代数曲线和黎曼面 英文 作者： （美）里克·米兰达（Rick Miranda）著 出版时间：2019年版 内容简介 《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。 目录 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
