內容大鋼
里克·米蘭達的《代數曲線和黎曼面(英文版)(精)/美國數學會經典影印系列》，作者認為複數域是與代數曲線酋次邂逅的最好地方，在那裡，讀者對於曲面、積分和其他概念的經典直覺可以發揮作用。因此，第一章列舉了代數曲線的許多例子。如此一來，本書便以復坐標圖表和亞純函數為中心舞台，開啟了一場對黎曼面的啟蒙教程。但是，本書主要的例子來自射影曲線，從而內容逐步而堅定地轉向了代數範疇。Riemann-Roch定理和Setre對偶定理的證明都以一種代數的方式給出，它是一種修改了的阿代爾證明(adelic proof)，借助於解Mittag-Leffler問題來完全表達。作為後面幾章的統一構架引進了層和上同調，它們的用處和自然性直接可見。
本書要求讀者有一學期的複變函數和一年的抽象代數的學習背景，從而很適合作為第二學期的複變函數課或一年期的代數幾何課的參考書。
目錄
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 III. More Examples of Riemann Surfaces
1.More Elementary Examples of Riemann Surfaces
Lines and Conics
Glueing Together Riemann Surfaces
Hyperelliptic Riemann Surfaces
Meromorphic Functions on Hyperelliptic Riemann Surfaces
Maps Between Complex Tori
Problems III.1
2.Less Elementary Examples of Riemann Surfaces
Plugging Holes in Riemann Surfaces
Nodes of a Plane Curve
Resolving a Node of a Plane Curve
The Genus of a Projective Plane Curve with Nodes
……
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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