代数曲线和黎曼面
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作者(美)里克·米兰达
出版社高等教育出版社
ISBN9787040469066
出版时间2019-03
装帧精装
开本16开
定价169元
货号1201934448
上书时间2024-11-02
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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 I.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 I.2
3.Projective Curves
The Projective Plane P2
Smooth Projective Plane Curves
Higher-Dimensional Projective Spaces
Complete Intersections
Local Complete Intersections
Problems I.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
Hurwitzs 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. Abels 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
内容摘要
里克·米兰达的《代数曲线和黎曼面(英文版)(精)/美国数学会经典影印系列》，作者认为复数域是与代数曲线酋次邂逅的优选地方，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，本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，本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问题来接近表达。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。
本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。
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