代数曲线和黎曼面
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作者Rick,Miranda
出版社高等教育出版社
ISBN9787040469066
出版时间2019-03
版次1
装帧精装
开本16开
纸张胶版纸
页数390页
定价169元
上书时间2024-06-01
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基本信息
书名:代数曲线和黎曼面
定价：169.00元
作者:Rick,Miranda
出版社：高等教育出版社
出版日期：2019-03-01
ISBN：9787040469066
字数：
页码：390
版次：
装帧：精装
开本：16开
商品重量：
编辑推荐
内容提要
《代数曲线和黎曼面（）》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，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
PrefaceChapter Ⅰ． Riemann Surfaces： Basic Definitions1．Complex Charts and Complex StructuresComplex ChartsComplex AtlasesThe Definition of a Piemann SurfaceReal 2-ManifoldsThe Genus of a Compact Riemann SurfaceComplex ManifoldsProblems 1．12．First Examples of Riemann SurfacesA Remark on Defining Riemann SurfacesThe Projective LineComplex ToriGraphs of Holomorphic FunctionsSmooth Affine Plane CurvesProblems 1．23．Projective CurvesThe Projective Plane P2Smooth Projective Plane CurvesHigher-Dimensional Projective SpacesComplete IntersectionsLocal Complete IntersectionsProblems 1．3Further ReadingChapter Ⅱ． Functions and Maps1．Functions on Riemann SurfacesHolomorphic FunctionsSingularities of Functions； Meromorphic FunctionsLanrent SeriesThe Order of a Meromorphic Function at a PointC∞o FunctionsHarmonic FunctionsTheorems Inherited from One Complex VariableProblems Ⅱ．12．Examples of Meromorphic FunctionsMeromorphic Functions on the Riemann SphereMeromorphic Functions on the Projective LineMeromorphic Functions on a Complex TorusMeromorphic Functions on Smooth Plane CurvesSmooth Projective CurvesProblems Ⅱ．23．Holomorphic Maps Between Riemann SurfacesThe Definition of a Holomorphic MapIsomorphisms and AutomorphismsEasy Theorems about Holomorphic MapsMeromorphic Functions and Holomorphic Maps to the Riemann SphereMeromorphic Functions on a Complex Torus， AgainProblems Ⅱ．34．Global Properties of Holomorphic MapsLocal Normal Form and MultiplicityThe Degree of a Holomorphic Map between Compact Riemann Sur-facesThe Sum of the Orders of a Meromorphic FunctionMeromorphic Functions on a Complex Torus， Yet AgainThe Euler Number of a Compact SurfaceHurwitz's FormulaProblems Ⅱ．4Further Reading……Chapter Ⅳ． Integration on Riemann SurfacesChapter Ⅴ． Divisors and Meromorphic FunctionsChapter Ⅵ． Algebraic Curves and the Riemann-Roch TheoremChapter Ⅶ． Applications of Riemann-RochChapter Ⅷ． Abel's TheoremChapter Ⅸ． Sheaves and Cech CohomologyChapter Ⅹ． Algebraic SheavesChapter Ⅺ． Invertible Sheaves， Line Bundles， and H1Further ReadingReferencesIndex of Notation
作者介绍
序言
PrefacenChapter Ⅰ． Riemann Surfaces： Basic Definitionsn1．Complex Charts and Complex StructuresnComplex ChartsnComplex AtlasesnThe Definition of a Piemann SurfacenReal 2-ManifoldsnThe Genus of a Compact Riemann SurfacenComplex ManifoldsnProblems 1．1n2．First Examples of Riemann SurfacesnA Remark on Defining Riemann SurfacesnThe Projective LinenComplex TorinGraphs of Holomorphic FunctionsnSmooth Affine Plane CurvesnProblems 1．2n3．Projective CurvesnThe Projective Plane P2nSmooth Projective Plane CurvesnHigher-Dimensional Projective SpacesnComplete IntersectionsnLocal Complete IntersectionsnProblems 1．3nFurther ReadingnnChapter Ⅱ． Functions and Mapsn1．Functions on Riemann SurfacesnHolomorphic FunctionsnSingularities of Functions； Meromorphic FunctionsnLanrent SeriesnThe Order of a Meromorphic Function at a PointnC∞o FunctionsnHarmonic FunctionsnTheorems Inherited from One Complex VariablenProblems Ⅱ．1n2．Examples of Meromorphic FunctionsnMeromorphic Functions on the Riemann SpherenMeromorphic Functions on the Projective LinenMeromorphic Functions on a Complex TorusnMeromorphic Functions on Smooth Plane CurvesnSmooth Projective CurvesnProblems Ⅱ．2n3．Holomorphic Maps Between Riemann SurfacesnThe Definition of a Holomorphic MapnIsomorphisms and AutomorphismsnEasy Theorems about Holomorphic MapsnMeromorphic Functions and Holomorphic Maps to the Riemann SpherenMeromorphic Functions on a Complex Torus， AgainnProblems Ⅱ．3n4．Global Properties of Holomorphic MapsnLocal Normal Form and MultiplicitynThe Degree of a Holomorphic Map between Compact Riemann Sur-facesnThe Sum of the Orders of a Meromorphic FunctionnMeromorphic Functions on a Complex Torus， Yet AgainnThe Euler Number of a Compact SurfacenHurwitz's FormulanProblems Ⅱ．4nFurther Readingn……nChapter Ⅳ． Integration on Riemann SurfacesnChapter Ⅴ． Divisors and Meromorphic FunctionsnChapter Ⅵ． Algebraic Curves and the Riemann-Roch TheoremnChapter Ⅶ． Applications of Riemann-RochnChapter Ⅷ． Abel's TheoremnChapter Ⅸ． Sheaves and Cech CohomologynChapter Ⅹ． Algebraic SheavesnChapter Ⅺ． Invertible Sheaves， Line Bundles， and H1nFurther ReadingnReferencesnIndex of Notation
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