代数曲面、代数流形和概型

IntroductionbyI.R.Shafarevich

Chapter1.PdemannSurfaces

§1.BasicNotions

1.1.ComplexChart;ComplexCoordinates

1.2.ComplexAnalyticAtlas

1.3.ComplexAnalyticManifolds

1.4.MappingsofComplexManifolds

1.5.DimensionofaComplexManifold

1.6.RiemannSurfaces

1.7.DifferentiableManifolds

§2.MappingsofPdemannSurfaces

2.1.NonconstantMappingsofRiemannSurfacesareDiscrete

2.2.MeromorphicFunctionsonaRiemannSurface

2.3.MeromorphicFunctionswithPrescribedBehaviouratPoles

2.4.MultiplicityofaMapping;OrderofaFunction

2.5.TopologicalPropertiesofMappingsofRiemannSurfaces

2.6.DivisorsonRiemannSurfaces

2.7.FiniteMappingsofRiemannSurfaces

2.8.UnramifiedCoveringsofRiemannSurfaces

2.9.TheUniversalCovering

2.10.ContinuationofMappings

2.11.TheRiemannSurfaceofanAlgebraicFunction

§3.TopologyofRiemannSurfaces

3.1.Orientability

3.2.Triangulability

3.3.Development;TopologicalGenus

3.4.StructureoftheFundamentalGroup

3.5.TheEulerCharacteristic

3.6.TheHurwitzFormulae

3.7.HomologyandCohomology;BettiNumbers

3.8.IntersectionProduct;PoincareDuality

§4.CalculusonRiemannSurfaces

4.1.TangentVectors;Differentiations

4.2.DifferentialForms

4.3.ExteriorDifferentiations;deRhamCohomology

4.4.KahlerandRiemannMetrics

4.5.IntegrationofExteriorDifferentials;GreensFormula

4.6.Periods;deRhamIsomorphism

4.7.HolomorphicDifferentials;GeometricGenus;RiemannsBilinearRelations

4.8.MeromorphicDifferentials;CanonicalDivisors

4.9.MeromorphicDifferentialswithPrescribedBehaviouratPoles;Residues

4.10.PeriodsofMeromorphicDifferentials

4.11.HarmonicDifferentials

4.12.HilbertSpaceofDifferentials;HarmonicProj

5.3.TypesofPiemannSurfaces

5.4.AutomorphismsofCanonicalRegions

5.5.RiemannSurfacesofEllipticType

5.6.RiemannSurfacesofParabolicType

5.7.RiemannSurfacesofHyperbolicType

5.8.AutomorphicForms;PoincareSeries

5.9.QuotientRiemannSurfaces;theAbsoluteInvariant

5.10.ModuliofRiemannSurfaces

§6.AlgebraicNatureofCompactRiemannSurfaces

6.1.FunctionSpacesandMappingsAssociatedwithDivisors

6.2.Riemann-RochFormula;ReciprocityLawforDifferentialsoftheFirstandSecondKind

6.3.ApplicationsoftheRiemann-RochFormulatoProblemsofExistenceofMeromorphicFunctionsandDifferentials

6.4.CompactRiemannSurfacesareProjective

6.5.AlgebraicNatureofProjectiveModels;ArithmeticRiemannSurfaces

6.6.ModelsofRiemannSurfacesofGenus1

Chapter2.AlgebraicCurves

§1.BasicNotions

1.1.AlgebraicVarieties;ZariskiTopology

1.2.RegularFunctionsandMappings

1.3.TheImageofaProjectiveVarietyisClosed

1.4.Irreducibility;Dimension

1.5.AlgebraicCurves

1.6.SingularandNonsingularPointsonVarieties

1.7.RationalFunctions,MappingsandVarieties

1.8.Differentials

1.9.ComparisonTheorems

1.10.LefschetzPrinciple

§2.Riemann-RochFormula

2.1.MultiplicityofaMapping;Ramification

2.2.Divisors

2.3.IntersectionofPlaneCurves

2.4.TheHurwitzFormulae

2.5.FunctionSpacesandSpacesofDifferentialsAssociatedwithDivisors

2.6.ComparisonTheorems(Continued)

2.7.Riemann-RochFormula

2.8.ApproachestotheProof

2.9.FirstApplications

2.10.RiemannCount

§3.GeometryofProjectiveCurves

3.1.LinearSystems

3.2.MappingsofCurvesintoPn

3.3.GenericHyperplaneSections

3.4.GeometricalInterpretationof

3.12.TheClebschFormula

3.13.DualCurves

3.14.PliickerFormulafortheClass

3.15.CorrespondenceofBranches;DualFormulae

Chapter3.JacobiansandAbelianVarieties

§1.AbelianVarieties

1.1.AlgebraicGroups

1.2.AbelianVarieties

1.3.AlgebraicComplexTori;PolarizedTori

1.4.ThetaFunctionandRiemannThetaDivisor

1.5.PrincipallyPolarizedAbelianVarieties

1.6.PointsofFiniteOrderonAbelianVarieties

1.7.EllipticCurves

§2.JacobiansofCurvesandofRiemannSurfaces

2.1.PrincipalDivisorsonRiemannSurfaces

2.2.Inversion-Problem

2.3.PicardGroup

2.4.PicardVarietiesandtheirUniversalProperty

2.5.PolarizationDivisoroftheJacobianofaCurve;PoincareFormulae

2.6.JacobianofaCurveofGenus1

References

　　本书是一部全面介绍代数曲线、代数流形的教程(全英文版)。主体内容有两部分组成：一部分以V.V.Shokurov所写的学术著作为蓝本，主要讲述黎曼面和代数曲面理论，深刻地揭示了黎曼面和其模型复射影面中的复代数曲面的相互关系；另外一部分以V.I.Danilov的学术论文为蓝本主要讨论了代数变量及其概型。本书结构框架清晰，叙述简明扼要，可以帮助读者在很短的时间内了解并掌握代数几何的精华。本书适合数学专业的高年级本科生、研究生以及相关的科研人员阅读参考。