基礎代數(shù)幾何（第2卷）

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作　者：	（俄）Igor R.Shafarevich著
出版社：	世界圖書出版公司北京公司
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ISBN：	9787506236201	出版時間：	1998-01-01	包裝：	出版日期：1998-3-1 版次：1
開本：	20cm	頁數(shù)：	269頁	字數(shù)：

內容簡介

　　Books 2 and 3 correspond to Chap. V-IX of the first edition. They study schemes and complex manifolds， two notions that generalise in different directions the varieties in projective space studied in Book 1. Introducing them leads also to new results in the theory of projective varieties. For example， it is within the framework of the theory of schemes and abstract varieties that we find the natural proof of the adjunction formula for the genus of a curve， which we have already stated and applied in Chap. IV， 2.3. The theory of complex analytic manifolds leads to the study of the topology of projective varieties over the field of complex numbers. For some questions it is only here that the natural and historical logic of the subject can be reasserted; for example， differential forms were constructed in order to be integrated， a process which only makes sense for varieties over the （mai or） complex fields. Changes from the First Edition.此書為英文版。

作者簡介

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圖書目錄

BOOK2.SchemesandVarieties
ChapterV.Schemes
1.TheSpecofaRing
1.1.DefinitionofSpecA
1.2.PropertiesofPointsofSpecA
1.3.TheZariskiTopologyofSpecA
1.4.Irreducibility,Dimension
Exercisesto~1
2.Sheaves
2.1.Presheaves
2.2.TheStructurePresheaf
2.3.Sheaves
2.4.StalksofaSheaf
Exercisesto~2
3.Schemes
3.1.DefinitionofaScheme
3.2.GlueingSchemes
3.3.ClosedSubschemes
3.4.ReducedSchemesandNilpotents
3.5.FinitenessConditions
Exercisesto~3
4.ProductsofSchemes
4.1.DefinitionofProduct
4.2.GroupSchemes
4.3.Separatedness
Exercisesto~4
ChapterVI.Varieties
1.DefinitionsandExamples
1.1.Definitions
1.2.VectorBundles
1.3.VectorBundlesandSheaves
1.4.DivisorsandLineBundles
Exercisesto~1
2.AbstractandQuasiprojectiveVarieties
2.1.Chow'sLemma
2.2.BlowupAlongaSubvariety
2.3.ExampleofNon-QuasiprojectiveVariety
2.4.CriterionsforProjectivity
Exercisesto~2
3.CoherentSheaves
3.1.SheavesofOx-modules
3.2.CoherentSheaves
3.3.DevissageofCoherentSheaves
3.4.TheFinitenessTheorem
Exercisesto~3
4.ClassificationofGeometricObjectsandUniversalSchemes
4.1.SchemesandFunctors
4.2.TheHilbertPolynomial
4.3.FlatFamilies
4.4.TheHilbertScheme
Exercisesto~4
BOOK3.ComplexAlgebraicVarietiesandComplexManifolds
ChapterVII.TheTopologyofAlgebraicVarieties
1.TheComplexTopology
1.1.Definitions
1.2.AlgebraicVarietiesasDifferentiableManifolds;
Orientation
1.3.HomologyofNonsingularProjectiveVarieties
Exercisesto~1
2.Connectedness
2.1.PreliminaryLemmas
2.2.TheFirstProofoftheMainTheorem
2.3.TheSecondProof
2.4.AnalyticLemmas
2.5.Connectednes8ofFibres
Exercisesto~2
3.TheTopologyofAlgebraicCurves
3.1.LocalStructureofMorphisms
3.2;TriangulationofCurves
3.3.TopologicalClassificationofCurves
3.4.CombinatorialClassificationofSurfaces
3.5.TheTopologyofSingularitiesofPlaneCurves
Exercisesto~3
4.RealAlgebraicCurves
4.1.ComplexConjugation
4.2.ProofofHarnack'sTheorem
4.3.OvalsofRealCurves
Exercisesto~4
ChapterVIII.ComplexManifolds
1.DefinitionsandExamples
1.1.Definition
1.2.QuotientSpaces
1.3.CommutativeAlgebraicGroupsasQuotientSpaces
1.4.ExamplesofCompactComplexManifoldsnot
IsomorphictoAlgebraicVarieties
1.5.ComplexSpaces
Exercisesto~1
2.DivisorsandMeromorphicFunctions
2.1.Divisors
2.2.MeromorphicFunctions
2.3.TheStructureoftheFieldM(X)
Exercisesto~2
3.AlgebraicVarietiesandComplexManifolds
3.1.ComparisonTheorems
3.2.ExampleofNonisomorphicAlgebraicVarietiesthat
AreIsomorphicasComplexManifolds
3.3.ExampleofaNonalgebraicCompactComplex
ManifoldwithMaximalNumberofIndependent
MetamorphicFunctions
3.4.TheClassificationofCompactComplexSurfaces
Exercisesto~3
4.KahlerManifolds
4.1.KaihlerMetric
4.2.Examples
4.3.OtherCharacterisationsofKahlerMetrics
4.4.ApplicationsofKahlerMetrics
4.5.HodgeTheory
Exercisesto~4
ChapterIX.Uniformisation
1.TheUniversalCover
1.1.TheUniversalCoverofaComplexManifold
1.2.UniversalCoversofAlgebraicCurves
1.3.ProjectiveEmbeddingofQuotientSpaces
Exercisesto~1
2.CurvesofParabolicType
2.1.Thetafunctions
2.2.ProjectiveEmbedding
2.3.EllipticFunctions,EllipticCurvesandElliptic
Integrals
Exercisesto~2
3.CurvesofHyperbolicType
3.1.PoincareSeries
3.2.ProjectiveEmbedding
3.3.AlgebraicCurvesandAutomorphicFunctions
Exercisesto~3
4.UniformisingHigherDimensionalVarieties
4.1.CompleteIntersectionsareSimplyConnected
4.2.ExampleofManifoldwithaGivenFiniteGroup
4.3.Remarks
Exercisesto~4
HistoricalSketch
1.EllipticIntegrals
2.EllipticFunctions
3.AbelianIntegrals
4.RiemannSurfaces
5.TheInversionofAbelianIntegrals
6.TheGeometryofAlgebraicCurves
7.HigherDimensionalGeometry
8.TheAnalyticTheoryofComplexManifolds
9.AlgebraicVarietiesoverArbitraryFieldsandSchemes
References
ReferencesfortheHistoricalSketch
Index