基础代数几何（第2卷）

　　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.此书为英文版。

暂缺《基础代数几何（第2卷）》作者简介

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