Basic Algebraic Geometry 2: Schemes and Complex Manifolds

271 Pages·2013·1.58 MB·English

by  Igor R. Shafarevich (auth.)

Igor R. Shafarevich
Basic Algebraic
Geometry 2
Schemes and Complex Manifolds
Third Edition
Basic Algebraic Geometry 2
Igor R. Shafarevich
Basic Algebraic
Geometry 2
Schemes and Complex Manifolds
Third Edition
IgorR.Shafarevich Translator
AlgebraSection MilesReid
SteklovMathematicalInstitute MathematicsInstitute
oftheRussianAcademyofSciences UniversityofWarwick
Moscow,Russia Coventry,UK
ISBN978-3-642-38009-9 ISBN978-3-642-38010-5(eBook)
DOI10.1007/978-3-642-38010-5
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013945857
MathematicsSubjectClassification(2010): 14-01
Translation of the 3rd Russian edition entitled “Osnovy algebraicheskoj geometrii”. MCCME,
Moscow2007,originallypublishedinRussianinonevolume
©Springer-VerlagBerlinHeidelberg1977,1994,2013
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Preface to Books 2–3
Books2–3correspondtoChaptersV–IXofthefirstedition.Theystudyschemesand
complexmanifolds,twonotionsthatgeneraliseindifferentdirectionsthevarieties
in projective space studied in Book 1. Introducing them leads also to new results
inthetheoryofprojectivevarieties.Forexample,itiswithintheframeworkofthe
theoryofschemesandabstractvarietiesthatwefindthenaturalproofoftheadjunc-
tionformulaforthegenusofacurve,whichwehavealreadystatedandappliedin
Section2.3,Chapter4.Thetheoryofcomplexanalyticmanifoldsleadstothestudy
ofthetopologyofprojectivevarietiesoverthefieldofcomplexnumbers.Forsome
questionsitisonlyherethatthenaturalandhistoricallogicofthesubjectcanbere-
asserted;forexample,differentialformswereconstructedinordertobeintegrated,
aprocesswhichonlymakessenseforvarietiesoverthe(realor)complexfields.
Changesfrom theFirstEdition
AsintheBook1,thereareanumberofadditionstothetext,ofwhichthefollowing
two are the most important. The first of these is a discussion of the notion of the
algebraic variety classifying algebraic or geometric objects of some type. As an
exampleweworkoutthetheoryoftheHilbertpolynomialandtheHilbertscheme.
IamverygratefultoV.I.Danilovforaseriesofrecommendationsonthissubject.
In particular the proof of Theorem 6.7 is due to him. The second addition is the
definitionandbasicpropertiesofaKählermetric,andadescription(withoutproof)
ofHodge’stheorem.
Prerequisites
Varietiesinprojectivespacewillprovideuswiththemainsupplyofexamples,and
thetheoreticalapparatusofBook1willbeused,butbynomeansallofit.Differ-
ent sections use different parts, and there is no point in giving exact indications.
ReferencestotheAppendixaretotheAlgebraicAppendixattheendofBook1.
V
VI PrefacetoBooks2–3
PrerequisitesforthereaderofBooks2–3areasfollows:forBook2,thesameas
forBook1;forBook3,thedefinitionofdifferentiablemanifold,thebasictheoryof
analyticfunctionsofacomplexvariable,andaknowledgeofhomology,cohomol-
ogyanddifferentialforms(knowledgeoftheproofsisnotessential);forChapter9,
familiaritywiththenotionoffundamentalgroupandtheuniversalcover.References
forthesetopicsaregiveninthetext.
RecommendationsforFurther Reading
For the reader wishing to go further in the study of algebraic geometry, we can
recommendthefollowingreferences.
For the cohomology of algebraic coherent sheaves and their applications: see
Hartshorne[37].
AnelementaryproofoftheRiemann–RochtheoremforcurvesisgiveninW.Ful-
ton,Algebraiccurves.Anintroductiontoalgebraicgeometry,W.A.Benjamin,Inc.,
New York–Amsterdam, 1969. This book is available as a free download from
http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf.
ForthegeneralcaseofRiemann–Roch,seeA.BorelandJ.-P.Serre,Lethéorème
deRiemann–Roch,Bull.Soc.Math.France86(1958)97–136,
Yu.I.Manin,LecturesontheK-functorinalgebraicgeometry,UspehiMat.Nauk
24:5 (149) (1969) 3–86, English translation: Russian Math. Surveys 24:5 (1969)
1–89,
W. Fulton and S. Lang, Riemann–Roch algebra, Grundlehren der mathematis-
chenWissenschaften277,Springer-Verlag,NewYork,1985.
Moscow,Russia I.R.Shafarevich
Contents
Book2:SchemesandVarieties
5 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 TheSpecofaRing. . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1 DefinitionofSpecA . . . . . . . . . . . . . . . . . . . . . 5
1.2 PropertiesofPointsofSpecA . . . . . . . . . . . . . . . . 7
1.3 TheZariskiTopologyofSpecA . . . . . . . . . . . . . . . 9
1.4 Irreducibility,Dimension . . . . . . . . . . . . . . . . . . 11
1.5 ExercisestoSection1 . . . . . . . . . . . . . . . . . . . . 14
2 Sheaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Presheaves . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 TheStructurePresheaf. . . . . . . . . . . . . . . . . . . . 17
2.3 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 StalksofaSheaf . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 ExercisestoSection2 . . . . . . . . . . . . . . . . . . . . 24
3 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1 DefinitionofaScheme . . . . . . . . . . . . . . . . . . . 25
3.2 GlueingSchemes . . . . . . . . . . . . . . . . . . . . . . 30
3.3 ClosedSubschemes . . . . . . . . . . . . . . . . . . . . . 32
3.4 ReducedSchemesandNilpotents . . . . . . . . . . . . . . 35
3.5 FinitenessConditions . . . . . . . . . . . . . . . . . . . . 36
3.6 ExercisestoSection3 . . . . . . . . . . . . . . . . . . . . 38
4 ProductsofSchemes . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.1 DefinitionofProduct . . . . . . . . . . . . . . . . . . . . 40
4.2 GroupSchemes . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 Separatedness . . . . . . . . . . . . . . . . . . . . . . . . 43
4.4 ExercisestoSection4 . . . . . . . . . . . . . . . . . . . . 46
6 Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1 DefinitionsandExamples . . . . . . . . . . . . . . . . . . . . . . 49
1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 49
VII
VIII Contents
1.2 VectorBundles . . . . . . . . . . . . . . . . . . . . . . . . 53
1.3 VectorBundlesandSheaves . . . . . . . . . . . . . . . . . 56
1.4 DivisorsandLineBundles. . . . . . . . . . . . . . . . . . 63
1.5 ExercisestoSection1 . . . . . . . . . . . . . . . . . . . . 67
2 AbstractandQuasiprojectiveVarieties . . . . . . . . . . . . . . . 68
2.1 Chow’sLemma . . . . . . . . . . . . . . . . . . . . . . . 68
2.2 BlowupAlongaSubvariety . . . . . . . . . . . . . . . . . 70
2.3 ExampleofNon-quasiprojectiveVariety . . . . . . . . . . 74
2.4 CriterionsforProjectivity . . . . . . . . . . . . . . . . . . 79
2.5 ExercisestoSection2 . . . . . . . . . . . . . . . . . . . . 81
3 CoherentSheaves . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.1 SheavesofO -Modules . . . . . . . . . . . . . . . . . . . 81
X
3.2 CoherentSheaves . . . . . . . . . . . . . . . . . . . . . . 85
3.3 DévissageofCoherentSheaves . . . . . . . . . . . . . . . 88
3.4 TheFinitenessTheorem . . . . . . . . . . . . . . . . . . . 92
3.5 ExercisestoSection3 . . . . . . . . . . . . . . . . . . . . 93
4 ClassificationofGeometricObjectsandUniversalSchemes . . . . 94
4.1 SchemesandFunctors . . . . . . . . . . . . . . . . . . . . 94
4.2 TheHilbertPolynomial . . . . . . . . . . . . . . . . . . . 100
4.3 FlatFamilies . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.4 TheHilbertScheme . . . . . . . . . . . . . . . . . . . . . 107
4.5 ExercisestoSection4 . . . . . . . . . . . . . . . . . . . . 110
Book3:ComplexAlgebraicVarietiesandComplexManifolds
7 TheTopologyofAlgebraicVarieties . . . . . . . . . . . . . . . . . . 115
1 TheComplexTopology . . . . . . . . . . . . . . . . . . . . . . . 115
1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 115
1.2 Algebraic Varieties as Differentiable Manifolds;
Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . 117
1.3 HomologyofNonsingularProjectiveVarieties . . . . . . . 118
1.4 ExercisestoSection1 . . . . . . . . . . . . . . . . . . . . 121
2 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
2.1 PreliminaryLemmas. . . . . . . . . . . . . . . . . . . . . 121
2.2 TheFirstProofoftheMainTheorem . . . . . . . . . . . . 122
2.3 TheSecondProof . . . . . . . . . . . . . . . . . . . . . . 124
2.4 AnalyticLemmas . . . . . . . . . . . . . . . . . . . . . . 126
2.5 ConnectednessofFibres . . . . . . . . . . . . . . . . . . . 127
2.6 ExercisestoSection2 . . . . . . . . . . . . . . . . . . . . 128
3 TheTopologyofAlgebraicCurves . . . . . . . . . . . . . . . . . 129
3.1 LocalStructureofMorphisms . . . . . . . . . . . . . . . . 129
3.2 TriangulationofCurves . . . . . . . . . . . . . . . . . . . 131
3.3 TopologicalClassificationofCurves . . . . . . . . . . . . 133
3.4 CombinatorialClassificationofSurfaces . . . . . . . . . . 137
3.5 TheTopologyofSingularitiesofPlaneCurves . . . . . . . 140
3.6 ExercisestoSection3 . . . . . . . . . . . . . . . . . . . . 142
Contents IX
4 RealAlgebraicCurves . . . . . . . . . . . . . . . . . . . . . . . . 142
4.1 ComplexConjugation . . . . . . . . . . . . . . . . . . . . 143
4.2 ProofofHarnack’sTheorem . . . . . . . . . . . . . . . . 144
4.3 OvalsofRealCurves . . . . . . . . . . . . . . . . . . . . 146
4.4 ExercisestoSection4 . . . . . . . . . . . . . . . . . . . . 147
8 ComplexManifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
1 DefinitionsandExamples . . . . . . . . . . . . . . . . . . . . . . 149
1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 149
1.2 QuotientSpaces . . . . . . . . . . . . . . . . . . . . . . . 152
1.3 CommutativeAlgebraicGroupsasQuotientSpaces . . . . 155
1.4 ExamplesofCompactComplexManifoldsnotIsomorphic
toAlgebraicVarieties . . . . . . . . . . . . . . . . . . . . 157
1.5 ComplexSpaces . . . . . . . . . . . . . . . . . . . . . . . 163
1.6 ExercisestoSection1 . . . . . . . . . . . . . . . . . . . . 165
2 DivisorsandMeromorphicFunctions . . . . . . . . . . . . . . . . 166
2.1 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
2.2 MeromorphicFunctions . . . . . . . . . . . . . . . . . . . 169
2.3 TheStructureoftheFieldM(X) . . . . . . . . . . . . . . 171
2.4 ExercisestoSection2 . . . . . . . . . . . . . . . . . . . . 174
3 AlgebraicVarietiesandComplexManifolds . . . . . . . . . . . . 175
3.1 ComparisonTheorems . . . . . . . . . . . . . . . . . . . . 175
3.2 ExampleofNonisomorphicAlgebraicVarietiesthatAre
IsomorphicasComplexManifolds . . . . . . . . . . . . . 178
3.3 ExampleofaNonalgebraicCompactComplexManifold
with MaximalNumberofIndependentMeromorphic
Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 181
3.4 TheClassificationofCompactComplexSurfaces . . . . . 183
3.5 ExercisestoSection3 . . . . . . . . . . . . . . . . . . . . 185
4 KählerManifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 185
4.1 KählerMetric . . . . . . . . . . . . . . . . . . . . . . . . 186
4.2 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4.3 OtherCharacterisationsofKählerMetrics . . . . . . . . . 190
4.4 ApplicationsofKählerMetrics . . . . . . . . . . . . . . . 193
4.5 HodgeTheory . . . . . . . . . . . . . . . . . . . . . . . . 196
4.6 ExercisestoSection4 . . . . . . . . . . . . . . . . . . . . 198
9 Uniformisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
1 TheUniversalCover . . . . . . . . . . . . . . . . . . . . . . . . . 201
1.1 TheUniversalCoverofaComplexManifold . . . . . . . . 201
1.2 UniversalCoversofAlgebraicCurves . . . . . . . . . . . 203
1.3 ProjectiveEmbeddingofQuotientSpaces . . . . . . . . . 205
1.4 ExercisestoSection1 . . . . . . . . . . . . . . . . . . . . 206
2 CurvesofParabolicType . . . . . . . . . . . . . . . . . . . . . . 207
2.1 ThetaFunctions . . . . . . . . . . . . . . . . . . . . . . . 207
2.2 ProjectiveEmbedding . . . . . . . . . . . . . . . . . . . . 209
X Contents
2.3 EllipticFunctions,EllipticCurvesandEllipticIntegrals . . 210
2.4 ExercisestoSection2 . . . . . . . . . . . . . . . . . . . . 213
3 CurvesofHyperbolicType . . . . . . . . . . . . . . . . . . . . . 213
3.1 PoincaréSeries. . . . . . . . . . . . . . . . . . . . . . . . 213
3.2 ProjectiveEmbedding . . . . . . . . . . . . . . . . . . . . 216
3.3 AlgebraicCurvesandAutomorphicFunctions . . . . . . . 218
3.4 ExercisestoSection3 . . . . . . . . . . . . . . . . . . . . 221
4 UniformisingHigherDimensionalVarieties . . . . . . . . . . . . 221
4.1 CompleteIntersectionsareSimplyConnected . . . . . . . 221
4.2 ExampleofManifoldwithπ aGivenFiniteGroup . . . . 222
1
4.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
4.4 ExercisestoSection4 . . . . . . . . . . . . . . . . . . . . 227
HistoricalSketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
1 EllipticIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
2 EllipticFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 231
3 AbelianIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
4 RiemannSurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 235
5 TheInversionofAbelianIntegrals . . . . . . . . . . . . . . . . . 237
6 TheGeometryofAlgebraicCurves . . . . . . . . . . . . . . . . . 239
7 HigherDimensionalGeometry . . . . . . . . . . . . . . . . . . . 241
8 TheAnalyticTheoryofComplexManifolds . . . . . . . . . . . . 243
9 AlgebraicVarietiesoverArbitraryFieldsandSchemes . . . . . . . 244
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
ReferencesfortheHistoricalSketch. . . . . . . . . . . . . . . . . . . . 250
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253