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CAMBRIDGESTUDIESINADVANCEDMATHEMATICS209

EditorialBoard

COMPLEXALGEBRAICTHREEFOLDS

Thefirstbookontheexplicitbirationalgeometryofcomplexalgebraicthreefoldsarising fromtheminimalmodelprogram,thistextissuretobecomeanessentialreferencein thefieldofbirationalgeometry.Threefoldsremaintheinterfacebetweenlow-andhighdimensionalsettings,andagoodunderstandingofthemisnecessaryinthisactively evolvingarea.

Intendedforadvancedgraduatestudentsaswellasresearchersworkinginbirational geometry,thebookisasself-containedaspossible.Detailedproofsaregiventhroughout, andmorethan100exampleshelptodeepenunderstandingofbirationalgeometry.

Thefirstpartofthebookdealswiththreefoldsingularities,divisorialcontractions andflips.AfterathoroughexplanationoftheSarkisovprogram,thesecondpartis devotedtotheanalysisofoutputs,specificallyminimalmodelsandMorifibrespaces. Thelatteraredividedintoconicalfibrations,delPezzofibrationsandFanothreefolds accordingtotherelativedimension.

MasayukiKawakita isAssociateProfessorattheResearchInstituteforMathematical Sciences,KyotoUniversity.Hehasestablishedaclassificationofthreefolddivisorial contractionsandisaleadingexpertinalgebraicthreefolds.

CAMBRIDGESTUDIESINADVANCEDMATHEMATICS

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ComplexAlgebraicThreefolds

MASAYUKIKAWAKITA

KyotoUniversity

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Formyfamily

Prefacepage xi

1TheMinimalModelProgram 1

1.1Preliminaries2 1.2NumericalGeometry11 1.3TheProgram19

1.4LogarithmicandRelativeExtensions28 1.5ExistenceofFlips36

1.6TerminationofFlips46 1.7Abundance52

2Singularities 59

2.1AnalyticGerms60

2.2QuotientsandCoverings64

2.3TerminalSingularitiesofIndexOne76

2.4TerminalSingularitiesofHigherIndex86

2.5SingularRiemann–RochFormula96

2.6CanonicalSingularities107

3DivisorialContractionstoPoints 116

3.1IdentificationoftheDivisor117

3.2NumericalClassification121

3.3GeneralElephantsforExceptionalType128

3.4GeneralElephantsforOrdinaryType136

3.5GeometricClassification145

3.6Examples157

4DivisorialContractionstoCurves 162

4.1ContractionsfromGorensteinThreefolds162

4.2ContractionstoSmoothCurves168

4.3ContractionstoSingularCurves176

4.4ConstructionbyaOne-Parameter Deformation186

5Flips 199

5.1StrategyandFlops200

5.2NumericalInvariants207

5.3PlanarityofCoveringCurves213

5.4DeformationsofanExtremal Neighbourhood221

5.5GeneralElephants231

5.6Examples244

6TheSarkisovCategory 249

6.1LinksofMoriFibreSpaces250

6.2TheSarkisovProgram255

6.3RationalityandBirationalRigidity265

6.4Pliability275

7ConicalFibrations 285

7.1StandardConicBundles286

7.2 Q-ConicBundles294

7.3Classification303

7.4Rationality309

7.5BirationalRigidity319

8DelPezzoFibrations 324

8.1StandardModels325

8.2SimpleModelsandMultipleFibres333

8.3Rationality343

8.4BirationalRigidity348

9FanoThreefolds 359

9.1Boundedness360

9.2GeneralElephants371

9.3ClassificationinSpecialCases378

9.4ClassificationofPrincipalSeries388

9.5BirationalRigidityandK-Stability398

10MinimalModels 410

10.1Non-Vanishing411

10.2Abundance421

10.3BirationalMinimalModels428

Preface

Thepresentbooktreatsexplicitaspectsofthebirationalgeometryofcomplex algebraicthreefolds.Itisafundamentalprobleminbirationalgeometrytofind andanalyseagoodrepresentativeofeachbirationalclassofalgebraicvarieties. Theminimalmodelprogram,ortheMMPforshort,conjecturallyrealisesthis bycomparingthecanonicaldivisorsofvarieties.AccordingastheMMPhas developed,thebookhasarisenfromtwoperspectives.

Firstly,sincetheMMPindimensionthreewasestablishedaboutaquarter centuryago,itisdesirabletounderstandindividualthreefoldsexplicitlyby meansoftheMMP.Theinitialstepistodescribebirationaltransformationsin theMMP.Nowwehaveapracticalclassificationofthemindimensionthree. TheensuingimportantsubjectistoanalysethethreefoldsoutputbytheMMP. InthisdirectiononecanmentiontheSarkisovprogram,whichdecomposes everybirationalmapofMorifibrespaces.

Secondly,theMMPinhigherdimensionsisstillevolvingactivelyafterthe existenceofflipswasproved,astypifiedinthesettlementoftheBorisov–Alexeev–Borisovconjecture.Agoodknowledgeofthreefoldsisusefulandwill benecessaryinfurtherdevelopmentofhigherdimensionalbirationalgeometry. Thisiscomparabletothenaturethatmostresultsonthreefoldsarebasedupon theclassicaltheoryofsurfaces.

Thebookconcentratesontheexplicitstudyofalgebraicthreefoldsbythe MMP.Theauthorhastriedtoelucidatetheproofsrigorouslyandtomakethe bookasself-containedaspossible.Anumberofexampleswillhelptodeepen theunderstandingofthereader.Thereaderisstronglyencouragedtoverify thecomputationsinexamples.Thoughitdoesnotcoverimportanttopicssuch asaffinegeometry,derivedcategoriesandpositivecharacteristicaspects,the bookwillsupplyenoughknowledgeofthreefoldbirationalgeometrytoenter thefieldofhigherdimensionalbirationalgeometry.

Preface

Thebookisintendedforadvancedgraduatestudentswhoareinterestedin thebirationalgeometryofalgebraicvarieties.Itcanalsobeusedbyresearchers asareferencefortheclassificationresultsonthreefolds.Thereadershouldbe familiarwithbasicalgebraicgeometryatthelevelofHartshorne’stextbook [178].SomeknowledgeoftheMMPishelpfulbutitisnotaprerequisite.One canlearnthegeneraltheoryoftheMMPfromastandardbooksuchasthatby KollárandMori[277]orbyMatsuki[307].Whilstitroughlycorrespondsto Chapter1ofthebook,themainbodystartsfromChapter2andconcentrates onthreefolds.

ThevolumeeditedbyCortiandReid[97]isanoutstandingcollectionfrom thesamestandpoint.Itplayedaguidingroleintheexplicitstudyofalgebraic threefoldswhenitwaspublished.However,greatprogresshasbeenmadesince then.Thepresentbookaimsatanorganisedtreatmentofthreefolds,including recentresults.Itseekstobesomewhatofathreefoldversionofthebookon surfacesbyBarth,Hulek,PetersandVandeVen[30]orbyBeauville[35].

Chapter1summarisesthetheoryoftheMMPinanarbitrarydimensionbut excludesdetailedproofs.Thefirstpart,Chapters2to5,ofthemainbodydeals withobjectswhichappearinthecourseoftheMMP.Chapter2classifiesthreefoldsingularitiesintheMMPcompletely.ThenChapters3,4and5describe threefoldbirationaltransformationsintheMMP,thatis,divisorialcontractions andflips.Theycontainallthenecessaryargumentsbyomittingonlytheparts whichrepeattheprecedingarguments.

Thesecondpart,Chapters6to10,isdevotedtotheanalysisofoutputs oftheMMP,whichareMorifibrespacesandminimalmodels.AfterChapter6explainsthegeneraltheoryoftheSarkisovprogram,Chapters7,8and9 investigatethegeometryofthreefoldMorifibrespacesaccordingtotherelativedimensionofthefibrestructure.FinallyChapter10discussesminimal threefoldsfromthepointofviewofabundance.

Theauthorwouldliketoexpressthankstoallthecolleagueswithwhomhe helddiscussions.YujiroKawamataintroducedhimtothesubjectofbirational geometryashisacademicsupervisor.AlessioCortiandMilesReidcommunicatedtheirprofoundknowledgeofthreefoldstohimduringhisvisittothe UniversityofCambridge.WhenhestayedattheInstituteforAdvancedStudy, hereceivedwarmhospitalityfromJánosKollár.Healsolearntagreatdealfrom theregularseminarorganisedbyShigefumiMori,ShigeruMukaiandNoboru Nakayama.FinallyhewouldliketothankPhilipMeylerandJohnLingleiMeng atCambridgeUniversityPressfortheirsupportforthepublication.

TheMinimalModelProgram

Thischapteroutlinesthegeneraltheoryoftheminimalmodelprogram.Weshall studyalgebraicthreefoldsthoroughlyinthesubsequentchaptersinalignment withtheprogram.Thereaderwhoisnotfamiliarwiththeprogrammaygrasp thebasicnotionsatfirstandreferbacklater.

Blowingupasurfaceatapointisnotanessentialoperationfromthebirational pointofview.Itsexceptionalcurveischaracterisednumericallyasa (−1)-curve. Asisthecaseinthisobservation,theintersectionnumberisabasiclineartool inbirationalgeometry.Theminimalmodelprogram,ortheMMPforshort, outputsarepresentativeofeachbirationalclassthatisminimalwithrespectto thenumericalclassofthecanonicaldivisor.

TheMMPgrewoutofthesurfacetheorywithallowingmildsingularities. Foragivenvariety,itproducesaminimalmodeloraMorifibrespaceafter finitelymanybirationaltransformations,whicharedivisorialcontractionsand flips.Nowtheprogramisformulatedinthelogarithmicframeworkwherewe treatapairconsistingofavarietyandadivisor.

TheMMPfunctionssubjecttotheexistenceandterminationofflips.Hacon andMcKernanwithBirkarandCasciniprovedtheexistenceofflipsinan arbitrarydimension.Consideringafliptobetherelativecanonicalmodel,they establishedtheMMPwithscalinginthebirationalsetting.Theterminationof threefoldflipsfollowsfromthedecreaseinthenumberofdivisorswithsmall logdiscrepancy.Shokurovreducedtheterminationinanarbitrarydimension tocertainconjecturalpropertiesoftheminimallogdiscrepancy.

ItisalsoimportanttoanalysetherepresentativeoutputbytheMMP.The SarkisovprogramdecomposesabirationalmapofMorifibrespacesintoelementaryones.Foraminimalmodel,weexpecttheabundancewhichclaimsthe freedomofthelinearsystemofamultipleofthecanonicaldivisor.Itdefines amorphismtotheprojectivevarietyassociatedwiththecanonicalring,which weknowisfinitelygenerated.

1.1Preliminaries

Weshallfixthenotationandrecallthefundamentalsofalgebraicgeometry. Thebook[178]byHartshorneisastandardreference.

The naturalnumbers beginwithzero.Thesymbol �� ≥�� for �� = N, Z, Q or R standsforthesubset {�� ∈ �� | �� ≥ �� } andsimilarly ��>�� = {�� ∈ �� | ��>�� }

Forinstance, N = Z≥0.Thequotient Z�� = Z/��Z isthe cyclicgroup oforder �� The round-down �� ofarealnumber �� isthegreatestintegerlessthanorequal to ��,whilstthe round-up �� isdefinedas �� = − −��

Schemes A scheme isalwaysassumedtobeseparated.Itissaidtobe integral ifitisirreducibleandreduced.

Weworkoverthefield C ofcomplexnumbersunlessotherwisementioned. An algebraicscheme isaschemeoffinitetypeover Spec �� forthealgebraically closedgroundfield ��,whichistacitlyassumedtobe C.Wecallita complex scheme whenweemphasisethatitisdefinedover C.Analgebraicschemeis saidtobe complete ifitisproperover Spec ��.A point inanalgebraicscheme usuallymeansaclosedpoint.

A variety isanintegralalgebraicscheme.A complexvariety isavariety over C.A curve isavarietyofdimensiononeanda surface isavarietyof dimensiontwo.An ��-fold isavarietyofdimension ��.The affinespace A�� is Spec �� [��1,...,����] andthe projectivespace P�� is Proj �� [��0,...,����].The originof A�� isdenotedby ��.

The germ �� ∈ �� ofaschemeisconsideredataclosedpointunlessotherwise specified.Itisanequivalenceclassofthepair ( ��,��) ofascheme �� andapoint �� in �� where ( ��,��) isequivalentto ( �� ,�� ) ifthereexistsanisomorphism �� �� ofopenneighbourhoods �� ∈ �� ⊂ �� and �� ∈ �� ⊂ �� sending �� to ��

Bya singularity,wemeanthegermatasingularpointasarule.

Foralocallyfreecoherentsheaf E onanalgebraicscheme ��,the projective spacebundle P(E ) = Proj�� ��E over �� isdefinedbythesymmetric O�� -algebra ��E = �� ∈N ���� E of E .Itisa P��-bundle if E isofrank �� + 1.Inparticular,the projectivespace P�� = Proj ���� isdefinedforafinitedimensionalvectorspace �� .Itisregardedasthequotientspace (�� ∨ \ 0)/�� × ofthedualvectorspace �� ∨ minuszerobytheactionofthemultiplicativegroup �� × = �� \{0} oftheground field ��.Asusedabove,thesymbol ∨ standsforthedualand × forthegroupof units.

Morphisms Foramorphism �� : �� → �� ofschemes,the image �� ( ��) ofa closedsubset �� of �� andthe inverseimage �� 1 (��) ofaclosedsubset �� of �� areconsideredset-theoretically.When �� isproperand �� isaclosedsubscheme, weregard �� ( ��) asareducedscheme.Wealsoregard �� 1 (��) foraclosed

subscheme �� asareducedschemeanddistinguishitfromthescheme-theoretic fibre �� �� ��.

A rationalmap �� : �� �� ofalgebraicschemesisanequivalenceclassof amorphism �� → �� definedonadenseopensubset �� of ��.The image �� ( ��) of �� istheimage �� (Γ) ofthegraph Γ of �� asaclosedsubschemeof �� × �� bytheprojection �� : �� × �� → �� .Wesaythatamorphismorarationalmap is birational ifithasaninverseasarationalmap.Twoalgebraicschemesare birational ifthereexistsabirationalmapbetweenthem.Bydefinition,two varietiesarebirationalifandonlyiftheyhavethesamefunctionfield.

Let �� : �� → �� beamorphismofalgebraicschemes.Wesaythat �� is projective ifitisisomorphicto Proj�� R → �� byagraded O�� -algebra R = �� ∈N R�� generatedbycoherent R1,with R0 = O�� .When �� isquasi-projective,the projectivityof �� meansthatitisrealisedasaclosedsubschemeofarelative projectivespace P�� × �� → �� .Aninvertiblesheaf L on �� is relativelyvery ample (or veryample over �� or ��-veryample)ifitisisomorphicto O (1) byan expression �� Proj�� R asabove.Wesaythat L is relativelyample (��-ample) if L ⊗�� isrelativelyveryampleforsomepositiveinteger ��

Supposethat �� : �� → �� isproper.Wesaythat �� hasconnectedfibres ifthe naturalmap O�� → ��∗O�� isanisomorphism.Thisimpliesthatthefibre �� ×�� �� atevery �� ∈ �� isconnectedandnon-empty[160,IIIcorollaire4.3.2].Theproof foraprojectivemorphismisin[178,IIIcorollary11.3].Ingeneral, �� admits the Steinfactorisation �� = �� ◦ �� with �� : �� → �� and �� : �� → �� definedby �� = Spec�� ��∗O�� ,forwhich �� isproperwithconnectedfibresand �� isfinite.If �� isaproperbirationalmorphismfromavarietytoanormalvariety,thenthe factor �� intheSteinfactorisationisanisomorphismandhence �� hasconnected fibres.Thisisreferredtoas Zariski’smaintheorem.

Lemma1.1.1 Let �� : �� → �� and �� : �� → �� bemorphismsofalgebraic schemessuchthat �� isproperandhasconnectedfibres.Ifeverycurvein �� contractedtoapointby �� isalsocontractedby ��,then �� factorsthrough �� as �� = �� ◦ �� foramorphism �� : �� → ��.

Proof Let �� �� and �� �� denotethesetsofclosedpointsin �� and �� respectively. For �� ∈ �� ��,theinverseimage �� 1 (��) isconnectedand ��(�� 1 (��)) isonepoint. Define �� �� : �� �� → �� �� by �� �� (��) = ��(�� 1 (��)).Since �� isproperandsurjective, foranyclosedsubset �� of ��, �� (�� 1 (��)) isclosedin �� and ( �� ��) 1 (��| �� �� ) = �� (�� 1 (��))|�� �� .Thus �� �� extendstoacontinuousmap �� : �� → ��,whichisa morphismofschemesbythenaturalmap O�� → ��∗O�� = ��∗��∗O�� = ��∗O�� .

Chow’slemma [160,II§5.6]replacesthepropermorphism �� : �� → �� byaprojectivemorphism.Itassertstheexistenceofaprojectivebirational

TheMinimalModelProgram

morphism �� : �� → �� suchthat �� ◦ �� : �� → �� isprojective.The projection formula andthe Lerayspectralsequence,formulatedforringedspacesin[160, 0§12.2],willbefrequentlyused.Thereference[198,section3.6]explains spectralsequencesfromourperspective.

Theorem1.1.2 (Projectionformula) Let �� : �� → �� beamorphismofringed spaces.Let F bean O�� -moduleandlet E beafinitelocallyfree O�� -module. Thenthereexistsanaturalisomorphism ���� ��∗F ⊗ E ���� ��∗ (F ⊗ ��∗E )

Theorem1.1.3 (Lerayspectralsequence) Let �� : �� → �� and �� : �� → �� bemorphismsofringedspaces.Let F bean O�� -module.Thenthereexistsa spectralsequence

Inpracticeforaspectralsequence ��

,weassumethat �� ��,�� 2 is zerowhenever �� or �� isnegative.Thenthereexistsanexactsequence

Iffurther �� ��,�� 2 = 0 forall �� ≥ 0

Cohomologies Wewrite ���� (F ) forthecohomology ���� ( ��, F ) ofasheaf F ofabeliangroupsonatopologicalspace �� whenthereisnoconfusion.If �� is noetherian,then ���� (F ) vanishesforall �� greaterthanthedimensionof ��.

Let F beacoherentsheafonanalgebraicscheme ��.If �� isaffine,then ���� (F ) = 0 forall �� ≥ 1.If �� : �� → �� isapropermorphism,thenthehigher directimage ���� ��∗F iscoherent[160,IIIthéorème3.2.1].Inparticularif �� is complete,then ���� (F ) isafinitedimensionalvectorspace.Thedimensionof ���� (F ) isdenotedby ℎ�� (F ).Thealternatingsum ��(F ) = �� ∈N (−1)�� ℎ�� (F ) iscalledthe Eulercharacteristic of F .

Let �� beacompleteschemeofdimension ��.Foracoherentsheaf F and aninvertiblesheaf L on ��,the asymptoticRiemann–Rochtheorem definesthe intersectionnumber (L �� · F )∈ Z bytheexpression ��(L ⊗�� ⊗ F ) = (L �� F ) ��! �� �� + �� (�� �� 1), whereby Landau’ssymbol ��, �� (��) = �� (��(��)) meanstheexistenceofaconstant �� suchthat | �� (��)|≤ ��|��(��)| foranylarge ��.Bythis,Grothendieck’s dévissage yieldstheestimate ℎ�� (F ⊗ L ⊗�� ) = �� (�� ��) forall �� [266,sectionVI.2].

If �� isprojectivewithaveryamplesheaf O�� (1),thentheEulercharacteristic ��(F ⊗ O�� (��)) isdescribedasapolynomialin Q[��],calledthe Hilbert

polynomial of F .Thevanishingof ���� (F ⊗ O�� (��)) belowisknownas Serre vanishing.

Theorem1.1.4 (Serre) Let F beacoherentsheafonaprojectivescheme ��.Thenforanysufficientlylargeinteger ��,thetwistedsheaf F ⊗ O�� (��) is generatedbyglobalsectionsandsatisfies ���� (F ⊗ O�� (��)) = 0 forall �� ≥ 1.

Wehavethe cohomologyandbasechangetheorem forflatfamiliesofcoherentsheaves[160,III§§7.6–7.9],[361,section5].Seealso[178,sectionIII.12].

Theorem1.1.5 (Cohomologyandbasechange) Let �� : �� → �� beaproper morphismofalgebraicschemes.Let F beacoherentsheafon �� flatover �� . Taketherestriction F�� of F tothefibre ���� = �� ×�� �� ataclosedpoint �� in �� andconsiderthenaturalmap

, where �� (��) istheskyscrapersheafoftheresiduefieldat ��.

(i) Thedimension ℎ�� (F�� ) isuppersemi-continuouson �� andtheEulercharacteristic ��(F�� ) islocallyconstanton �� .

(ii) Fix �� and �� andsupposethat ���� �� issurjective.Then ���� �� isanisomorphism forall �� inaneighbourhoodat ��.Further, ���� ��∗F islocallyfreeat �� if andonlyif ���� 1 �� issurjective.

(iii) (Grauert) Supposethat �� isreduced.Fix ��.If ℎ�� (F�� ) islocallyconstant, then ���� ��∗F islocallyfreeand ���� �� isanisomorphism.

Divisors Let �� beanalgebraicscheme.Wewrite K�� forthesheafoftotal quotientringsof O�� .If �� isavariety,thenitistheconstantsheafofthe functionfield �� ( ��) of ��.A Cartierdivisor �� on �� isaglobalsectionofthe quotientsheaf K × �� /O × �� ofmultiplicativegroupsofunits.Itisassociatedwith aninvertiblesubsheaf O�� (��) of K�� .If �� isrepresentedbylocalsections ���� ∈ K × ���� with ���� �� 1 �� ∈ O × ���� ∩�� �� ,then O�� (��)|���� = �� 1 �� O���� .Wesaythat �� is principal ifitisdefinedbyaglobalsectionof K × �� orequivalently O�� (��) O�� .Theprincipaldivisorgivenby �� ∈ Γ( ��, K × �� ) isdenotedby ( �� )�� .If ���� belongsto O���� ∩ K × ���� forall ��,then �� definesaclosedsubscheme of �� andwesaythat �� is effective.

The Picardgroup Pic �� of �� isthegroupofisomorphismclassesofinvertible sheaveson ��.Ithasanisomorphism

Pic �� ��1 (O × �� )

InfactthisholdsforanyringedspaceviaČechcohomology.Theproofisfound in[440,section5.4].Theisomorphismforavariety �� isderivedatoncefrom thevanishingof ��1 (K × �� ) fortheflasquesheaf K × ��

By Serre’scriterion,analgebraicscheme �� isnormalifandonlyifitsatisfies theconditions ��1 and ��2 definedas

(���� ) forany �� ∈ ��, O��,�� isregularif O��,�� isofdimensionatmost �� and (���� ) forany �� ∈ ��, O��,�� isCohen–Macaulayif O��,�� isofdepthlessthan ��, inwhichweconsiderscheme-theoreticpoints �� ∈ ��.Let �� beanormalvariety.

Aclosedsubvarietyofcodimensiononein �� iscalleda primedivisor.A Weil divisor �� on ��,orsimplycalleda divisor,isanelementinthefreeabelian group �� 1 ( ��) generatedbyprimedivisorson ��.ACartierdivisoronanormal varietyisaWeildivisor.EveryWeildivisoronasmoothvarietyisCartier.The divisor �� isexpressedasafinitesum �� = �� ���� ���� ofprimedivisors ���� with non-zerointegers ���� .The support of �� istheunionof ���� .Thedivisor �� is effective ifall ���� arepositive,anditis reduced ifall ���� equalone.Wewrite �� ≤ �� if �� �� iseffective.The linearequivalence �� ∼ �� ofdivisors meansthat �� �� isprincipal.

Thedivisor �� isassociatedwithadivisorialsheaf O�� (��) on ��.A divisorial sheafisareflexivesheafofrankone,whereacoherentsheaf F issaidtobe reflexive ifthenaturalmap F → F ∨∨ tothedoubledualisanisomorphism. Thesheaf O�� (��) isthesubsheafof K�� definedby Γ(��, O�� (��)) = { �� ∈ �� ( ��)|( �� )�� + �� |�� ≥ 0}, inwhichzeroiscontainedinthesetontherightbyconvention.The divisor classgroup Cl �� isthequotientofthegroup �� 1 ( ��) ofWeildivisorsdividedby thesubgroupofprincipaldivisors.Itisregardedasthegroupofisomorphism classesofdivisorialsheaveson �� andhasaninjection Pic ��↩→ Cl ��.

Linearsystems Let �� beanormalcompletevariety.Let �� beaWeildivisor on �� andlet �� beavectorsubspaceofglobalsectionsin ��0 (O�� (��)).The projectivespace Λ= P�� ∨ = (�� \ 0)/�� × where �� isthegroundfieldiscalleda linearsystem on ��.Itdefinesarationalmap �� P�� .When �� = ��0 (O�� (��)), wewrite |�� | = P��0 (O�� (��))∨ andcallita complete linearsystem.Bythe inclusion O�� (��)⊂ K�� ,thelinearsystem |�� | isregardedasthesetofeffective divisors �� linearlyequivalentto ��,and Λ isasubsetof |�� |.Thatis, Λ ⊂|�� | = {�� ≥ 0 | �� ∼ �� }

The baselocus of Λ meansthescheme-theoreticintersection �� = �� ∈Λ �� in ��.Wesaythatthelinearsystem Λ is free if �� isempty.Wesaythat Λ is mobile if �� isofcodimensionatleasttwo.Thedivisor �� issaidtobe free (resp.

mobile)if |�� | isfree(resp.mobile).Bydefinition,afreeWeildivisorisCartier. When ∅ ≠Λ ⊂|�� |, Λ isdecomposedas Λ=Λ + �� withamobilelinear system Λ ⊂|�� �� | andthemaximaleffectivedivisor �� suchthat �� ≤ ��1

forall ��1 ∈ Λ.Theconstituents Λ and �� arecalledthe mobilepart andthe fixedpart of Λ respectively.Therationalmapdefinedby Λ isisomorphicto �� P�� .Thelinearsystem Λ ismobileifandonlyif �� iszero.

Evenif �� isnotcomplete,thelinearsystem Λ= P�� ∨ isdefinedforafinite dimensionalvectorsubspace �� of ��0 (O�� (��)).Weconsider |�� | tobethedirect limit lim −−→�� Λ oflinearsystems.

A general pointinavariety �� meansapointinadenseopensubset �� of ��.A verygeneral pointin �� meansapointintheintersection �� ∈N ���� ofcountably manydenseopensubsets ���� .Thusbythegeneralmemberofthelinearsystem Λ,wemeanageneralpointin Λ asaprojectivespace.Bertini’stheoremasserts thatafreelinearsystemonasmoothcomplexvarietyhasasmoothmember. Thestatementforthehyperplanesectionholdseveninpositivecharacteristic.

Theorem1.1.6 (Bertini’stheorem) Let Λ= P�� ∨ beafreelinearsystemona smoothvariety �� andlet �� : �� → P�� betheinducedmorphism.Supposethat �� isaclosedembeddingorthegroundfieldisofcharacteristiczero.Thenthe generalmember �� of Λ isasmoothdivisoron ��,andiftheimage ��( ��) isof dimensionatleasttwo,then �� isasmoothprimedivisor.

Thecanonicaldivisor Itisthecanonicaldivisorthatplaysthemostimportant roleinbirationalgeometry.The sheafofdifferentials onanalgebraicscheme �� isdenotedby �� .When �� issmooth, �� �� denotesthe ��-thexteriorpower �� �� .

Definition1.1.7 The canonicaldivisor ���� onanormalvariety �� isthe divisordefineduptolinearequivalencebytheisomorphism O�� (���� )|�� �� �� onthesmoothlocus �� in ��,where �� isthedimensionof ��.

Example1.1.8 Theprojectivespace P�� hasthecanonicaldivisor ��P�� ∼ −(�� + 1)�� forahyperplane ��.Thisfollowsfromthe Eulersequence 0 → ΩP�� → OP�� (−1) ⊕(��+1) → OP�� → 0.

Onecandescribe ��P�� inanexplicitway.Takehomogeneouscoordinates ��0,...,���� of P��.Let ���� A�� denotethecomplementofthehyperplane ���� definedby ���� .Thechart ��0 admitsanowherevanishing ��-form ����1 ∧···∧ ������ withcoordinates ��1,...,���� for ���� = ���� �� 1 0 .Itisexpressedonthechart ��1 havingcoordinates ��0,��2,...,���� for ���� = ���� �� 1 1 astherational ��-form ���� 1 0 ∧ �� (��2 �� 1 0 )∧···∧ �� (���� �� 1 0 ) = ��−(��+1) 0 ����0 ∧ ����2 ∧···∧ ������,whichhas poleoforder �� + 1 along ��0.Thus ��P�� ∼−(�� + 1)��0.

Inspiteoftheambiguityconcerninglinearequivalence,itisstandardtotreat thecanonicaldivisorasifitwereaspecifieddivisor.

Foraclosedsubscheme �� ofanalgebraicscheme ��,thereexistsanexact sequence I /I 2 → Ω�� ⊗ O�� → Ω�� → 0,where I istheidealsheafin O�� defining ��.Thisinducesthe adjunctionformula,whichconnectsthecanonical divisortothatonaCartierdivisor.

Theorem1.1.9 (Adjunctionformula) Let �� beanormalvarietyandlet �� be areducedCartierdivisoron �� whichisnormal.Then ���� = (���� + ��)|�� in thesensethat O�� (���� ) O�� (���� + ��)⊗ O��

Duality AlbeitGrothendieck’sdualitytheoryworksinthederivedcategoryfor propermorphisms[177],itisextremelyhardtoobtainthedualisingcomplex andatracemapinacompatiblemanner.Thetheorybecomesefficientif itisrestrictedtotheCohen–Macaulayprojectivecaseasexplainedin[178, sectionIII.7]and[277,section5.5].Forexample,thedualisingcomplexona Cohen–Macaulayprojectivescheme �� ofpuredimension �� istheshift ���� [��] ofthedualisingsheaf ���� .

Definition1.1.10 Let �� beacompleteschemeofdimension �� overanalgebraicallyclosedfield ��.The dualisingsheaf ���� for �� isacoherentsheafon �� endowedwitha tracemap �� : �� �� (���� )→ �� suchthatforanycoherentsheaf F on ��,thenaturalpairing

(F ,���� )×

inducesanisomorphism Hom(F ,���� ) �� �� (F )∨ .

Thedualisingsheafisuniqueuptoisomorphismifitexists.Theprojective space P�� hasthedualisingsheaf ��P�� �� ΩP�� .ThiswithLemma1.1.11yields theexistenceof ���� foreveryprojectivescheme �� bytakingafinitemorphism �� → P�� knownasprojective Noethernormalisation.If �� isembeddedinto aprojectivespace �� withcodimension ��,then ���� Ext�� �� (O�� ,���� ) [178,III proposition7.5].If �� isanormalprojectivevariety,then ���� coincideswith thesheaf O�� (���� ) associatedwiththecanonicaldivisor.

Forafinitemorphism �� : �� → �� ofalgebraicschemes,thepush-forward ��∗ definesanequivalenceofcategoriesfromthecategoryofcoherent O�� -modules tothatofcoherent ��∗O�� -modules.Thisassociateseverycoherentsheaf G on �� functoriallywithacoherentsheaf ��!G on �� satisfying ��∗ H om�� (F ,��!G )

H om�� (��∗F , G ) foranycoherentsheaf F on ��.

Lemma1.1.11 Let �� : �� → �� beafinitemorphismofcompleteschemesof thesamedimension.Ifthedualisingsheaf ���� for �� exists,then ���� = ��!���� is thedualisingsheaffor ��

Proof Let �� denotethecommondimensionof �� and �� .Foracoherentsheaf F on ��, Hom�� (F ,��!���� ) = Hom�� (��∗F ,���� ) isdualto �� �� (F ) = �� �� (��∗F ) bythepropertyof ���� ,wherethelatterequalityfollowsfromtheLerayspectral sequence �� �� (���� ��∗F )⇒ �� ��+�� (F )

ThedualityforCohen–Macaulaysheavesonaprojectiveschemeisderived fromthatontheprojectivespaceviaprojectiveNoethernormalisation.See [277,theorem5.71].

Theorem1.1.12 (Serreduality) Let �� beaprojectiveschemeofdimension ��.Let F beaCohen–Macaulaycoherentsheafon �� withsupportofpure dimension ��.Then ���� (H om�� (F ,���� )) isdualto �� �� �� (F ) forall ��

The adjunctionformula ���� ���� ⊗ O�� (��)⊗ O�� holdsforaCohen–Macaulayprojectivescheme �� ofpuredimensionandaneffectiveCartier divisor �� on ��.CompareitwithTheorem1.1.9.

Resolutionofsingularities Aprojectivebirationalmorphismisdescribedasa blow-up.The blow-up ofanalgebraicscheme �� alongacoherentidealsheaf I in O�� ,oralongtheclosedsubschemedefinedby I ,istheprojectivemorphism �� : �� = Proj�� �� ∈N I �� → ��.Thepull-back IO�� = �� 1I · O�� in O�� isan invertibleidealsheaf.Noticethat IO�� isdifferentfrom ��∗I .Theblow-up �� hastheuniversalpropertythateverymorphism �� : �� → �� thatmakes IO�� invertiblefactorsthrough �� as �� = �� ◦ �� foramorphism �� : �� → ��

Let �� : �� �� beabirationalmapofvarieties.The exceptionallocus of �� isthelocusin �� where �� isnotbiregular.Let �� beaclosedsubvarietyof �� notcontainedintheexceptionallocusof �� .The stricttransform ��∗ �� in �� of �� istheclosureoftheimageof �� �� .When �� and �� arenormal,the strict transform ��∗ �� in �� ofanarbitraryprimedivisor �� on �� isdefinedasadivisor insuchamannerthat ��∗ �� iszeroif �� isintheexceptionallocusof �� .Bylinear extension,wedefinethestricttransform ��∗ �� in �� foranydivisor �� on ��.

Resolutionofsingularitiesisafundamentaltoolincomplexbirationalgeometry.Wesaythatareduceddivisor �� onasmoothvariety �� is simplenormal crossing,or snc forshort,if �� isdefinedateverypoint �� in �� bytheproduct ��1 ���� ofapartofaregularsystem ��1,...,���� ofparametersin O��,�� .

Definition1.1.13 A resolution ofavariety �� isaprojectivebirationalmorphism �� : �� → �� fromasmoothvariety.Theresolution �� issaidtobe strong ifitisisomorphiconthesmoothlocusin ��.

Definition1.1.14 Let �� beanormalvariety,let Δ beadivisoron �� andlet I beacoherentidealsheafin O�� .A logresolution of ( ��, Δ, I ) isaresolution �� : �� → �� suchthat

• theexceptionallocus �� of �� isadivisoron �� ,

• thepull-back IO�� isinvertibleandhencedefinesadivisor �� and

• �� + �� + �� 1 ∗ �� hassncsupportforthesupport �� of Δ.

Thelogresolution �� issaidtobe strong ifitisisomorphiconthemaximal locus �� in �� suchthat �� issmooth, I |�� definesadivisor ���� and ���� + ��|�� hassncsupport.A(strong)logresolutionof �� meansthatof ( ��, 0, O�� ),and thoseof ( ��, Δ) and ( ��, I ) arelikewisedefined.

TheexistenceoftheseresolutionsforcomplexvarietiesisduetoHironaka. Theitems(i)and(ii)belowarederivedfromthemaintheoremsIandIIin [187]respectively.

Theorem1.1.15 (Hironaka[187]) (i) Astrongresolutionexistsforevery complexvariety.

(ii) Astronglogresolutionexistsforeverypair ( ��, I ) ofasmoothcomplex variety �� andacoherentidealsheaf I in O��

Hironaka’sconstructionincludestheexistenceofastronglogresolution �� → �� equippedwithaneffectiveexceptionaldivisor �� on �� suchthat O�� (−�� ) isrelativelyample.

Analyticspaces Weshalloccasionallyconsideracomplexschemetobean analyticspaceintheEuclideantopology.Whilstanalgebraicschemeisobtained bygluingaffineschemesin A��,ananalyticspaceisconstructedbygluing analyticmodelsinadomainin C��.Areferenceis[151].Theringofconvergent complexpowerseriesisdenotedby C{��1,...,���� }

Let �� beadomaininthecomplexmanifold C��.Let O�� denotethesheafof holomorphicfunctionson ��.Let I beanidealsheafin O�� generatedbya finitenumberofglobalsections.Thelocally C-ringedspace (��, (O�� /I )|��) forthesupport �� ofthequotientsheaf O�� /I iscalledan analyticmodel,where being C-ringed meanshavingthestructuresheafof C-algebras.An analytic space isalocally C-ringedHausdorffspacesuchthateverypointhasanopen neighbourhoodisomorphictoananalyticmodel.

Everycomplexscheme �� isassociatedwithananalyticspace ��ℎ .This definesafunctor ℎ fromthecategoryofcomplexschemestothecategoryof analyticspaces.Thereexistsanaturalmorphism ��ℎ → �� oflocally C-ringed spaceswhichmaps ��ℎ bijectivelytothesetofclosedpointsin ��.Itpullsback acoherentsheaf F on �� toacoherentsheaf Fℎ on ��ℎ .When �� iscomplete,it

inducesanequivalenceofcategories.Thisisknownasthe GAGAprinciple, whichtakestheacronymfromthetitleofSerre’spaper[414].

Theorem1.1.16 (GAGAprinciple[163,exposéXII],[414]) Let �� bea completecomplexschemeandlet ��ℎ betheanalyticspaceassociatedwith �� Thenthefunctor ℎ inducesanequivalenceofcategoriesfromthecategoryof coherentsheaveson �� tothecategoryofcoherentsheaveson ��ℎ .

Forananalyticspace �� ,theexponentialfunction exp(2��√ 1��) definesa grouphomomorphism O�� → O × �� .Theinducedexactsequence

iscalledthe exponentialsequence

Inprinciple,onecandealwithanalyticspacesanalogouslytocomplex schemesasin[29].Forananalyticspace �� ,the Oka–Cartantheorem assertsthecoherenceofeveryidealsheafin O�� thatdefinesananalyticsubspace of �� .Forapropermap �� : �� → �� ofanalyticspaces,thehigherdirectimage ���� ��∗F ofacoherentsheaf F on �� iscoherenton �� .Inparticular,theimage �� (�� ) isthesupportoftheanalyticsubspaceof �� definedbythekernelofthe map O�� → ��∗O�� ,whichisreferredtoasthe propermappingtheorem.

Thecanonicaldivisoronanormalanalyticspacemaynotbedefinedasa finitesumofprimedivisors.Somenotionssuchasprojectivityofresolutionof singularitiesonlymakesenseonasmallneighbourhoodaboutafixedcompact subsetofananalyticspace.Thesewillposenoobstaclesaswemainlyworkon thegermatapointintheanalyticcategory.

Notation1.1.17 Thesymbol ���� denotesadomaininthecomplexspace C�� whichcontainstheorigin ��.Forexample,wewrite �� ∈ ���� foragermofa complexmanifold.

1.2NumericalGeometry

Theintersectionnumberisabasiclineartoolinbirationalgeometry.Weshall defineitintherelativesettingofapropermorphism �� → ��.Thissectionworks overanalgebraicallyclosedfield �� ofanycharacteristic.

Oneencountersdivisorswithrationalcoefficientsnaturally.Forexamplefor afinitesurjectivemorphism �� → �� ofsmoothvarietiestamelyramifiedalong asmoothprimedivisor �� on �� ,theramificationformulawhichwillbeproved inTheorem2.2.20expresses ���� asthepull-backof ���� +(1 1/��) �� with theramificationindex �� along ��.Onealsohasdivisorswithrealcoefficients takinglimits.Webeginwithformulationofthesenotions.

Let �� beanormalvariety.Let �� 1 ( ��) denotethegroupofWeildivisorson ��.A Q-divisor isanelementintherationalvectorspace �� 1 ( ��)⊗ Q.Inlike manner,an R-divisor isanelementin �� 1 ( ��)⊗ R.An R-divisor �� isexpressed asafinitesum �� = �� ���� ���� ofprimedivisors ���� withrealcoefficients ���� ,and �� isa Q-divisorif ���� arerational.Itis effective if ���� ≥ 0 forall �� and �� ≤ �� meansthat �� �� iseffective.The round-down �� andthe round-up �� are definedas �� = �� ���� ���� and �� = �� ���� ���� .Wesometimessaythata usualdivisoris integral todistinguishitfroma Q-divisorandan R-divisor.

Let ��1 ( ��) denotethesubgroupof �� 1 ( ��) generatedbyCartierdivisorson ��.A Q-Cartier Q-divisorisanelementintherationalvectorspace ��1 ( ��)⊗ Q

Inotherwords,a Q-divisor �� is Q-Cartierifandonlyifthereexistsanon-zero integer �� suchthat ���� isintegralandCartier.Likewisean R-Cartier R-divisor isanelementin ��1 ( ��)⊗ R.An R-Cartier Q-divisorisalways Q-Cartierbuta Q-CartierintegraldivisorisnotnecessarilyCartier.

Example1.2.1 Considertheprimedivisor �� = (��1 = ��2 = 0) onthesurface �� = (��2 1 ��2��3 = 0)⊂ A3 withcoordinates ��1,��2,��3.Then 2�� istheCartier divisordefinedby ��2 andthescheme-theoreticintersection 2�� ∩ �� withtheline �� = (��1 = ��2 = ��3) in �� isoflengthone.Itfollowsthat �� isnotCartier.

Let �� : �� → �� beamorphismofnormalvarieties.The pull-back ��∗ �� ofan R-Cartier R-divisor �� on �� isdefinedasan R-Cartier R-divisoron �� bythe naturalmap ��∗ : ��1 ( ��)⊗ R → ��1 (�� )⊗ R.If �� isa Q-divisor,thensois ��∗ ��.

Definition1.2.2 Let �� beanormalvariety.Wesaythat �� is Q-Gorenstein if thecanonicaldivisor ���� is Q-Cartier.Wesaythat �� is Q-factorial ifalldivisors on �� are Q-Cartier,thatis, Cl ��/Pic �� istorsion.Itissaidtobe factorial ifall divisorsareCartier,thatis, Pic �� = Cl ��

The Q-factorialityisnotananalyticallylocalproperty.

Example1.2.3 Thealgebraicgerm �� ∈ �� = (��1��2 + ��3��4 = 0)⊂ A4 isnot Q-factorial.Theprimedivisor �� = (��1 = ��4 = 0) on �� isnot Q-Cartierand thedivisorclassgroup Cl �� is Z[��] Z.Indeed,theblow-up �� of �� at �� resolvestheprojection �� P3 from �� asamorphism �� → P3 andityields alinebundle �� → �� overthesurface �� = (��1��2 + ��3��4 = 0) P1 × P1 ⊂ P3 . Bythisstructure, Pic �� isgeneratedbythestricttransforms �� �� and �� �� of �� and �� = (��2 = ��4 = 0).Theysatisfytherelation �� �� + �� �� + �� ∼ 0 forthe exceptionaldivisor �� of �� → ��.Thus Cl �� Pic(�� \ ��) = Z[�� �� \ ��].

Ontheotherhand,thealgebraicgerm �� ∈ �� = (��1��2 + ��3��4 + �� = 0)⊂ A4 isfactorialforageneralcubicform �� in ��1,...,��4.Toseethis,wecompactify �� to �� = (��0 (��1��2 + ��3��4)+ �� = 0)⊂ P4.Theblow-up �� of �� at �� resolves

theprojectionfrom �� as �� → P3,andthisistheblow-upof P3 alongthe sexticcurve (��1��2 + ��3��4 = �� = 0).Bythisstructure, Pic �� isgeneratedbythe exceptionaldivisor �� of �� → �� andthestricttransform ���� of �� = �� \ �� .Thus

Cl �� Pic(�� \(�� + ���� )) = 0

Thetwogerms �� ∈ �� and �� ∈ �� becomeisomorphicintheanalyticcategory aswillbeseeninProposition2.3.3.SeeRemark3.1.11forfurtherdiscussion.

Weshallfixthebasescheme �� andwork relatively onapropermorphism �� : �� → �� ofalgebraicschemes,whichisfrequentlydenotedby ��/��.Every terminologyisaccompaniedbythereferencetotherelativesetting.Thereferenceisomittedwhenweconsideracompletescheme �� withthestructure morphism �� → �� = Spec ��.

A relativesubvariety �� of ��/�� meansaclosedsubvarietyof �� suchthat �� (��) isapointin ��.A relative ��-cycle on ��/�� isanelementinthefreeabelian group ���� ( ��/��) generatedbyrelativesubvarietiesofdimension �� in ��/��.For invertiblesheaves L1,..., L�� andarelative ��-cycle �� on ��,the intersection number (L1 ··· L�� · ��) isdefinedbythemultilinearmap (Pic ��) ⊕�� × ���� ( ��/��)→ Z suchthat (L �� ��) forarelativesubvariety �� coincideswith (L �� O�� ) in theasymptoticRiemann–Rochtheorem ��(L ⊗�� ⊗ O�� ) = (L �� O�� )�� ��/��! + �� (�� �� 1).The intersectionnumber (��1 �� �� ��) withCartierdivisors ���� on �� isdefinedas (O�� (��1)··· O�� (�� ��)· ��).If ���� areeffectiveandintersect properlyonarelativesubvariety ��,then (��1 �� �� ��) equalsthelengthofthe structuresheaf O�� oftheartinianscheme �� = ��1 ∩···∩ �� �� ∩ ��.Thelengthof O��,�� for �� ∈ �� isreferredtoasthe localintersectionnumber at �� anddenoted by (��1 ··· �� �� · ��)�� .When �� isacompletevarietyofdimension �� withthe structuremorphism �� → �� = Spec ��,wewrite (L1 ··· L��) = (L1 ··· L�� · ��) and ��1 ··· �� �� = (��1 ··· �� ��)�� = (��1 ··· �� �� · ��)

Bytheextension (Pic �� ⊗ R)×(��1 ( ��/��)⊗ R)→ R,the relativenumerical equivalence ≡�� isdefinedinboththerealvectorspaces Pic �� ⊗R and ��1 ( ��/��)⊗ R insuchawaythatitinducesaperfectpairing �� 1 ( ��/��)× ��

( ��/��)→ R ofvectorspacesonthequotients �� 1 ( ��/��) = (Pic �� ⊗ R)/≡�� and ��1 ( ��/��) = (��1 ( ��/��)⊗ R)/≡�� .When �� = Spec ��,wejustwrite ≡, �� 1 ( ��) and ��1 ( ��) withoutreferenceto �� asremarkedabove.

Definition1.2.4 Thespaces �� 1 ( ��/��) and ��1 ( ��/��) arefinitedimensional [254,IV§4,proposition3].Theequaldimensionof �� 1 ( ��/��) and ��1 ( ��/��) iscalledthe relativePicardnumber of ��/�� anddenotedby �� ( ��/��).When

�� = Spec ��,thisnumberiscalledthe Picardnumber ofthecompletescheme �� anddenotedby �� ( ��).

Let �� : �� → �� beapropermorphism.Itinducesthe pull-back ��∗ :Pic �� → Pic �� andthe push-forward ��∗ : ���� (�� /��)→ ���� ( ��/��) asgrouphomomorphisms.Thepush-forward ��∗ �� ofarelativesubvariety �� of �� /�� is ����(��) if themorphism �� → ��(��) isgenericallyfiniteofdegree ��,and ��∗ �� iszeroif ��(��) isofdimensionlessthanthatof ��.Thesesatisfythe projectionformula (�� ∗L1 �� ∗L�� ��) = (L1 L�� ��∗ ��) forinvertiblesheaves L�� on �� andarelative ��-cycle �� on �� .Theyyield ��∗ : �� 1 ( ��/��)→ ��

(�� /��) anddually ��

(�� /��)→ ��1 ( ��/��).Onealso hasthenaturalsurjection �� 1 (�� /��) �� 1 (�� /��) andinjection ��1 (�� /��) ↩→ ��1 (�� /��).If �� issurjective,then ��∗ : �� 1 ( ��/��)→ �� 1 (�� /��) isinjectiveand ��∗ : ��1 (�� /��)→ ��1 ( ��/��) issurjective. Henceforthwefixapropermorphism �� : �� → �� fromanormalvarietyto avarietyandmakebasicdefinitionsforan R-Cartier R-divisor �� on ��.We saythatintegraldivisors �� and �� on �� are relativelylinearlyequivalent and write �� ∼�� �� ifthedifference �� �� islinearlyequivalenttothepull-back ��∗ �� ofsomeCartierdivisor �� on ��.Namely, �� �� iszerointhequotient Cl ��/��∗ Pic ��.For R-divisors �� and �� on ��,the relative R-linearequivalence �� ∼R,�� �� meansthat �� �� iszeroin (Cl ��/��∗ Pic ��)⊗ R.When �� and �� are Q-divisors,thisisreferredtoasthe relative Q-linearequivalence and denotedby �� ∼Q,�� �� .Thespace Pic �� ⊗ R isregardedasthatof R-linear equivalenceclassesof R-Cartier R-divisorson ��.Theintersectionnumber (�� · ��) isdefinedforapairofan R-Cartier R-divisor �� on �� andarelative one-cycle �� on ��/��.Thismakesthenotionof relativenumericalequivalence �� ≡�� �� for R-Cartier R-divisors �� and �� on ��.

Definition1.2.5 An R-Cartier R-divisor �� on ��/�� issaidtobe relativelynef (or nef over �� or ��-nef )if (�� · ��)≥ 0 foranyrelativecurve �� in ��/��.When �� = Spec ��,wejustsaythat �� is nef asusual.

ACartierdivisor �� on �� issaidtobe relativelyample (��-ample)if O�� (��) is arelativelyampleinvertiblesheaf.Itissaidtobe relativelyveryample (��-very ample)if O�� (��) isarelativelyveryampleinvertiblesheaf.Inspiteofthe geometricdefinition,theamplenessischaracterisednumerically.

Theorem1.2.6 (Nakai’scriterion) Let �� : �� → �� beapropermorphismof algebraicschemes.Aninvertiblesheaf L on �� isrelativelyampleifandonly if (L dim �� ��) > 0 foranyrelativesubvariety �� of ��/��.

Kleiman’samplenesscriterion rephrasesNakai’scriterionintermsofthe conesofdivisorsandcurves.A convexcone ��,orsimplycalleda cone,ina finitedimensionalrealvectorspace �� isasubsetof �� suchthatif ��,�� ∈ �� and �� ∈ R>0,then �� + �� ∈ �� and ���� ∈ ��

Definition1.2.7 The amplecone ��( ��/��) istheconvexconein �� 1 ( ��/��) spannedbytheclassesofrelativelyampleCartierdivisorson ��.The closed cone NE( ��/��) ofcurves istheclosureoftheconvexconein ��1 ( ��/��) spanned bytheclassesofrelativecurvesin ��/��

Noticethat ��( ��/��) isanopenconesinceforarelativelyampledivisor �� andaCartierdivisor ��,thesum �� + ���� isrelativelyampleforlarge ��

Theorem1.2.8 (Kleiman’samplenesscriterion[254]) Let �� : �� → �� be apropermorphismofalgebraicschemes.ThenaCartierdivisor �� on �� isrelativelyampleifandonlyiftheclassof �� belongstotheamplecone ��( ��/��).If �� isprojective,then ��( ��/��) and NE( ��/��) aredualwithrespect totheintersectionpairing �� 1 ( ��/��)× ��1 ( ��/��)→ R inthesensethat

) >

�� ∈ NE( ��/��)\ 0}, NE( ��/��)\ 0 = {�� ∈ ��1 ( ��/��)|(��,��) > 0 forall �� ∈ ��( ��/��)}.

Thetheoremshowsthatif �� isprojective,thentheclosureoftheample cone ��( ��/��) coincideswiththe nefcone Nef ( ��/��) in �� 1 ( ��/��) spannedby relativelynef R-Cartier R-divisors.Thedualityof ��( ��/��) and NE( ��/��) still holdsfora Q-factorialcompletevariety ��/�� = Spec �� asstudiedin[254],but itfailsforapropermorphismingeneral.

Example1.2.9 Fujino[128]constructedanexampleofanon-projective completetoricthreefold �� with �� ( ��) = 1 suchthat NE( ��) isahalf-line R≥0.Thebook[140]byFultonisastandardintroductiontotoricvarieties. Let ��1 = (1, 0, 1), ��2 = (0, 1, 1), ��3 = (−1, 1, 1) and ��1 = (1, 0, 1), ��2 = (0, 1, 1), ��3 = (−1, 1, 1) in �� = Z3.Takethefan Δ whichconsistsoffaces ofthecones ��1,��2,��3 , ��1,��2,��3 , ��1,��2,��1 , ��2,��1,��2 , ��2,��3,��2,��3 , ��3,��1,��3,��1 .Thetoricvariety �� associatedwith (��, Δ) istheexample.

Thenumericalnatureextendsthenotionofamplenessto R-divisors.

Definition1.2.10 An R-Cartier R-divisor �� on ��/�� issaidtobe relatively ample (��-ample)iftheclassof �� belongstotheamplecone ��( ��/��).Inother words, �� isexpressedasafinitesum �� = �� ���� ���� ofrelativelyampleCartier divisors ���� with ���� ∈ R>0.

Wekeep �� : �� → �� beingapropermorphismfromanormalvarietytoa variety.ForaCartierdivisor �� on ��,thenaturalmap ��∗��∗O�� (��)→ O�� (��) definesarationalmap �� Proj�� ����∗O�� (��)

over �� forthesymmetric O�� -algebra ����∗O�� (��) of ��∗O�� (��).The relative baselocus of �� istheclosedsubscheme �� of �� givenbytheidealsheaf I�� in O�� suchthattheabovemapinducesthesurjection ��∗��∗O�� (��) I�� O�� (��). Wesaythat �� is relativelyfree (��-free)if �� isempty.Wesaythat �� is relatively mobile (��-mobile)if �� isofcodimensionatleasttwo.Thedefinitionscoincide withthoseonanormalcompletevariety.Unless �� = ��,thereexistsamaximal effectivedivisor �� suchthat I�� ⊂ O�� (−��).Thedivisors �� �� and �� are calledthe relativemobilepart (��-mobilepart)andthe relativefixedpart (��fixedpart)of �� respectively.

Definition1.2.11 ACartierdivisor �� on ��/�� issaidtobe relativelysemiample (��-semi-ample)if ���� isrelativelyfreeforsomepositiveinteger ��.An R-Cartier R-divisor �� on �� issaidtobe relativelysemi-ample (��-semi-ample) ifitisexpressedasafinitesum �� = �� ���� ���� ofrelativelysemi-ampleCartier divisors ���� with ���� ∈ R≥0.Thedefinitionisconsistentbythenextlemma.

Lemma1.2.12 Let �� : �� → �� beapropermorphismfromanormalvarietyto avariety.Let �� and ��1,...,���� beCartierdivisorson �� suchthat �� = �� ���� ���� with ���� ∈ R≥0.Ifall ���� arerelativelyfree,then ���� isrelativelyfreeforsome positiveinteger ��.

Proof Let �� 1 ( ��)Q denotetherationalvectorspaceof Q-divisorson ��.Let �� bethevectorsubspaceof �� 1 ( ��)Q spannedby ��1,...,����.Then �� belongs to (�� ⊗Q R)∩ �� 1 ( ��)Q = �� andhencewemayassumethat ���� ∈ Q andfurther ���� ∈ Z bymultiplying ��.Theassertioninthiscasefollowsfromtheexistence ofthenaturalmap �� (��∗��∗O�� ( ���� )) ⊕���� → ��

O�� (��). Weprovideanalternativecharacterisationofsemi-ampleness.

Lemma1.2.13 Let �� : �� → �� beapropermorphismfromanormalvariety toavariety.An R-divisor �� on �� isrelativelysemi-ampleifandonlyifthere existsaprojectivemorphism ���� : �� → �� fromanormalvarietythroughwhich �� factorsas �� = ���� ◦ �� for �� : �� → �� suchthat �� ∼R ��∗ �� byarelatively ample R-divisor �� on �� /��.

Proof Theifpartisobvious.Weshallprovetheonly-ifpartforarelatively semi-ample R-divisor ��.Write �� asafinitesum �� = �� ���� ���� ofrelativelyfree

divisors ���� with ���� ∈ R>0.Themorphism ���� : �� → ���� = Proj�� ����∗O�� (���� ) providesarelation ���� ∼ ��∗ �� ���� byarelativelyampledivisor ���� on ���� /��.

Let �� : �� → �� betheSteinfactorisationof �� → Proj�� ����∗O�� ( �� ���� ).A relativecurve �� in ��/�� iscontractedtoapointby �� ifandonlyif ( �� ���� ��) = 0. Thisisequivalentto (���� ��) = 0 forall �� since ���� arerelativelynef.By

Lemma1.1.1,every ���� factorsthrough �� as ���� = ���� ◦�� for ���� : �� → ���� and �� ∼R ��∗ �� ���� ��∗ �� ���� .Then �� iscontractedby �� ifandonlyif (��∗ ( �� ���� ��∗ �� ���� )·��) = 0 Thisshowstherelativeamplenessof �� ���� ��∗ �� ���� on �� /��

Definition1.2.14 ACartierdivisor �� on ��/�� issaidtobe relativelybig (��-big)ifthereexistsapositiveinteger �� suchthattherationalmap �� Proj�� ����∗O�� (����) isbirationaltotheimage.

Assumingthat �� isprojective, Kodaira’slemma characterisesthebigness numerically.

Theorem1.2.15 (Kodaira’slemma) Let �� : �� → �� beaprojectivemorphism fromanormalvarietytoaquasi-projectivevariety.ACartierdivisor �� on �� isrelativelybigifandonlyifthereexistarelativelyample Q-divisor �� andan effective Q-divisor �� suchthat �� = �� + ��

Proof Theifpartisobvious.Weshallprovetheonly-ifpartforarelatively bigdivisor ��.BySteinfactorisation,wemayassumethat �� hasconnected fibres.Multiplying ��,wemayassumethat �� Proj�� ����∗O�� (��) isbirational totheimage �� .Wewrite ���� : �� → ��.Takeanopensubset �� of �� suchthat �� : �� → �� isamorphismandsuchthatthecomplement �� \�� isofcodimension atleasttwo.Then O�� (��)|�� ��∗O�� (1) and

(��)⊂

forany �� ∈ Z.Hencethereexistsapositiverationalconstant �� suchthatthe rankofthe O�� -module ��∗O�� (����) isgreaterthan ���� �� forsufficientlylarge ��, where �� = dim �� dim �� isthedimensionofthegeneralfibreof ���� . Takeageneralveryampleeffectivedivisor �� on ��.Sincetherankof ��∗O�� (���� | �� ) isestimatedas �� (�� �� 1),theexactsequence 0 → ��∗O�� (���� ��)→ ��∗O�� (����)→ ��∗O�� (���� | �� ) yieldsthenon-vanishing ��∗O�� (���� ��) ≠ 0 forlarge ��.Hence ��0 (O�� (���� �� + ��∗��)) = ��0 (��∗O�� (���� ��)⊗ O�� (��)) ≠ 0 byasufficientlyveryample divisor �� on ��.Thusonecanwrite ���� �� + ��∗�� = ��1 +( �� )�� withan effectivedivisor ��1 andaprincipaldivisor ( �� )�� on ��.Then �� = �� + �� with �� = �� 1 (�� ��∗�� +( �� )�� ) and �� = �� 1 ��1.

Remark1.2.16 Withoutthequasi-projectivityofthebasevariety,Kodaira’s lemmagivesthedecomposition �� = �� + �� intoarelativelyample Q-divisor �� anda Q-divisor �� suchthat ��∗O�� (����) ≠ 0 forsomepositiveinteger �� Insomeliterature,aCartierdivisor �� on ��/�� with ��∗O�� (��) ≠ 0 issaid tobe relativelyeffective.Providedthat �� isquasi-projective,thismeansthat �� isrelativelylinearlyequivalenttoaneffectivedivisor.Wedonotusethis terminologyforthereasonthatarelativelyeffectivedivisorover �� = Spec C is notnecessarilyeffectivebuteffectiveuptolinearequivalence.

Bydefinition,aCartierdivisor �� on �� isrelativelybigifandonlyifsoisthe restriction �� | �� 1 (�� ) oversomeopensubset �� of �� containingthegenericpoint of �� ( ��).ThusKodaira’slemmawithKleiman’scriterionimpliesthatbigness onaprojectivemorphismisanumericalcondition.Thisprovidesgroundsfor consideringtheconeofbigdivisors.

Definition1.2.17 Assumethat �� : �� → �� isprojective.The bigcone ��( ��/��) istheconvexconein �� 1 ( ��/��) spannedbytheclassesofrelativelybigCartier divisorson ��.Thisisanopenconecontainingtheamplecone ��( ��/��).An R-Cartier R-divisor �� on �� issaidtobe relativelybig (��-big)iftheclassof �� belongstothebigcone ��( ��/��).Namely, �� isexpressedasafinitesum �� = �� ���� ���� ofrelativelybigCartierdivisors ���� with ���� ∈ R>0.

OnecanformulateKodaira’slemmafor R-divisors.

Corollary1.2.18 Let �� : �� → �� beaprojectivemorphismfromanormal varietytoaquasi-projectivevariety.An R-Cartier R-divisor �� on �� isrelatively bigifandonlyifthereexistarelativelyample Q-divisor �� andaneffective R-divisor �� suchthat �� = �� + ��

Finallyweintroducethenotionofnumericallimitofeffective R-divisors.

Definition1.2.19 The pseudo-effectivecone ��( ��/��) istheclosureofthe convexconein �� 1 ( ��/��) spannedbytheclassesofCartierdivisors �� on �� with ��∗O�� (��) ≠ 0.An R-Cartier R-divisor �� on �� issaidtobe relatively pseudo-effective (��-pseudo-effective)iftheclassof �� belongstothepseudoeffectivecone ��( ��/��).

ByKodaira’slemma,if �� isprojective,thenthepseudo-effectivecone ��( ��/��) coincideswiththeclosureofthebigcone ��( ��/��).If �� isquasiprojective,thenarelativelypseudo-effective R-divisoron �� isrealisedinthe space �� 1 ( ��/��) asalimitofasequenceofeffective R-divisors.

Example1.2.20 Let �� beasmoothprojectivecurveofgenus ��.The Jacobian �� (��) of �� representsthesubgroup Pic0 �� of Pic �� whichconsistsofinvertible

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