ABELIANVARIETIESASSOCIATEDTOCLIFFORDALGEBRASByCaseyMachenADISSERTATIONSubmittedtoMichiganStateUniversityinpartialentoftherequirementsforthedegreeofMathematics|DoctorofPhilosophy2016ABSTRACTABELIANVARIETIESASSOCIATEDTOCLIFFORDALGEBRASByCaseyMachenTheKuga-SatakeconstructionisaconstructioninalgebraicgeometrywhichassociatesanabelianvarietytoapolarizedK3-surfaceX.Thisabelianvariety,A,iscreatedfromthealgebraarisingfromthequadraticspaceH2(X;Z)=torsionwithitsnaturalco-homologypairing.Furthermore,thereisaninclusionofHodgestructuresH2(X;Q),!H1(A;Q)H1(A;Q)relatingthecohomologyoftheoriginalK3-surfacewiththatoftheabelianvariety.Weinvestigatewhenthisconstructioncanbegeneralizedtobotharbitraryquadraticformsaswellashigherdegreeforms.Sp,weassociateanabelianvari-etytothealgebraofanarbitraryquadraticforminawaywhichgeneralizestheKuga-Satakeconstruction.Whenthequadraticformarisesastheintersectionpairingonthemiddle-dimensionalcohomologyofanalgebraicvarietyY,weinvestigatewhentheco-homologyoftheabelianvarietycanberelatedtothatofY.Additionally,weexplorewhenfamiliesofalgebraicvarietiesgiverisetofamiliesofabelianvarietiesviathisconstruction.Weusethesetechniquestobuildananalogousmethodforconstructinganabelianvarietyfromthegeneralizedalgebraofahigherdegreeform.Wecertainfamiliesofcomplexprojective3-foldsand4-foldsforwhichanabelianvarietycanbeconstructedfromtherespectivecubicandquarticformsonH2.Therelationsbetweenthecohomologyoftheabelianvarietyandtheoriginalvarietyarealsodiscussed.Tomyfamily.iiiACKNOWLEDGMENTSIhavereceivedatremendousamountofsupportduringmytimeatMichiganStateUniversity.Byfar,themostsupporthascomefrommymom,dad,andPT.Withoutyoualwaysbeingthereforme,IwouldnotbewhereIamtoday.Icannotthankyouenough.IamtrulygratefulformyadvisorProfessorRajeshKulkarni.Heisaninspirationalteacherandgreatmotivator.Isincerelyappreciatedhishumorandnon-math-relatedcon-versations.Duringmytwoyears,IwasabletodedicatemoretimetoresearchbecauseIwaspartiallyfundedbyhisgrants.Myfriends,bothinandoutofthemathdepartment,playedakeyrolethroughoutthisprocess.Thoseoutsideofthedepartmenthelpedkeepmesane.Inthedepartment,Iespeciallywanttothankaspecialgroupofgoodfriendsandcollaborators:AdamChapman,AkosNagy,SebastianTroncoso,andCharlotteUre.Additionally,IreceivedagreatdealofsupportfromMichiganStateUniversity,whichhashelpedmeincompletingthisproject:CollegeofNaturalScienceFellowship,DresselScholarship,andDissertationCompletionFellowship.Thosenotmentionedherearenotforgotten.Iamthankfulforeveryoneelsewhohashelpedmealongtheway.ivTABLEOFCONTENTSChapter1Introduction...............................1Chapter2BackgroundMaterial..........................62.1HodgeStructures.................................62.2AbelianVarieties.................................102.3Algebras.................................132.4TheKuga-SatakeConstruction.........................14Chapter3AbelianVarietyfromtheAlgebra............183.1RepresentationsofAlgebras.......................183.2ConstructionoftheAbelianVariety.......................203.3MapsBetweenAbelianVarieties.........................273.4ExamplesandApplications...........................303.4.1Examples.................................303.4.2RelationtoKuga-SatakeConstruction.................31Chapter4FamiliesofAbelianVarieties.....................354.1VariationsofHodgeStructure..........................364.2FamiliesofAbelianVarieties...........................394.2.1Examples.................................43Chapter5AbelianVarietiesAssociatedtoHigherDegreeForms.....485.1GeneralizedAlgebras..........................485.1.1GradedRepresentations.........................505.2AbelianVarieties.................................555.3Lattices......................................585.4Applications....................................595.4.1Example..................................615.4.2Example..................................615.4.3Example..................................625.5ProductsofSurfaces...............................635.6ConstructioninFamilies.............................69BIBLIOGRAPHY....................................71vChapter1IntroductionTheconstructionoftheJacobianvarietyassociatedtoacomplexprojectivecurveisclassicalinalgebraicgeometry.GeometricquestionsaboutprojectivealgebraiccurvescanoftenbetranslatedtoJacobianvarieties.Thisisattractivebecausevarioustoolsdevelopedforabelianvarietiescanbeusedinaddressingthesequestions.Infact,Torelli'sTheoremtellsusthattwocurvesareisomorphicifandonlyiftheirJacobianvarietiesareisomorphic.RephrasedintermsofHodgestructures,wehavethattwocurvesC1andC2areisomorphicifandonlyifthereisanisomorphismofintegralHodgestructuresH1(C1;Z)˘=H1(C2;Z)whichrespectstheintersectionpairing.Forhigherdimensionalalgebraicvarieties,itisthusdesirabletohavenaturallyassociatedabelianvarietieswhichretainsomegeometricinformationoftheinitialvariety.TheAlbanesevarietyassociatedtoasurfaceissuchanexample.However,veryoftentheAlbanesevarietyisapoint,andnottoouseful.OtherconstructionsofthisvoraretheintermediateJacobianforcubic3-folds[CG72],andtheKuga-SatakevarietyassociatedtoaK3surface[KS67].Inthispaper,westudytheextensionsoftheKuga-Satakeconstructiontoarbitraryquadraticformsandhigherdegreeforms.LetVbeanintegral,weight2polarizedHodgestructurewithdimV2;0=1.TheKuga-SatakeconstructionassociatesanabelianvarietyA(whichwecalltheKuga-Satakevariety)tosuchaV.Furthermore,thereisaninclusionofweight2HodgestructuresVQ,!H1(A;Q)H1(A;Q):()1ThiswasappliedtoV=H2(X;Z)=torsionforapolarizedK3surfaceX.However,itworksingeneralforcomplexprojectivesurfaceswithh2;0=1.Weexplainthisconstruction,seeChapter2formoredetail.RecallthatthereisaalgebraassociatedtotheintersectionformqonV=H2(X;Z)=torsion.TheevenalgebraC+q(VR)overRcanbegivenacomplexstructure,andthishasafulllatticegivenbytheinclusionoftheevenalgebraC+q(V)overZintoC+q(VR).Thusthequotientisacomplextorus.Thiscomplextorusadmitsaveryamplelinebundle,andhenceanembeddingintoprojectivespace,makingitanabelianvariety.ThisconstructionreliesheavilyuponthefactthatdimV2;0=1(see[KS67]or[vG00,Proposition5.9]foramoderntreatment).TheKuga-SatakeconstructionhashelpedagreatdealinstudyingquestionsrelatedtoK3surfaces.DeligneusedthisconstructiontoprovetheWeilconjecturesforK3surfaces[Del72].Furthermore,thereisaTorelli-typetheoremforK3surfacesduetoPyatetskii-ShapiroandShafarevich[IIPS71].ItsaysthattwopolarizedK3surfacesX1andX2areisomorphicifandonlyifthereisanisomorphismofHodgestructuresH2(X1;Z)˘=H2(X2;Z)whichiscompatiblewiththeintersectionpairing.Inotherwords,theKuga-SatakevarietycompletelydeterminestheK3surface.ThisconstructionhasbeengeneralizedbeyondtheK3-surfacecase.In[Mor85],MorrisonnotesthatthemiddledimensionalcohomologyofabeliansurfacesthehypothesisoftheKuga-Satakeconstruction,andcomputestheabelianvarietyassociatedtoanabeliansurface.Voison[Voi05]providesanalternativeviewpointoftheKuga-SatakeconstructionbynotingthattheKuga-Satakevarietycanbeobtainedfromtheweight2HodgestructureontheexterioralgebraVV.ThebasicingredientoftheKuga-SatakeconstructionisaquadraticformonafreeZ-2moduleofrankV.WhenVisasintheKuga-Satakeconstruction,thesignatureofqis(dimV2;2).Notingthis,weconsiderthefollowingquestions:canwecreateanabelianvarietyAfromthealgebraofanarbitraryquadraticforminameaningfulway?Ifso,canwerecovertheKuga-Satakevarietywhenqhassignature(dimV2;2)?Finally,arethereotherpolarizedHodgestructuresonVforwhichwecanconstructanabelianvarietywithaninclusionofHodgestructuresoftheform()above?Chapter3providesaneanswertothesequestions.Namely,inTheorem3.2.8weshowthatforaquadraticformintwoormorevariableswithsignatureotherthan(1;1),wecancreateanabelianvarietyfromthealgebra.Furthermore,Proposition3.4.4discussesaninclusionofHodgestructuresasin().Finally,Proposition3.4.3showsthatwedoinfactrecovertheKuga-SatakevarietywhenthequadraticformisasintheKuga-Satakeconstruction.InChapter4,weshowthatourconstructionworksinfamilies.Webeginwithacontin-uouslyvaryingfamilyofpolarizedHodgestructures(variationofHodgestructure),insteadofjustasinglepolarizedHodgestructure.InTheorem4.2.2weprove,undersomehy-potheses,thatthecorrespondingabelianvarietiesobtainedfromourconstructionalsovarycontinuouslyinafamily.Theorem4.2.8providesasituationinwhichthehypothesesareandweconcludewithgeometricexamples,includingaproofthattheKuga-Satakeconstructionworksinfamilies.Theremainderofthepaperappliesourtechniquestohigherdegreeforms.TheLefshetzHyperplaneTheoremsuggeststhatforacomplexprojectivevarietyofdimensionn,themostinterestingcohomologyresidesinHn.ForK3surfacesthisiscertainlythecase,asonlyH2containsnontrivialinformation.Ontheotherhand,intermediateJacobianscanbeconstructedfromanyHkforkodd,sointerestinginformationcanbefoundoutside3themiddle-dimensionalcohomologygroups.OurideaistoconsiderhigherdegreeformsassociatedtoavarietyfromtheintersectionformonH2.Forexample,consideracomplex,projective3-foldY.Thenthe(trilinear)intersectionformH2(Y;Q)H2(Y;Q)H2(Y;Q)!QyieldsacubicformonthevectorspaceH2(Y;Q).Ingeneral,theintersectionformonH2ofad-dimensionalcomplexvarietyyieldsadegreedform.Thequestionweaskis:canweconstructanabelianvarietyfromadegreedforminamannersimilartothatoftheKuga-Satakeconstruction?AndcanwesayanythingmeaningfulinregardstotheHodgestructure?Thereisanotionofa\generalized"algebraCfforanarbitrarydegreedformfwhichhasbeenstudiedbyvariousauthorsincludingRoby[Rob69],Revoy[Rev77],andChilds[Chi78].ItistheZ=dZ-gradedassociativealgebraCf(K)=Khx1;:::xni=(a1x1++anxn)df(a1;:::;an):ai2KAsimilarscenariototheKuga-SatakeconstructionwouldbetosaythatC0f(Z)isafulllatticeinC0f(R),putacomplexstructureonC0f(R)sothatthequotientisacomplextorus,andshowthatthecomplextorusisinfactanabelianvariety.Unfortunately,whend3andtheformhastwoormorevariables,thegeneralizedalgebraisdimensional(thisiswell-known,seeforexample[BHS88,Theorem1.8]).Thisideaneedsmosincewedon'twantandimensionaltorus.Toremedythis,wewilltakeadimensional,gradedrepresentationofCf(C)intoMN(C)forsomeN,andlookattheinducedrepresentationsonCf(R)andCf(Z).Wedenotetheimagesbycf(R)andcf(Z),respectively.Sincewehavechosenagradedrepresentation,c0f(R)andc0f(Z)willplaytherolesofthevectorspaceandlattice,respectively.4TheideaofconsideringrepresentationsofCfisnotnew.VandenBergh[VdB87]showsthatthereisaone-to-onecorrespondencebetweendimensionalrepresentationsofCfandcertainvectorbundles(calledUlrichbundles)onthehypersurfaceXf=Z(wdf).Thereisanautomorphism˙ofXfbyw7!wforaprimitivedthrootofunity.UnderVandenBergh'scorrespondence,gradedrepresentationscorrespondtovectorbundleswhichareinvariantundertheautomorphism˙ofXf.RepresentationsofCfandtheirpropertiesarealsodiscussedin[HT88,CKM12].Chapter5discussesbackgroundmaterialandexistenceofgradedrepresentationsofCf.Theinitialquestioniswhetherornotc0f(R)canbegivenacomplexstructureforwhichthequotientc0f(R)=c0f(Z)isanabelianvariety.Section5.2putsconditionsontherepresentationwhichallowfortheconstructionofanabelianvariety.Underadimensional,gradedrepresentationofthealgebra,thereisnoguaranteethatc0f(Z)isafulllatticeinsideofc0f(R).WediscussthisinSection5.3.Thisisthereasonwerestricttocubicandquarticforms.Section5.4providesseveralexamplesof3-foldsand4-foldsforwhichanabelianvarietycanbeconstructedfromtheintersectionformonH2.Section5.5specializesto4-foldswithareproductsofsurfaces.WediscusshowtheabelianvarietyassociatedtoH2ofasurfacerelatestotheabelianvarietyassociatedtoH2ofthe4-fold.Inparticular,weshowthattheKuga-SatakevarietyofaK3surfaceisasubquotientoftheabelianvarietyassociatedtotheproductofK3surfaces.Weconcludewithadiscussionofwhenthisconstructionworksinfamilies.5Chapter2BackgroundMaterialThischaptercontainsthebackgroundinformationwhichwillbeusedthroughoutpaper.WediscussHodgestructuresandthewayinwhichtheyariseingeometry.Wethentreatabelianvarietiesandalgebras.ThechapterconcludeswiththeKuga-Satakeconstruction.2.1HodgeStructures2.1.1.LetVbeafreeZ-moduleofrank.A(pure)HodgestructureofweightkonVisadecompositionVC=Lp+q=kVp;qofVC:=VZCintosubspacesVp;qwithVp;q=Vq;p,wherevz:=vz.Givensuchadecomposition,wetheHodgeltrationVC˙Fp˙Fp+1˙˙0byFp=LrpVr;rp.NotethatVp;q=Fp\Fkp+1,sothatequivalentdataisobtainedfromeitheraHodgestructureoraHodgeation.InthecasewhereVisadimensionalQ-vectorspace,theissimilar.Todistinguishbetweenthetwocases,wewillsayintegralHodgestructurewhenVisafreeZ-moduleofrank,andrationalHodgestructurewhenVisaQ-vectorspace.WesayaHodgestructureisoftypeT=f(p1;q1);:::;(pn;qn)gifVr;s=0unless(r;s)2T.When6wedon'tspecifythetypeTofaHodgestructure,weassumethatthepi;qj0.Thisassumptionisduetothefollowingexample.Example2.1.2.LetXbeasmooth,complexprojectivevariety.ThusXcanbeconsideredasanonsingularcomplexmanifold.ThesingularcohomologygroupVQ=Hk(X;Q)hasanaturalHodgestructureofweightkwithVp;q=Hq(X;p)wherepbethesheafofentialp-forms.ThisistheexampleofaHodgestructurethatwewillhaveinmindthroughoutthepaper.Similarly,VZ=Hk(X;Z)=torsionhasanintegralHodgestructureofweightk.WenowlistseveralimportantnotionswhichpertaintoHodgestructures.(i)TheTatestructureZ(k)istobethefreeZ-moduleV=(2ˇi)kZˆCwithVk;k=VC.ThisisaHodgestructureoftypef(k;k)gandweight2k.WeQ(k)analogously,byreplacingZwithQ.Forourpurposes,wewillnotneedthefactorof(2ˇi)kintheofZ(k),sowewilldropittosimplifynotation.(ii)IfVhasanintegralHodgestructureofweightk,weaweightkintegralHodgestructureonV_=HomZ(V;Z)bysetting(V_)p;q=HomC(Vp;q;C).(iii)IfVandWhaveHodgestructuresofweightnandm,respectively.ThenVWhasaHodgestructureofweightn+mgivenby(VW)p;q=(Vp1;q1Wp2;q2)wherep1+p2=pandq1+q2=q:SinceHom(V;W)˘=V_W,weseethatHom(V;W)hasanaturalHodgestructureofweightmn.WewriteV(k)forVZ(k),whichhasweightn2kforanyk.Inparticular,V(0)=V.7(iv)WeamorphismofHodgestructurestobealinearmapf:V!WsuchthattheC-linearextensionfC(Vp;q)ˆWp;q.Suchamorphismisnecessarily0,unlessVandWhavethesameweight.IfVhasweightn,Whasweightn+2k,andwehavealinearmapf:V!WsatisfyingfC(Vp;q)ˆWp+k;q+k,thenweobtainamorphismofHodgestructuresV!W(k)givenbyv7!f(v).2.1.3.AnalgebraicrepresentationofConVisdtobeamorphismofrealalgebraicgroupsˆ:C!GL(VR).Proposition2.1.4.ThereisabijectionbetweenrationalHodgestructuresofweightkonVandalgebraicrepresentationsˆ:C!GL(VR)withˆ(t)=tkfort2R.TheHodgestructuredbyˆisthedecompositionVp;q:=fv2VC:ˆ(z)v=zpzqvg.Proof.See[vG00,Proposition1.4].Usingthiscorrespondence,wenowtranslate(i)-(iv)above:(i)TheTatestructureZ(k)correspondstoV=ZˆCwithˆ(z)(v)=(zz)kv,forv2Z(k).(ii)IfVhasrepresentationˆ:C!GL(VR),V_hasrepresentationˆ:C!GL(V_)byˆ(z)('):v7!'(ˆ1(z)(v)),for'2V_andv2V.(iii)IfVandWhaveˆVandˆW,respectively,thenVWhasrepresentationˆVˆW.(iv)AmorphismofHodgestructures(ofsameweight)f:V!Wmeansf(ˆV(z)(v))=ˆW(z)(f(v))2.1.5.LetVbeanintegralHodgestructureofweightkwithcorrespondingrepresentationˆ:C!GL(VR).ApolarizationisamorphismofHodgestructures8Q:VV!Z(k)suchthatQ(v;ˆ(i)w)isasymmetric,positiveformonVR.Thepair(V;Q)iscalledapolarizedHodgestructure.NotethatQ(ˆ(z)v;ˆ(z)w)=(zz)kQ(v;w)forallv;w2VRbyofamorphismofHodgestructures(see(iv)above).Whenk=1,thedofpolarizationtranslatestotheRiemannbilinearrelations.Furthermore,apolarizationgivesanisomorphismoftheweightkrationalHodgestructuresV(k)˘=V_viav17!Q(v;).Example2.1.6.LetXbeacomplex,projectivevarietyofdimensionnasinExample2.1.2.Denoteby!theclassinH2(X;Q)correspondingtoanampledivisoronX.For;2Hk(X;Q)withkn,Q(;)=RX!nk^^:ThisisabilinearformonHk.ThentheprimitivecohomologygroupsPk(X;Q),togetherwithQ,formapolarizedHodgestructureofweightk(thisiswell-known,seeforexample[VS02,Theorem6.32]).Asbefore,identicalstatementsholdwithHk(X;Q)replacedbyHk(X;Z)=torsion.WhenXisaprojectivesurface,let!bethedivisorcorrespondingtoOX(1)relativetosomeprojectiveembedding.ThenP2(X;Q)istheorthogonalcomplementof!inH2(X;Q)andthepolarizationistheintersectionpairingonH2.Lemma2.1.7.Let(V;Q)beapolarizedHodgestructureofevenweight2k.ThenQis(1)kponthesubspaceVR\(Vp;qVq;p).Proof.LetˆdenotetherepresentationcorrespondingtotheHodgestructureonV.Recallthatˆ(i)actsasipqonthesubspaceVp;qbyProposition2.1.4.SinceVhasweight2k,9ipq=ip(2kp)=i2p2k=(1)pk=(1)kp.Similarly,iqp=(1)kp.SinceQisapolarization,weknowthatQ(v;ˆ(i)v)>0forallv2VR.HenceQ(v;ˆ(i)v)=(1)kpQ(v;v)>0onthesubspaceVR\(Vp;qVq;p).If(V;Q)isapolarizedHodgestructureofweight2k,wecanapplythefollowinglemmatothepolarizedweight0HodgestructureonV(k).Lemma2.1.8.Let(V;Q)beapolarizedHodgestructureofweightk=0.LetSO(V)denotethespecialorthogonalgroupwithrespecttoQ.Thentheimageofthecorrespondingrepresentationˆ:C!GL(VR)liesinSO(VR).Proof.WehavethatQ(ˆ(z)v;ˆ(z)w)=(zz)kQ(v;w)=Q(v;w)sincek=0.Therefore,theimageliesinO(VR).SinceCisconnected,theimagelandsinSO(VR).2.2AbelianVarietiesWerecallsomebasicfactsaboutcomplexabelianvarieties.Allthestatementsherecanbefoundin[BL04].LetVbeacomplexvectorspaceofdimensionnandˆVafulllattice.ThismeansisafreeZ-submoduleofVofrank2nandR=V.ThelatticeactsonVbyadditionandweacomplextorustobethequotientX:=V=AdditioninVanabeliangroupstructureonX.Wenotethat˘=ˇ1(X)˘=H1(X;Z);sinceˇ1(X)isabelian.TheUniversalCoetTheoremgives10H1(X;Z)˘=Hom(H1(X;Z);Z)˘=;Z):Furthermore,theKunnethformulatellsusthatHn(X;Z)˘=^nH1(X;Z):2.2.1.FortwocomplextoriXandX0,ahomomorphismisaholomorphicmapf:X!X0whichisalsoagrouphomomorphism.WesaytwocomplextoriXandX0areisogenousifthereisasurjectivehomomorphismX!X0withkernel.Suchamapiscalledanisogeny.Isogeniesinduceanequivalencerelationonthesetofcomplextori.WerecalltheoftheChernclassofalinebundle.Associatedtotheexponentialsequence0!Z!OXe2ˇi!OX!0isthelongexactsequenceincohomology!H1(X;OX)!H1(X;OX)c1!H2(X;Z)!IdentifyingH1(X;OX)withPic(X),wetheChernclassofalinebundletobeitsimageinH2(X;Z)underthemapc1.SinceH2(X;Z)˘=;Z)^;Z)fromabove,wecanidentifytheChernclassofalinebundleonXwithaZ-valuedalternatingformonthelatticeProposition2.2.2.LetE:VV!RbeanR-linearalternatingform.ThenErepresentstheChernclassofalinebundleifandonlyifE;ˆZ,andE(iv;iw)=E(v;w)forallv;w2V.11WecallalinebundleonXpositiveifE(v;iv)>0forallv2V,whereEisthealternatingformassociatedtothelinebundleasinProposition2.2.2.2.2.3.ApolarizationonacomplextorusXistheChernclassofapositivelinebundleonX.Inparticular,apolarizationisanalternatingformE:!ZwhoseR-linearextensionE:VV!RE(iv;iw)=E(v;w)E(v;iv)>0forallv;w2V.ThesearecalledtheRiemannbilinearrelations.Proposition2.2.4.OnacomplextorusX,apositivelinebundleisequivalenttoanamplelinebundle.Thethirdpowerofsuchalinebundleisveryample,andhenceanembeddingofXintoprojectivespace.Therefore,apolarizationonXhasacorrespondinglinebundlewhosethirdpowerisveryample.2.2.5.Anabelianvarietyisacomplextoruswithapositivelinebundle.Theabovediscussionshowsthatanabelianvarietyisacomplex,projectivevarietywhichisalsoanalgebraicgroup.Lemma2.2.6.ThereisabijectionbetweenthesetofisomorphismclassesofcomplextoriandthesetofisomorphismclassesofintegralHodgestructuresofweight1.8><>:ComplexTori9>=>;1-1 !8><>:IntegralHodgestructuresofweight19>=>;Furthermore,underthiscorrespondence,abelianvarietiescorrespondtopolarized,integralHodgestructuresofweight1.128><>:AbelianVarieties9>=>;1-1 !8><>:Polarized,integralHodgestructuresofweight19>=>;Proof.LetX=V=beacomplextorus.ThenR=VhasacomplexstructureJ.Let1;0and0;1bethesubspacesofZC=VRConwhichJactsasiandi,respectively.ThisanintegralHodgestructureofweight1onConvsersely,supposeVZhasaweight1HodgestructurewithVC=V1;0V0;1.ThenwecanprojectVZisomorphicallyintoV1;0andV1;0=VZisacomplextorus.Notethattheofapolarizationonacomplextorusandthatofapolarizationonaweight1integralHodgestructureareidentical.Thecorrespondencebetweenabelianvarietiesandpolarized,integralHodgestructuresofweight1followsimmediately.Remark2.2.7.Wenotethatifwechange\integral"to\rational"inthepreviouslemma,werecovercomplextori(andabelianvarieties)uptoisogeny.2.3Algebras2.3.1.Letq(X1;:::;Xn)beaquadraticformwithcoinZ.WethealgebraassociatedtoqtobetheassociativealgebraCq(Z)=Zhx1;:::xni=(a1x1++anxn)2q(a1;:::;an):ai2ZwhereZhx1;:::xnidenotesthetensoralgebrainnvariables.Fornotationalconvenience,weusecapitallettersforthevariablesoftheformqandlowercaselettersforthecorrespondinggeneratorsofthealgebra.WewilloftenconsiderCq(K)˘=Cq(Z)KwhereK2fQ;R;Cg.ThetensoralgebraisZ=2Z-graded(intoevenandodddegreepieces),andsincethetwo-sidedidealthealgebra13isgeneratedbyevendegreeelements,thegradingdescendstoaZ=2Z-gradingontheClalgebra:Cq(Z)=C+q(Z)Cq(Z)Thesesummandsarecalledtheevenandoddalgebras,respectively.Forsomesituationsinwhichweareinterested,wewillhaveaquadraticformqonafreeZ-moduleVofrank.Inthiscase,thealgebraisCq(V)=T(V)=vvq(v):v2VwhereT(V)isthetensoralgebraonV.TheoriginalofCq(Z)isrecoveredbychoosingabasisofV.WehaveCq(VK)˘=Cq(V)KwhereVK=VK.WealsodealwithVadimensionalvectorspaceoverQ.Foraquadraticspace(V;q),wehaveananti-involutionofthetensoralgebra:T(V)!T(V)whichswapscomponents:v1vn7!vnv1.Sincepreservestheidealthealgebra,itdescendstoananti-automorphism,stilldenoted,ofthealgebra(andalsotheevenClalgebra).ThedgroupisthealgebraicgroupCSpin(VR)=fg2C+q(VR):gVRg1=VRg.InthisweidentifyVRwithitsimageinCq(VR).NotethatCSpin(VR)naturallyactsonVRbyconjugation.ThisamapCSpin(VR)!O(VR)intotheorthogonalgrouponVR,andtheimageliesinSO(VR).2.4TheKuga-SatakeConstructionWenowoutlinetheclassicalKuga-Satakeconstructionfrom[Sat66,KS67].ThisconstructionwascreatedwithK3-surfacesinmind,sincethemiddledimensional(primitive)cohomologyofaK3-surfacetheassumptionslistedbelow.14LetVbeafreeZ-moduleofrankn.SupposeVhasaweight2HodgestructuresuchthatdimV2;0=1.SuchaHodgestructureissaidtobeofK3-type.Furthermore,supposeqpolarizesthisHodgestructure.Wehavethatthesignatureofqis(n2;2).ThenthereisabasisofVRsuchthatq=X21X22+X23++X2n.LetxirepresentthegeneratorsoftheClalgebracorrespondingtoXi.ThenJ:=x1x2hasJ2=1,andsoleft-multiplicationbyJinducesacomplexstructureonC+q(R).ItiseasilyvdthatJ(uptosign)isindependentofthechoiceoforthonormalbasisforVR.ThequotientC+q(R)=C+q(Z)isthenacomplextorus.Equivalently,C+q(Z)hasaweight1HodgestructuredeterminedbyJ.Wearenowleftwithshowingthatthiscomplextorusisanabelianvariety.Inotherwords,theweight1HodgestructureonC+q(Z)aboveispolarized.BydiagonalizingqoverQ,oneseesthatthereexistsat>0suchthattJ2C+q(Z).Let=tJforthesmallestsucht.ThenthemapC+q(Z)C+q(Z)!Z(1)byxy7!tr((x)y)apolarizationoftheweight1HodgestructureonC+q(Z).Heretr(L)meanstraceoftheendomorphismofC+q(Z)whichisleft-multiplicationbyL.WesummarizetheseresultsinthefollowingtheoremTheorem2.4.1.Let(V;q)beanintegral,polarizedHodgestructureofK3-type.ThenC+q(R)canbegivenacomplexstructureforwhichthequotientC+q(R)=C+q(Z)isanabelianvariety.WecallthisquotienttheKuga-Satakevarietyof(V;q).Onemaywonderhowtheweight1HodgestructureonC+q(Z)isrelatedtotheoriginalweight2HodgestructureonV.Wehavetheresultmentionedintheintroduction:15Proposition2.4.2.LetVbearational,polarizedHodgestructureofK3-typewithpolar-izationq.Thereisaninclusionofweight2HodgestructuresV,!C+q(V)C+q(V):Furthermore,theHodgestructureonVcanberecoveredfromthatofC+q(V).Proof.Foraconstructiveproof,see[Huy15,Proposition4.2.4].ThereisanequivalentviewpointoftheKuga-SatakeconstructiontakenbyDeligne[Del72]whichweoutlinenow.Byanabuseofnotation,wesayaHodgestructureisofK3-typeifsomeTatetwistofitisaweight2HodgestructureofK3-type.For(V;q)arational,polarizedweight0HodgestructureofK3-type,thereisacommutativediagramCSpin(VR)Ch>~h>SO(VR);_wherehisthemaptheHodgestructureonVfromLemma2.1.8,~h(a+bi)=a+bJwhereJ=x1x2wasabove,andtheverticalmapistheorthogonalrepresentationbyconjugationofCSpin(VR)onVR.SinceCSpin(VR)actsonC+q(VR)byleft-multiplication,thisgivesrisetoamap˙:CSpin(VR)!GL(C+q(VR)).Thecomposition˙~haHodgestructureonC+q(V)byProposition2.1.4.ThisHodgestructureonC+q(V)isbothpolarizedandofweight1[Del72,Proposition4.5].BytheequivalenceinLemma2.2.6,theresultisanabelianvariety.Furthermore,Deligneshowsthatthisconstructionworksinfamilies(see4.1.4fortheofavariationofHodgestructure):Theorem2.4.3.Let(V; )beapolarizedvariationofHodgestructureofK3-typeoverasmoothandconnectedschemeSoftypeoverC.Thenthereisaetaleextension16ˇ:S0!SandanabelianschemeAoverS0satisfying:fors2S0,theerAsistheKuga-SatakevarietyassociatedtothepolarizedHodgestructureonVˇ(s).17Chapter3AbelianVarietyfromtheAlgebraInthischapter,weshowthatwecanconstructanabelianvarietyfromthealgebraofanarbitraryquadraticforminawaythatgeneralizestheKuga-Satakeconstruction,providedthesignatureoftheformisotherthan(1;0);(0;1);or(1;1).Ourconstructioninvolvesviewingthealgebraasasubalgebraofamatrixalgebra.Thisapproachallowsustobuildananalogoustheoryforcubicandquarticformsinthechapterofthispaper.3.1RepresentationsofAlgebrasWenowdiscussrepresentationsofalgebrasstartingwiththediagonalformq=Pni=1X2i.Theconstructionisasfollows:set1=0B@01101CA2=0B@10011CA3=0B@01101CANoticethat2=113andthat31=13.Whenniseven,weeamapCq(C)!n=2z}|{M2(C)M2(C)˘=M2n=2(C)18byx17!3x27!i1x37!23x47!i21...xn17!223xn7!i221;where3means311insideofthe(n=2)-foldtensorproduct.Forsimplicity,letE1;:::;Endenotetheimagesofx1;:::;xn,respectively.ItiseasilyvdthatEjEk=EkEjforanyj<kandthatE2i=I.Hence(c1E1++cnEn)2=f(c1;:::;cn)Isothemapiswell.Since1and3generateallofM2(C),thismapisanisomorphismfordimensionreasons.Remark3.1.1.Thismatrixrepresentationcomesfromtheunique(inthesenseof[BHS88,Theorem3.9])indecomposablematrixfactorizationofX20+qoverCconstructedin[BHS88,Example3.12].Nowwhennisodd,Cq(C)isnotisomorphictoamatrixalgebra.However,byconsideringtherepresentationofCg(C)whereg=q+X2n+1constructedabove,werealizeCq(C)asasubalgebraofthematrixalgebraM2(n+1)=2(C).Wesummarizetheabovediscussioninthefollowingproposition:19Proposition3.1.2.Letn>0bearbitrary,setN=b(n+1)=2c,andsupposeq=Pni=1aiX2iisaquadraticformwithainonzerointegers.LetibearootofX2ai.ThemapCq(C)!M2N(C)dbyxi7!Fi:=iEi(3.1)isanisomorphismofalgebraswhennisevenandisafaithfulalgebrarepresentationwhennisodd.Underthisrepresentation,theimageofC+q(K)isKDnYi=1Fmiimi2f0;1g;nXi=1miisevenE(3.2)whereK2fZ;Q;R;Cg.3.2ConstructionoftheAbelianVarietyInthissection,weshowthatwecanobtainanabelianvarietyfromthealgebraofanyquadraticformqin2variables,aslongasthesignatureofqisnot(1;1).Therestrictiononthesignaturecomesfromthefollowinglemma.Lemma3.2.1.Leta;bbothbepositiveintegers.Considerthebinaryquadraticformsq1=aX2+bY2q2=aX2bY2q3=aX2bY2Then1.Uptosign,thereisauniqueelementJ2C+q1(R)withJ2=1.ThesameresultholdsforC+q2(R).2.ThereisnoelementofC+q3(R)withsquare1.20Proof.Part1:Toprovethepart,wenotethatCq1(R)=R[x;y:x2=a;y2=b;yx=xy]:sothatC+q1(R)=h1;xyiR.SupposewehaveJ=c+dxy2C+q1(R)withc;d2RsuchthatJ2=1.Computing,wegetJ2=c2+d2(xy)2+2cd(xy)=(c2abd2)+2cd(xy)=1since(xy)2=x2y2=ab.Equatingbothsides,wegetc2abd2=1and2cd=0.Thiscanonlyhappenifc=0andd=1pab.Thus,wegetuptosignJ=1pabxy:NowwhenwelookatC+q2(R),weagainletJ=c+dxyandcomputeJ2=(c2abd2)+2cd(xy)because(xy)2=abasbefore.InordertohaveJ2=1,wemusthavethesameconditionsoncandd,soagainJ=1pabxy.Part2:Thistime,however,wewillshowthatthereisnoelementJ2C+q3(R)withJ2=I.SupposewehaveJ=cI+dAB2C+q3(R).ThenJ2=(c2+abd2)+2cd(xy)since(xy)2=x2y2=ab.IfJ2=I,wearetosolvetheequationsc2+abd2=1and2cd=0.Thisisimpossiblesincea;barepositive.Therefore,thereisnosuchJ.Remark3.2.2.Considercase(1)fromthelemma.LeftmultiplicationbyJonC+q(R)acomplexstructure;furthermore,C+q(Z)isafulllatticeinsidethecomplexvectorspaceC+q(R),andsothequotientC+q(R)=C+q(Z)isacomplextorus.RecallfromLemma2.2.6thatabelianvarieties(respectively,abelianvarietiesuptoisogeny)areequivalenttopolarizable,integral(resp.rational)Hodgestructuresofweight1.Asweproceedtoshowthiscomplextorusisanabelianvariety,ourassumptionthatfis21diagonalisaveryminorhypothesis.WeworkwithadiagonalquadraticformwithcocientsinZandconstructanabelianvariety.However,anyquadraticformwithcoinZcanbetransformed,viaaQ-linearchangeofvariables,intoadiagonalformwithcoef-inQ.ThisamountstopassingfromanintegralHodgestructuretoarationalone,whichmeansinsteadofcreatinganabelianvarietyonthenose,wearecreatinganisogenyclassofabelianvarieties.Inwhatfollows,allstatementsremaintrueifwereplaceZbyQ.Lemma3.2.3.Letnbearbitrary,N=b(n+1)=2c,andq=Pni=1aiX2iwithai2Z.Letv:=Qni=1Fmii2M2N(C)andw:=Qni=1Flii2M2N(C)formi;li2f0;1gwithPni=1miandPni=1lieven,asin(3.2)ofProposition3.1.2.LetTrdenotethetraceofamatrixandletdenoteconjugatetransposeofamatrix.(a)Tr(v)=8>>><>>>:2Nifmi=0foralli(i.e.v=I)0else:(b)Tr(x)=Tr(x)forallxintheimageofC+q(R)underthemapin(3.1).(c)vv=Qni=1jaijmiI:(d)Tr(vw)=8>>><>>>:2NQni=1jaijmiifv=w0otherwise.Proof.(a)RecallfromSection3.1thatTr(1)=Tr(2)=Tr(3)=0:Ifv6=I,thenchoosethelargestisuchthatmi=1.Thentheb(i+1)=2c-thelementofthetensorproductthatmakesupvisaconstantmultipleof228>>>>>>>><>>>>>>>>:1ifiisevenandmi1=02ifiisevenandmi1=13ifiisoddfromtheconstructioninSection3.1.Sincethetraceofatensorproductofmatricesistheproductofthetracesofthematricesthatmakeupthetensorproduct,thismakesTr(v)=0.Inthecasev=I,Tr(v)=2N,sinceIisa2N2Nmatrix.For(b),notethatconjugatetransposeandtracearebothR-linear,soittoprovetheclaimforbasiselementsoftheformin(3.2)ofProposition3.1.2.SowewillshowTr(v)=Tr(v).ButFi=8>>><>>>:FiifaiispositiveFiifaiisnegative.SinceFiFj=FjFifori6=jand(AB)=BA,wethatviseithervorv.Ifv=v,thenTr(v)=Tr(v).Ifv=v,thenpart(a)impliesthatTr(v)=0,andhenceTr(v)=Tr(v).For(c),notethatEkEk=Iforallk,sinceii=Ifori=1;2;3.SoFkFk=(ii)EkEk=jaijI.Therefore,vv=Fm11Fmnn(Fn)mn(F1)m1=Qni=1(ii)miI=Qni=1jaijmiI:Finally,(d)isjustacombinationof(a)and(c).Remark3.2.4.Itispreferabletohaveabasis-independentanti-involutiondonthedalgebrawhich,undertherepresentation(3.1),agreeswiththeconjugatetransposeofamatrix.Forthediagonalformq=Pni=1aiX2iwithainonzerointegers,letxidenote23thegeneratorsofCqcorrespondingtoXi.Under(3.1),xi7!Fi.AsnotedintheproofofLemma3.2.3part(b),Fi=8>>><>>>:FiifaiispositiveFiifaiisnegative.':Cq!Cqby'(xi)=aijaijxi.Then'isananti-involutionofCq,whereistheanti-involutionofthedalgebrafromSection2.3.Thisclearlyagreeswithundertherepresentation(3.1).Unfortunately,byassumingtheformisdiagonal,weareimplicitlychoosingabasis.Thefollowinglemmagivesasituationinwhichthisissueisresolved.Lemma3.2.5.Let(V;Q)beanintegral,polarizedHodgestructureofevenweight2k.Thenthereisabasis-independentanti-involutiononCQ(V)which,undertherepresentation(3.1),agreeswiththeconjugatetransposeofamatrix.Proof.WehavethatQis(1)kponthesubspaceV\(Vp;qVq;p)byLemma2.1.7.':V!Vbyv7!(1)kpvforv2V\(Vp;qVq;p).Then'extendstoamapofthealgebraCQ(V).Togetadiagonalform,wediagonalizeQQonthesubspacesVQ\(Vp;qVq;p).Oncewehaveadiagonalform,weapplyRemark3.2.4,whichshowsthat'agreeswithonCQ(V),whereistheanti-involutionofthealgebrafromSection2.3.Remark3.2.6.Itisalsopreferabletodiscuss\trace"ofanelementofC+qwithoutreferringtothetraceofamatrixundertherepresentation(3.1).Forx2C+q,wecouldconsiderthetraceoftheendomorphismLxofC+qdbyleft-multiplicationbyx.Inthecasethatqis24asintheKuga-Satakeconstruction,Lemma3.2.3agreeswiththeKuga-Satakeconstruction(see[vG00,Lemma5.8])uponreplacing2Nwith2n1.Moregenerally,let(V;Q)beanintegral,polarizedHodgestructureofevenweight2k,andsupposetherankofVisn.Letdenotetheanti-involutionofCQ(V)guaranteedbyLemma3.2.5.ThentheresultsofLemma3.2.3holdfortraceofamatrixreplacedbythetraceoftheendomorphismLx(with2Nreplacedby2n1).Wenowarriveatthemainresultofthissection:Theorem3.2.7.Letq=Pni=1aiX2ibeaquadraticformwithcoinZ.SupposethereisanelementJ2C+q(R)satisfyingthefollowingproperties:1.Left-multiplicationbyJinducesacomplexstructureonC+q(R),i.e.J2=I.2.J=J(onceweidentifythedalgebrawithmatricesvia(3.1)).3.:=tJ2C+q(Z)forsomet>0.ThenthebilinearformQ:C+q(R)C+q(R)!R;Q(v;w):=Tr(vw)theRiemannbilinearrelations(2.2.3),andhenceC+q(R)=C+q(Z)isanabelianva-rietywithpolarizationQ.Proof.First,wenotethatQisR-bilinearsincethetracemapandtakingconjugatetransposeofamatrixbothare.ByLemma3.2.3part(a),therestrictionofQtoC+q(Z)C+q(Z)takesvaluesinZ.NotethatQ(Jv;Jw)=Q(v;w):Q(Jv;Jw)=Tr((Jv)(Jw))25=Tr(JvwJ)=Tr(JJvw)J=JandTr(AB)=Tr(BA)=Tr(vw)JcommuteswithandJ2=I=Q(v;w):Thebilinearform(v;w)7!Q(v;Jw)isalsosymmetric:Q(v;Jw)=Tr(v(Jw))=Tr((vwJ))Lemma3.2.3part(b)=Tr(Jwv)=Tr(wvJ)=andJcommuteswith=Tr(wvJ)J=J=Tr(w(Jv))=Q(w;Jv):ItremainstocheckthatQ(v;Jv)>0forallv2C+q(R),v6=0.Wecheckthisforthebasiselementv=Qni=1FmiiwithF2i=ai,mi2f0;1gandPni=1mieven.WehaveQ(v;Jv)=Tr(v(Jv))=Tr(Jvv)=tTr(vv)sinceJ=t,recallt>0=2NtnYi=1jaijmi>026byLemma3.2.3part(d),whereN=b(n+1)=2c.SinceQ(x;Jx)=tTr(xx),bytheR-linearityofTrandLemma3.2.3part(d),wehavepositiveforallx2C+q(R).Theorem3.2.8.Letq=Pni=1aiX2ibeaquadraticforminn2variableswithsignaturenot(1;1).Forachoiceoflandmwithalam>0,thereisaunique(uptosign)ele-mentJinC+q(R)withsquare1.Furthermore,thecomplexstructuredbyJmakesC+q(R)=C+q(Z)anabelianvarietyofcomplexdimension2n2.Proof.Aslongasthesignatureofqisnot(1;1),weareabletol;m2f1;:::;ngwithalam>0.WemaythenapplyLemma3.2.1togetJ=1palamFlFm.VerifyingtheassumptionsofTheorem3.2.7isstraight-forward:J2=I,J=J,andt=palam.3.3MapsBetweenAbelianVarietiesInthissectionwediscusshowmapsbetweenalgebraspreservingthecomplexstruc-turegiverisetomapsbetweenabelianvarieties.Webeginwithamapbetweenvectorspaces(orfreeZ-modulesoferank)preservingthequadraticform,andshowhowitamapofalgebras.SupposewehavequadraticformsqVandqWonfreeZ-modulesofrank,VandW,respectively.Furthermore,supposewehavealinearmapL:V!WwhichqV(x)=qW(Lx)forallx2V.LetiVandiWdenotetheinclusionsofVandWintotheirrespectivealgebras.TheninCqW(W)wehave(iW(Lx))2=qW(Lx)=qV(x)sobytheuniversalpropertyofthealgebra,thereisauniquemap,whichwealsodenotebyL,fromCqV(V)toCqW(W)whichmakesthefollowingdiagramcommute27VˆiV>CqV(V)WL_ˆiW>CqW(W)L_Moreconcretely,CqV(V)isgeneratedbyproductsv1vkwithvi2V,andthemaptakesv1vk7!L(v1)L(vk).ThismakesitclearthatLpreservesthegrading,sothatthemaprestrictstoL:C+qV(V)!C+qW(W).SupposethereexistsaJV2C+qV(VR)sothatleft-multiplicationbyJVacomplexstructureonC+qV(VR).ThenJ2V=1impliesL(JV)2=1,sothatleft-multiplicationbyJW:=L(JV)acomplexstructureonC+qW(WR).WehavethefollowingcommutativediagramC+qV(VR)JV>C+qV(VR)C+qW(WR)L_JW>C+qW(WR)L_Insuchasituation,wegetamapofcomplextoriC+qV(VR)=C+qV(V)!C+qW(WR)=C+qW(W):WhenLisinjective,thismapisaninclusionofcomplextori.Iffurthermore,JWthehypothesesofTheorem3.2.7,thenAW:=C+qW(WR)=C+qW(W)isanabelianvariety.Sinceacomplexsubtorusofanabelianvarietyisitselfanabelianvariety,AV:=C+qV(VR)=C+qV(V)isalsoanabelianvariety.Inthisway,thereisaninclusionofabelianvarietiesAV,!AW.Fortheremainderofthesection,wewillassumethatLinjectiveandthatVandWarevectorspacesoverQorR.NotethatifweassumeqVandqWarenondegeneratebilinearformspreservedbyL,thenLisautomaticallyinjective,forifLx=0,thenqV(x;y)=qW(Lx;Ly)=0forally,whichmeansx=0sinceqVisnondegenerate.ForthequadraticformqV,wehavetheassociatedbilinearform,alsodenotedqV,whichisby28qV(x;y)=12(qV(x+y)qV(x)qV(y));andsimilarlyforqW.TheconditionqV(x)=qW(Lx)impliesqV(x;y)=qW(Lx;Ly),whichmeansthatorthogonalelementsinVaremappedtoorthogonalelementsinW.Ifwechooseabasisfv1;:::;vngofVforwhichqVisdiagonal,thenqWrestrictedtothesubspacespannedbyfLv1;:::;Lvngwillalsobediagonal.ChoosingabasisoforthogonalcomplementofL(V)inWforwhichqWisdiagonal,wemayassumethatqVandqWarediagonalandqV(x)=qW(Lx)forallx2V.Wesummarizetheabovediscussioninthefollowingproposition:Proposition3.3.1.Letq=Pki=1aiX2iandg=q+Pni=k+1aiX2ibequadraticformswithcoinZ.SupposeJ2C+q(R)ˆC+g(R)acomplexstructureonbothC+q(R)andC+g(R).Furthermore,supposeJthehypothesesofTheorem3.2.7forC+g(R).ThenwehaveaninclusionofabelianvarietiesC+q(R)=C+q(Z),!C+g(R)=C+g(Z).Example3.3.2.Let':X!Ybeamapbetweenn-dimensionalcomplexprojectivevari-eties.ConsidertheintersectionformsqVandqWonV=Hn(Y;Q)andW=Hn(X;Q),respectively,withL=':V!W.ThenLisinjectivesinceqVandqWarenondegenerate.Iftheseintersectionformshavesignaturesotherthan(1;0);(0;1);(1;1),thenbyTheorem3.2.8wecanaJsatisfyingthehypothesesofProposition3.3.1.WeobtainaninclusionofabelianvarietiesAY,!AX.293.4ExamplesandApplications3.4.1ExamplesExample3.4.1(CubicSurface).LetXbeacubicsurfaceinP3.SinceXisobtainedbyblowingup6pointsintheplane,weknowthatPic(X)˘=Z7=hH;E1;:::;E6iwithH2=1,E2i=1,HEi=0,andEiEj=0fori6=j.HereHisthestricttransformofalinenotpassingthroughanyoftheblownuppoints,andEiaretheexceptionaldivisors.TheprimitivecohomologyP2(X;Z)=torsionisspannedbytheclassesoftheEi.TheintersectionformonP2(X;R)isq=x21x26,wherethexicorrespondtotheEi.Leteibetheelementsofthedalgebracorrespondingtoxi.ThenJ=e1e6thehypothesesofTheorem3.2.7,andsomakesC+q(R)=C+q(Z)anabelianvariety.Inthiscase,JiscentralinC+q(R).Furthermore,theweight2HodgestructureonP2(X;Z)=torsionisoftypef(1;1)g.Wecangetaweight0HodgestructureVoftypef(0;0)gbytakingthetwistP2(X;Z)(1)=torsion.ThisgivesacommutativediagramasinSection2.4:CSpin(VR)Ch>~h>SO(VR)_where~h(a+bi)=a+bJ.Ingeneral,letVbeanintegral,polarizedHodgestructureofweight0andtypef(0;0)gwithrank(V)=2n,nodd.ThenJ=e1e2niscentralinC+(R)andthecondi-tionsofTheorem3.2.7.Additionally,thediagramofthepreviousexampleabovecommuteswith~h(a+bi)=a+bJ.30Example3.4.2(K3CoverofEnriquesSurface).Letf:X!YbethedoublecoverofanEnriquessurfacebyaK3surfaceX.ThenH2(Y;Z)=torsionisisometrictoasublatticeinsideofH2(X;Z)=torsion(see[BHPVdV04,p.350]).UponchoosinganappropriateJ,Proposition3.3.1givesaninjectivemapofabelianvarietiesAY!AX,wherethecomplexstructureonAXisinducedfromthecomplexstructureonAYasabove.NotethattheinducedcomplexstructureonAXisnotthesameasthecomplexstructuredviatheKuga-SatakeconstructionforK3-surfaces,becauseh2;0(Y)=0.3.4.2RelationtoKuga-SatakeConstructionWenowdescribehowtheconstructionofSection3.2generalizestheKuga-Satakeconstruc-tion[KS67].AsanapplicationofTheorem3.2.8,wegetProposition3.4.3.Whenqhassignature(n2;2),werecovertheKuga-Satakevariety,uptoisogeny.Proof.RecallRemark3.2.2regardingabelianvarietiesuptoisogeny.Wemaywriteq=Pni=1aiX2iwithai2Qanda1;a2<0andallothercotspositive.LetJbetheuniquecomplexstructurecreatedfromthenegativecotsasinLemma3.2.1.ThisisthesamecomplexstructureasintheKuga-Satakeconstruction.Ifniseven,thepolarization(2n1=2n=2)Q(v;w)takestheexactsamevaluesasthepolarizationfromtheKuga-Satakeconstruction[vG00,Proposition5.9].Inthecasethatnisodd,(2n1=2(n+1)=2)Q(v;w)doesthetrick.SeealsoRemark3.2.6.NowifwebeginwithapolarizedHodgestructureVwithpolarizationq,forwhichC+q(VR)=C+q(V)isanabelianvariety,wewouldliketoknowifwecanrecovertheHodgestructureonVfromtheHodgestructureonC+q(V).Wehave:31Proposition3.4.4.SupposeVisapolarized,rationalHodgestructureoftypef(k;k)gwithpolarizationq.AlsosupposedimV2,sothatthereisJ2C+q(VR)forwhichtheconditionsofTheorem3.2.7ared.ForanysuchJ,wehaveaninclusionofweight2HodgestructuresVQ(k1),!C+q(V)C+q(V)Further,theHodgestructureonVcanberecoveredfromtheHodgestructureonC+q(V).Proof.SinceVhasaHodgestructureoftypef(k;k)g,weknowthatthequadraticformbyqisthisiswhydimV2allowsustoapplyTheorem3.2.8togetanabelianvarietyC+q(VR)=C+q(V).ThecomplexstructuregiventoC+q(VR)correspondstoaweightoneHodgestructureonC+q(V)viaLemma2.2.6.Notethat(C+q(V))C+q(V)˘=End(C+q(V))andsotheweightzeroHodgestructureon(C+q(V))C+q(V)givesEnd(C+q(V))aHodgestructureofweight0.TheHodgestructureisdeterminedbyanactionofConEnd(C+q(V)).Explicitly,itisgivenby(z')(w):=z('(z1w))(3.3)where'2End(C+q(V)),w2C+q(V),andtheactionofz=a+biontherighthandsideisgivenbyleft-multiplicationbya+bJonC+q(V).Weknowthatthepolarizationgivesanisomorphismofweight1HodgestructuresC+q(V)Q(1)˘=(C+q(V)),asmentionedinSection2.1.WewillshowthatVQ(k),!End(C+q(V)),thenEnd(C+q(V))˘=(C+q(V))C+q(V)˘=C+q(V)C+q(V)Q(1)32TensoringbyQ(1)willthenyieldtheproposition.Intheprocess,itwillbeclearthattheHodgestructureonC+q(V)determinestheHodgestructureonV.Fixanelementv02VwhichisinvertibleinCq(V).VQ(k)!End(C+q(V))byv7!fvwherefv(w)=wv0v.Thismapisclearlylinear.Toshowinjectivity,notethatsincev0isinvertible,v20=q(v0;v0)6=0.Givenv,wecanav12Vwithq(v1;v)6=0.Setw=v1v0,thenfv(w)=q(v0;v0)v1v6=0inC+q(V).Thus,themapisinjective.ToshowthatthisisamorphismofHodgestructures,wejustneedtoshowthatthemapv7!fvcommuteswiththeactionofC.Soweneedtoshowfzv=zfv;(3.4)wherezvistheactionofzonVQ(k),andtheactionofzfvistheactiondescribedin(3.3).SincetheactionofConVisgivenbyzv:=(zz)kv,CactstriviallyonVQ(k).Thismeansfzv=fv.Now,(zfv)(w)=z(fv(z1w))=z(z1wv0v)=wv0v=fv(w)sozfv=fvandequation(3.4)holds.TheHodgestructureonVcanbetriviallyobtainedfromtheHodgestructureonC+q(V).CombiningProposition3.4.4withtheanalogousoneforHodgestructuresofK3-type,wegetthefollowing:Corollary3.4.5.SupposeVandWarerational,polarizedweight2Hodgestructures(withpolarizationsqandr,respectively)ofeitherK3-typeoroftypef(1;1)g,andwithdimV;dimW2.Thenwehaveaninclusionofweight4HodgestructuresVW,!C+q(V)C+q(V)C+r(W)C+r(W)33andfurthermore,werecovertheHodgestructureonVWfromthatofC+q(V)andC+r(W).Ofcourse,thereisananalogousinclusionifweinsteadtaketensorproductsofthreesuchweight2Hodgestructures,etc.Hereisthegeometricexamplewehaveinmind:Example3.4.6.LetXandYbenonsingularcomplex,projectivesurfaceswithHodgestruc-turesonH2asinCorollary3.4.5.LetAXandAYbetheabelianvarietiesarisingfromthevectorspacesH2(X;Q)andH2(Y;Q),respectively.ThenthecorollaryshowsthatthereisaninclusionofHodgestructuresH2(X;Q)H2(Y;Q),!H1(AX;Q)H1(AX;Q)H1(AY;Q)H1(AY;Q)ˆH2(AXAY;Q)H2(AXAY;Q):Inparticular,thisappliestotheproductoftwoK3-surfaces,aK3-surfacewithanEn-riquessurfaceoracubicsurface,etc.(cf.Examples3.4.1and3.4.2).34Chapter4FamiliesofAbelianVarietiesThegoalofthissectionistoshowthat,undercertainhypotheses,ourconstructionfromthepreviouschapterworksinfamilies.WealsoprovideexamplesofwhenthehypothesesareFirst,wediscusssomelinearalgebrapreliminaries.LetV_=HomK(V;K)denotethedualvectorspaceofV.ThensinceEnd(V)˘=VV_,wemaythetraceofalinearmapasfollows.Forv2Vand˚2V_,tr(v˚)=˚(v).Clearly,tramaptr:VV_!K,andsothetrace(stilldenotedtr)ofaK-linearendomorphismofVisbythecompositionEnd(V)˘=!VV_tr!K:UponchoosingabasisforV,thisagreeswiththeusualnotionoftraceofamatrix.Theabovediscussioncarriesovertovectorbundles.ForavectorbundleV!X,wehavethedualbundleV_!XwhichisthesameastheHom-bundleHom(V;KX)!Xofbundlehomomorphisms(overX)ofVintothetrivialbundleKX.Furthermore,thereisacanonicalisomorphismofvectorbundlesEnd(V)˘=VV_.ThisallowsustothetraceofanendomorphismofvectorbundlesasthecompositionEnd(V)˘=>VV_tr>KXX_<>Notethatoners,thisagreeswiththeclassicalnotionoftraceofamatrix.35Recalltheanti-automorphismoftheevenrdalgebrafromSection2.3.Thismapalsocarriesovertovectorbundles.Sp,ifwehaveavectorbundleequippedwithaquadraticform(V; )overabaseX,thenwecanformthetensorbundle,bundle,andevenbundleasvectorbundlesoverX.Themap:C+(V; )!C+(V; )isgloballyasamorphismofbundlesoverXandonersitagreeswiththebefore.4.1VariationsofHodgeStructureThenotionofavariationofHodgestructureisdueto[Gri68,Gri70].Thissectionrecallssomeofthenecessarynitions.4.1.1.LetGbeanabeliangroupandletXbeatopologicalspace.AsheafGonXiscalledalocalsystemifitislocallyisomorphictotheconstantsheafwithstalkG.ForE!Xavectorbundle,wedenotebyO(E;U)thesetofsectionsoveranopensubsetUˆX.Welet1(U)denotethe1-formsoverU.4.1.2.AconnectiononavectorbundleE!Xisalinearmapr:O(E;U)!1(U)O(E;U)UˆXopenwhichtheLeibnizruler(f˙)=df˙+fr˙forf2O(U)and˙2O(E;U).Wesaythatasection˙2O(E;U)isifr˙=0.TheconnectioniscalledifthereisacoverofXforwhichthecorrespondinglocalframesconsistofsections.We36sayavectorbundleE!Xisifithasaconnection.Asillustratedbythefollowingproposition,vectorbundleshaveconstanttransitionfunctions.Proposition4.1.3.ThefollowingthreecategoriesareequivalentLocalsystemsofstalkCnoveraconnectedcomplexmanifoldX.Representationsofthefundamentalgroupˇ1(X;x)!GLn(C).Flat,holomorphicvectorbundlesE!Xofrankn.Underthiscorrespondence,locallyconstantsections,ˇ1(X)-invariantsections,andsec-tionsareallequivalent.Proof.Thisiswell-known,seeforexample[Cat14,x1.3].4.1.4.LetXbeaconnectedcomplexmanifold.AvariationofHodgestructure(VHS)ofweightkonXconsistsofthefollowingingredients:AlocalsystemVZoffreeZ-modulesonX.AdecreasingationoftheassociatedholomorphicvectorbundleVC˙˙Fp˙Fp+1˙˙0byholomorphicsubbundleswhichsatisfy:(i)Foreachpointx2X,theersFpxformaHodgeationoftheweightkHodgestructureontheerVC;x.(ii)rO(Fp)ˆ1XO(Fp1)whereristheconnectiononVCguaranteedbyProposition4.1.3.37Notethatwhenk=1andtheisVC˙F1˙0,condition(ii)inthedofVHSisautomatically4.1.5.TherankofaVHSistherankofthecorrespondingvectorbundle.ThetypeofaVHSisdtobethetypeoftheHodgestructureofaer(seeSection2.1).WeassumeXisconnectedsothattheseareindependentofthechoiceofer.4.1.6.WesaythattheVHSispolarizedifthereisanon-degeneratebilinearform onVZsatisfyingthefollowingproperties.First,thisformisrequiredtobesymmetricorskew-symmetric(ifkisevenorodd,respectively).Second,werequirethat xpolarizestheHodgestructureoftheerVC;xforeachx2X.ByabilinearformonavectorbundleE!XisasectionofE_E_.TosayabilinearformismeansthecorrespondingsectionofE_E_isExample4.1.7.Letf:Y!Xbeaproper,smoothmorphismofsmoothandconnectedschemesoftypeoverC.ThenRkfZisalocalsystemonXbyEhresmann'sTheorem.Furthermore,thelocalsystemsRkfRandRkfCcanalsobeviewedasC1andholomorphicvectorbundlesonX,respectively.Intheholomorphiccase,thesheafofholomorphicsectionsissimplyRkfCCOX.TheconnectiononthevectorbundleRkfCiscalledtheGauss-Maninconnection.Thestalksofthesesheavesoveranyx2XarethecohomologygroupsoftheerYx=f1(x):Hk(Yx;Z),Hk(Yx;R),andHk(Yx;C),respectively.EachHk(Yx;C)hasaHodgeation,andthesegluetogethertogiveholomorphicsubbundlesFpˆRkfCwhichsatisfytheconditionsoftheabove.HencewehaveaVHSofweightk.Furthermore,supposefisprojective,i.e.ffactorsthroughaclosedimmersionfollowedbyprojectionasinthediagram38Y,!PNXX_f>Thentheersoffcomeequippedwithprojectiveembeddings.Thisallowsustoprimitivecohomologyoneacher,whichgivesrisetothelocalsystemsPkfZandsimilarlywithRandC.ThepolarizationsoneacherapolarizationonPkfZ.ThisisthegeometricsettingthatyieldsapolarizedVHSofweightk.4.2FamiliesofAbelianVarietiesFromapolarizedVHS(V; ),wecanconstructalocalsystemC(V; ),whosecomplex,real,andintegralbundlesarebundlesofalgebras.Additionally,wehavethelocalsystemC+(V; )whosebundlescorrespondtotheevenalgebras.Weoutlinetheconstructionbelow.Since(V; )isaVHSonX,VZisalocalsystemoffreeZ-modulesonX,andhencecomesequippedwithaconnectionr.ThetensorbundleT(VZ)isthenalocalsystemoffreeZ-modulesonX,andrextendstoaconnectiononT(VZ).Sinceweareassuming(VZ; Z)ispolarizedby ,wehavethat iswithrespecttor.ThiscompatibilityguaranteesusthatthebundleC(VZ; Z)isaswell(thesameholdsfortheevenbundle).Thus,C+(VZ; Z)isalocalsystemoffreeZ-modulesonX.ThisalsoappliesforRandC.Proposition4.2.1.Let(V; )beapolarizedvariationofHodgestructureofweightkoverasmoothschemeXoftypeoverC.LetC+(V; )betheassociatedevendbundleoverX.Supposethereexistsalocally-constantsectionJofC+(VR; R)suchthatforevery39x2X,leftmultiplicationbyJxacomplexstructureontheerC+(VR; R)x.ThenC+(V; )aweight1variationofHodgestructureonX.Proof.ThefactthatC+(VZ; Z)isalocalsystemoffreeZ-modulesonXhasbeendiscussedabove.Forx2X,wehaveadecompositionC+(VR; R)xC=C+(VR; R)x1;0C+(VR; R)x0;1intotheeigenspacesonwhichJxactsbyiandi,respectively.SinceJisaglobalsectionoftheevenbundle,thisdecompositionC+(VR; R)1;0andC+(VR; R)0;1asholomorphic(sinceJCisholomorphic)subbundlesofC+(VC; C).SetF1=C+(VR; R)1;0.ThenwehaveadecreasingC+(VC; C)˙F1˙0andtheersofF1xaHodgeoftheweight1HodgestructureontheerC+(VC; C).Thislastpartisbyconstruction,asJxtheweight1Hodgestructureontheeroverx2X.Asnotedabove,condition(ii)intheofVHSisautomaticforweight1.Let(V; )beapolarizedVHSofweightkoverabaseX.WeC1-subbundlesofVCbyVp;q=Fp\Fq,wherecomplexconjugationistakenrelativetoVR.ThenthereisadecompositionVC=Lp+q=kVp;qasC1-vectorbundles[Sch73,2.15].Set40VpR=8>>><>>>:Vp;qVq;p\VRifp6=qVp;p\VRifp=qThenwehaveVR=LpVpR.WecananalogouslyVpZ.WeanautomorphismofC1-vectorbundlesVC'>VCX<>whichactsonVp;qasik(1)p.Whenkiseven,wecan':VR!VRby(1)k=2ponVpR,andthisagreeswiththeabove'map.ThismapcanalsobeonVZinananalogousfashion.Fromhereon,weassumekiseven,sothatwecanfreelymention'withoutspecifyingC;R;orZ.Thismapextendstoanautomorphism,whichwestilldenoteby',ofthetensorbundleT(V).Thepolarization isorthogonalwithrespectto(Vp;qVq;p)andtherefore'descendstoanautomorphismoftheevenbundleC+(V; )'>C+(V; )X<>Weananti-automorphismoftheevenbundleasthecomposition:='.When(V; )isaVHSofevenweightk,thisisjusttheglobalversionoftheanti-involutionfromLemma3.2.5.Theorem4.2.2.Let(V; )beapolarizedvariationofHodgestructureofevenweightkoverasmoothschemeXoftypeoverC.Supposethereisalocally-constantsectionJofC+(VR; R)whichthefollowingproperties:J2=1,J=,andtJisasectionofC+(VZ; Z)forsomet>0.ThenC+(V; )aweight1polarizedvariationofHodgestructureonX.41Proof.WemaythinkofthesectionJasanendomorphismofC+(V; )becauseofthealgebrastructure.Thus,Proposition4.2.1alreadytellsusthatC+(V; )aweight1VHSonX.Thenewparthereisthatitispolarized.Wethepolarizationasasection˙ofthebundleC+(VZ; Z)_C+(VZ; Z)_X˙^ˇ_by˙(x):=7!tr(LtJx)where;2C+(VZ; Z)x,andthetraceoccursintheerC+(VZ; Z)x.Recallthattr(Lf)meansthetraceoftheendomorphismwhichisleft-multiplicationbyf.Since˙iscontinuous,itisautomaticallylocally-constant,sinceweareconsideringitasasectionofabundleofZ-modules.Wewouldliketomakethismapmoreexplicit.IdentifysectionsandofC+(VZ; Z)withtheirimages(X)and(X),respectively.WehavethemapC+(VZ; Z)C+(VZ; Z)!End(C+(VZ; Z))7!LtJThenthemapdescribedaboveisthecompositionC+(VZ; Z)C+(VZ; Z)>End(C+(VZ; Z))tr>ZXX_<>whichonerspolarizestheHodgestructurebyLemma3.2.5,Remark3.2.6,andTheorem3.2.7.Weseethat˙isbothalternatingandnondegeneratebythesametheorems.42Lemma4.2.3.Aweight1polarizedVHSoverasmoothschemeXoverCisequivalenttoanabelianschemeoverX.Proof.See[Del72].Onedirectioniseasy:ifAf!Xisanabelianscheme,thenR1fZisaweight1polarizedVHS.Intheotherdirection,wecanseethisbylookingattheers.Theeroverx2Xhasthestructureofanintegralweight1Hodgestructurewhichispolarized.ThecategoryofsuchHodgestructuresisequivalenttothecategoryofabelianvarietiesoverC.ThefactthattheHodgestructure,polarization,etc.areallgloballymeansthattheseabelianvarietiesgluetogethertogiveaprojectivefamilyofabelianvarietiesoverX.Furthermore,anapplicationofBailey-BorelshowsthatthisisanabelianschemeoverX.4.2.1ExamplesTheorem4.2.4(RiemannExistenceTheorem).LetXbeaschemewhichislocallyoftypeoverCandletXandenotethecorrespondingcomplexanalyticspace.ThefunctorY7!YangivesanequivalenceofcategoriesbetweenthecategoryofetalecoveringsY=XandcoveringspacesYan=Xan.LetXbeasmoothschemewhichislocallyoftypeoverC.Let(V; )beapolarizedvariationofHodgestructureonX.Chooseabase-pointx2X.Thetopologicalfundamentalgroupˇ1(X;x)actsontheerVZ;xofVZabovex.ThisactionextendstoanactiononVR;xandVC;xbyscalarextension.Sincetheactionpreservesthepolarization,wegetamapˇ1(X;x)ˆ!O(VZ;x);whereO(VZ;x)=fg2Aut(VZ;x)j Z;x(gv;gw)= Z;x(v;w)forallv;w2VZ;xg:Lemma4.2.5.AfterapossibleetaleextensionofX,wemayassumethattheimageofˆliesinSO(VZ;x).43Proof.Indeed,iftheimagedoesnotlieinSO(VZ;x),thenthekernelofthecompositionˇ1(X;x)ˆ!O(VZ;x)det!Z=2Z=1ganindex2subgroupHEˇ1(X;x).LetYbethedoublecoverofXcorrespondingtoHandletWdenotethevariationofHodgestructureonYobtainedbypullingbackVunderthemapf:Y!X.Thenfisaetalemorphismandwehaveamapˇ1(Y;y)!SO(WZ;y)byconstruction.Furthermore,YisasmoothschemewhichislocallyoftypeoverCbytheRiemannExistenceTheorem.WewillneedthefollowingtoproveTheorem4.2.8below.Proposition4.2.6.Letn2mod4.LetVbearealvectorspaceofdimensionnwithpositivequadraticformq.TheninC+q(V)thereisaunique(uptosign)centralelementJwithJ2=1.Furthermore,foranyT2SO(V;q),T(J)=J.Proof.Chooseabasisfe1;:::;engofVsothatq=x21++x2n,whereeicorrespondstoxi.ThenabasisforC+q(V)isgivenbyfei1ei2jj0jn=2g:WewillshowthatJ=e1en(orJ=e1en)isthedesiredelement.First,Jiscentralsince(eiej)J=(1)n1eiJej=(1)n1(1)n1Jeiej=J(eiej)andsimilarlyfortheotherbasiselementsofC+q(V).WealsohaveJ2=(e1en)(e1en)=(1)n(n1)2e21e2n=1sincenisevenandn=2isodd.44Nowforuniqueness:thestructuretheoremforalgebrasshowsthatC+q(V)isacentralsimplealgebraoverC,hencethereisaunique(uptosign)centralelementwhichsquaresto1.Aswehaveshown,Jisthatelement.Finally,weshowthatT(J)=J.ForanyT2SO(V;q),TamaponC+q(V)bytheuniversalpropertyofalgebras.Inparticular,T(J)=T(e1)T(en).Sinceeiej=ejeifori6=j,wehavethatT(J)=T(e1)T(en)=(detT)e1en=J:Proposition4.2.7.Letn2mod4.Let(V; )beapolarizedvariationofHodgestructureofevenweightk,oftypef(k2;k2)g,andranknoverasmoothschemeXoftypeoverC.Then(afterpossiblyreplacingXbyaetaleextension)thereisalocally-constantsectionJofC+(VR; R)whichacomplexstructureoneacher.Proof.UponapplyingLemma4.2.5above,wemayassumethattheactionofˇ1(X;x)on(VZ; Z)landsinSO(VZ; Z).Bytheuniversalpropertyofthealgebra,theactionofˇ1(X;x)ontheerVZ;xextendstoanactionofˇ1(X;x)onC+(VZ; Z)xandsimilarlyforRandC.ChooseatrivializingcoverfUgofthelocalsystemC+(VR; R)overX.OverU,thelocalsystemistrivial,andwewritethisasUC+(VR; R).Thereisaunique(uptosign)J2C+(VR; R)guaranteedbyProposition4.2.6.NotethatthetypeoftheVHSguaranteesthat ispositiveWhenU\U6=;,weobtainanautomorphism45':C+(VR; R)!C+(VR; R).Thisautomorphismisconstant(doesn'tdependonpointsinU\U),sincewearedealingwithalocalsystem.Thisautomorphismpreservesthecenterandthealgebrastructure,andhenceJ='(J)istheunique(uptosign)elementofC+(VR; R)guaranteedbyProposition4.2.6.ForwithU\U\U6=;,wehavetwoelementsofC+(VR; R)whicharecentralandsquareto1:J='(J)andJ0='('(J)),andhenceJ=J0orJ=J0.Checkingthatthesetwoelementsarethesameisequivalenttocheckingthatforx2U\U\U,theactionofˇ1(X;x)onaerC+(VR; R)xpreservesJx.Sinceweareassumingtheactionofˇ1(X;x)landsinSO(VR;x; R;x),Proposition4.2.6assuresusthatJ=J0.HencethereisaconsistentchoiceofJwhichyieldsalocally-constantsectionJofC+(VR; R).Byconstruction,Jacomplexstructureoneacher.Theorem4.2.8.Letn2mod4.Let(V; )beapolarizedvariationofHodgestructureofevenweightk,oftypef(k2;k2)g,andranknoverasmoothschemeXoftypeoverC.Then(afterpossiblyreplacingXbyaetaleextension)wecanaweight1polarizedVHSonC+(V; ).Proof.WeneedtoshowtheassumptionsofTheorem4.2.2areProposition4.2.7givesusthesectionJwhichJ2=.ChooseabasisforVR;xforwhich R;xisdiagonal.UsingthenotationoftheproofofProposition4.2.6,wehavethatJx=e1en.Now(e1en)=(1)k=2k=2(ene1)=e1en.HenceJx=Jx.Therefore,J=sinceitholdsoners.46ItremainstoshowthattJisasectionofC+(VZ; Z)forsomet>0.UsingthenotationfromtheproofofProposition4.2.7,weknowthattJ2C+(VZ; Z)forsomet>0asdiscussedinSection2.4.Forx2U,theactionofˇ1(X;x)ontheerslandsinSO(VZ; Z),andhencethesameargumentasintheproofofProposition4.2.7showsthatthechoiceoftisconsistentandyieldsaglobalsectiontJofC+(VZ; Z).Example4.2.9.Asmooth,projectivefamilyofcubicsurfacesYf!Xtheassump-tionsofTheorem4.2.8withVZ=P2f(Z)and theintersectionpairing,asinExample4.1.7(uponpossiblyreplacingXbyaetaleextensionX0andYbytheeredproductYXX0).Therefore,weobtainanabelianschemeA!Xwhoseersaretheabelianvarietiescorrespondingtotheersoff(whicharecubicsurfaces,seeExample3.4.1).Example4.2.10(Kuga-SatakeConstructioninFamilies).Theorem2.4.3showsthattheKuga-Satakeconstructionworksinfamilies.47Chapter5AbelianVarietiesAssociatedtoHigherDegreeFormsInthischapter,weapplythetechniquesofChapter3toconstructabelianvarietiesfromthealgebrasofhigherdegreeforms.Duetothefactthatthesegeneralizedalgebrasaredimensional,weconsidertheirrepresentations.Thisisthereasonwetooktherepresentation-theoreticviewpointwheninvestigatingthealgebrasofquadraticforms.Wegivecriteriaforwhicharepresentationofageneralizedalgebrayieldsanabelianvariety.ThisisdoneinamannerwhichparallelstheKuga-Satakeconstruction.Itturnsoutthatcubicandquarticformsarethebestcandidatesforthisconstruction.Weprovidegeometricexamplesof3-foldsand4-foldsforwhichwecancreateanabelianvarietyfromthecubicandquarticforms,respectively,onH2.Thisincludes4-foldswhichariseastheproductoftwosurfaces.Finally,westudywhenthisconstructionisnaturalenoughtoworkinfamiliesasinChapter4.5.1GeneralizedAlgebrasWebeginbydiscussinggradedalgebrasoverC(thediscussionworksjustaswellforarbitraryWesayanalgebraAisZ=dZ-gradedifthereisadecompositionintosubspacesA=A0Ad1withAjAkˆAj+k,wherethesubscriptistakenmodd.Anelement48a2Akiscalledhomogeneousofdegreek.Amap':A!BofZ=dZ-gradedalgebrasiscalledagradedhomomorphismifitisahomomorphismwith'(Aj)ˆBj.Example5.1.1.ThematrixalgebraMd(C)canbegivenaZ=dZ-gradingforwhichthehomogeneouselementsofdegreekarethosematriceswhose(i;j)entryis0unlessjikmodd.Whend=4,wehavegrade0:000000000000grade1:000000000000grade2:000000000000grade3:000000000000WhendjN,wecangiveMN(C)aZ=dZ-gradingasfollows.SinceMN(C)˘=Md(C)Md(C),declareanelementofMN(C)tobehomogeneousifitstensorcomponentsunderthisdecompositionarehomogeneous.ThedegreeofahomogeneouselementofMN(C)isthesum(takenmodd)ofthedegreesofitstensorcomponents.LetfbeadegreedhomogeneouspolynomialinnvariablesoverK,whereKiseitherZorisaWethegeneralizeddalgebraofftobeCf(K)=Khx1;:::xni=(a1x1++anxn)df(a1;:::;an):ai2K;whereKhx1;:::xnidenotesthetensoralgebrainnvariables.IfweinsteadhaveadegreedformonavectorspaceVofdimensionn,theis:Cf(V)=T(V)=hvdf(v):v2Vi;whereT(V)isthetensoralgebraonVandvdmeansvv(d-times).ChoosingabasisforVrecoverstheion.ThetensoralgebraisZ=dZ-gradedbydegree,andsincetheidealthealgebrapreservesthisgrading,wethatCf(K)isZ=dZ-graded:49Cf(K)=C0f(K)C1f(K)Cd1f(K):Itiswell-knownthatwhend3andn2,thisisanalgebra.5.1.1GradedRepresentationsWenowdiscussgradedrepresentationsofalgebras.ForarepresentationCf(C)!MN(C),wherefisaformofdegreed,wenecessarilyhavethatdjN[HT88].Agradedrepresentationisjustagradedhomomorphismbetweenagradedalgebraandamatrixalgebra(whichwewilltaketobegradedasinExample5.1.1).Weconsiderdiagonalforms,thendiscussthegeneralcase.Throughout,weletˆ2Cbeaprimitivedthrootofunity.Supposewehaveadiagonalformofdegreedf(X1;:::;Xn)=a1Xd1++anXdn:LetxidenotethegeneratorsoftheClialgebracorrespondingtoXi.Foreachintegern1,wewillconstructarepresentationofthedalgebraCf(K).SetN=dnandwriteMN(C)=n-timesz}|{Md(C)Md(C):arepresentation'f:Cf(K)!MN(C)bysetting'f(xj)=Aj:=j1z}|{AIIwhere2Md(C)is=diag(1;ˆ;:::;ˆd1);50andA=0BBBBBBBBB@010............001ai001CCCCCCCCCA:(5.1)ItiseasilyvthatˆAkAj=AjAkforj<k.ThefactthatthismapisarepresentationofCffollowsfromrepeateduseofthenextlemma.ThisrepresentationisgradedsinceeachAjhasgrade1(seeExample5.1.1forgradingonmatrixalgebras)andbecausetherelationˆAkAj=AjAkforj<kpreservesthegrading.Lemma5.1.2.LetAbeanassociativeC-algebraandletˆbeaprimitivedthrootofunity.Supposex;y2Asatisfyˆyx=xy.Then(x+y)d=xd+yd.Proof.Wegivetheproofford=4,andsketchthegeneralcase.Byexpanding,weget(x+y)4=x4+(x3y+x2yx+xyx2+yx3)+(x2y2+xy2x+xyxy+yxyx+yx2y+y2x2)+(y3x+y2xy+yxy2+xy3)+y4=x4+y4;wherewehavenotedthat,becauseoftheassumptionsonxandy,thesumofthethreemiddletermsis0.Forthegeneralcase,uponexpanding(x+y)d,weelementsxkydk(monomialswithkx-termsand(dk)y-termsinanyorder)foreachk,1k<d.Themonomialsof(x+y)dcanbepartitionedintosetswhoseelementsareobtainedbycyclicallypermutingthevariables.Forexample,inthed=4case,wehavethatx2y2splitsupintothetwosets51S1=fx2y2;xy2x;y2x2;yx2ygandS2=fxyxy;yxyxg.Cyclicallypermutingthevariablesamountstomultiplyingbyapowerofˆ.Choosingarepresentativeofeachpartitionset,wewriteeachotherelementofthesetasˆtosomepowertimestherepresentative.Nowwesumeachpartitionedsetandthatthecotisjust1+!++!m1forsomeprimitivemthrootofunity!wheremjd.Thusthesumofeachpartitionedsetis0,andhencexkydk=0when1k<d.NowweconstructarepresentationofthealgebraofanarbitrarydegreedformusingthemethodofChapman[Cha13].Supposewehaveaformf(X1;:::;Xn)=Pij0ci1inXi11Xinnofdegreed=Pjij.LetIbethesetofindicesi=i1incorrespondingtothenonzeroci1:::in'softheformf.FixanorderingofthesetI.MM(C)=Ni2IMiwhereMiisacopyofMd(C).Fori=i1in2Iandk2f1;:::;ng,thematricesB;~B2Md(C)asfollows:Bhasci1ininthe(d;1)-entry,1intheentries(1;2);:::;(ik1;ik),and0elsewhere.~Bhas1intheentries(i1++ik1;i1++ik1+1);:::;(i1++ik1;i1++ik)and0elsewhere.Fori=i1in2Iandk2f1;:::;ng,wedematricesBk;i2MM(C)by:Ifik=0,setBk;i=0,thezeromatrix.Ifik6=0andi1==ik1=0,52Bk;i=BII;whereBisintheithcomponentofthetensorproduct.Otherwise,Bk;i=~BII;where~Bisintheithcomponentofthetensorproduct.amap'f:Cf!MM(C)by'f(xk)=Xi2IBk;i:(5.2)Notethatwhenfisdiagonal,thisagreeswiththepreviousof'ffromthebeginningofthissection.Toseewhatishappening,wegiveanon-diagonalexample:Example5.1.3.Letf(X;Y)=X4+2X3Y+5XY3+Y4.Thentherepresentationis'f(x)=0100001000011000+0100001000002000+0000000000005000and'f(y)=0000000000010000+0100001000010000+0100001000011000whereweleaveoutthetrailingI,forexample0100001000011000means0100001000011000IIISince(ax+by)4=f(a;b)inthedalgebra,thatrelationbetterholdintheimageforthistobearepresentation.Tocheckthis,noticethatax+bymapsto530a0000a0000aa000+0a0000a0000b2a000+ 0b0000b0000b5a000!+ 0b0000b0000bb000!NowusingLemma5.1.2,itiseasytoseethatthefourthpowerofthismatrixisf(a;b)I.Wenowproveingeneral,thatthemap(5.2)iswandisagradedrepresentationofthealgebra.Proposition5.1.4.Themap'f:Cf!MM(C)daboveisagradedrepresentation.Proof.Inorderfor'ftobearepresentation,weneedtoshowthata1'f(x1)++an'f(xn)d=f(a1;:::;an)I:Bya1'f(x1)++an'f(xn)d=Pi2Ia1B1;i++anBn;idButbyLemma5.1.2andtheaboveconstruction,Xi2Ia1B1;i++anBn;id=Xi2Ia1B1;i++anBn;id=Xci1in6=0ci1inai11ainnI=f(a1;:::;an)IWearenowleftwithshowingthatthisrepresentationisgraded.Byconstruction,thenonzeroBk;iarehomogeneousofdegree1.Foreachxk2CfcorrespondingtoXkoftheformf,'f(xk)containsatleastonenonzeroBk;iandhenceishomogeneousofdegree1.Byconstruction,therelationsamongtheBk;iaresuchthatthegradingispreserved,hencetherepresentation'fisgraded.54Althoughthisrepresentationisgraded,onemayrepresentationswhicharenotgraded.SinceweareinterestedingradedrepresentationsoftheClialgebra,wecalluponaresultfromChilds[Chi78,Lemma6,p.274].ItsaysthefollowingLemma5.1.5.If'f:Cf!MM(C)isarepresentationsendingxitoAi,thenCf7!MM(C)Md(C)dbyxi7!Ai 010.........110!isagradedrepresentation.5.2AbelianVarietiesOneofthethingsthatmakestheKuga-Satakeconstructionworkistheexistenceoftheanti-involutionofthe(quadratic)algebrabybasiselements:x0xn7!xnx0.Thus,forx=x0xn,(x)xisinthecenterofthealgebrasincex2iiscentral.ThisplaysakeyroleinshowingthattheKuga-Satakecomplextorusisanabelianvariety(see[vG00,Lemma5.8andProposition5.9]andLemma3.2.3).However,forthealgebrasofhigherdegreeforms,wedonothavethatx2iiscentral.Fordegreed3forms,thepping"anti-involutionseemstobeoflittleuse,sinceforexample(x0x1)x0x1=x1x20x1,whichisunlikelytobecentralinthealgebra.ThepurposeofTheorem3.2.7wastoshowthatonceweviewthealgebraofaquadraticformasamatrixalgebra,theconjugatetransposeisasubstitutefor.Wewillshowthatthesameideacarriesovertohigherdegreeforms.Wesomenotation.Fixadegreedformf.SupposewehaveaZ=dZ-gradedrepresentation'f:Cf(C)!MN(C).WemayconsidertheinducedrepresentationsonCf(K)forK2fZ;Q;RgviathenaturalinclusionCf(K)ˆCf(C).Letcf(K)denotethe55imageofCf(K)underthisrepresentation.Sincetherepresentationisgraded,weletc0f(K)denotethedegree0partoftheimage.LetdenotetheconjugatetransposeofamatrixinMN(C).Wemakethefollowingassumption:1.ThereisaJ2c0f(R)sothatleftmultiplicationbyJonc0f(R)acomplexstructure,and:=tJ2c0f(Z)forsomet>0.FurtherJ=J.Asastarttowardconstructinganabelianvariety,weQ:c0f(R)c0f(R)!RbyQ(v;w)=Re(tr(vw))whereisapositivemultipleofJwhichisinc0f(Z)asinassumption(1).NotethatQisR-bilinearsincetakingRe,tr,andallare.WenowprovesomefactsaboutQ.Proposition5.2.1.Q(Jv;Jw)=Q(v;w)forallv;w2c0f(R).Proof.Q(Jv;Jw):=Re(tr((Jv)(Jw)))=Re(tr(JvwJ))=Re(tr((J)Jvw))J=Jandtr(AB)=tr(BA)=Re(tr(vw))Jcommuteswith,andJ2=I=Q(v;w)56Proposition5.2.2.Q(v;Jw)isapositivesymmetricform.Proof.Firstnotethatforanysquare,complexmatrixA=(ai;j),wehaveRe(tr(A))=Re(tr(A))sinceRe(tr(A))=RePiai;i=RePiai;i=Re(tr(A)):NowwecomputeQ(v;Jw)=Re(tr(v(Jw)))=Re(tr((vwJ)))sinceRe(tr(A))=Re(tr(A))=Re(tr(Jwv))=Re(tr(wvJ))=andJcommuteswith=Re(tr(wvJ))J=J=Re(tr(w(Jv)))=Q(w;Jv)sowegetthesymmetricpart.Finally,weneedtocheckthatQ(v;Jv)>0forallnonzerov2c0f(R).WehaveQ(v;Jv)=Re(tr(v(Jv)))=Re(tr(Jvv))=tRe(tr(vv))sinceJ=t,forsomet>0Buttr(AA)>0foranycomplexmatrixA=(ai;j),sincetr(AA)=Pi;jjai;jj2.57Theorem5.2.3.Letfbeadegreedformsuchthatc0f(Z)ˆc0f(R)isafulllatticeandassumetheexistenceofaJasin(1).Furthermore,supposethatQrestrictedtothelatticec0f(Z)c0f(Z)takesvaluesinZ.Thenc0f(R)=c0f(Z)isanabelianvariety.Proof.TheassumptionthatQrestrictedtothelatticetakesvaluesinZcombinedwithPropositions5.2.1and5.2.2showthatQtheRiemannbilinearrelations.Hencethecomplextorusc0f(R)=c0f(Z),withcomplexstructurebyJ,isanabelianvariety.AlthoughthereareseveralassumptionsforTheorem5.2.3,theremainderofthepapershowsthattherearecasesinwhichtheassumptionsare5.3LatticesWenowjustifytherestrictiontothecaseofcubicandquarticformsmadeintheintroductiontothischapter.Proposition5.1.4givesagradedrepresentationofthealgebraCf(overC;R;Q,andZ)intomatricesoverC.Asbefore,letc0f(K)denotethedegree0partoftheimageoftherepresentationofCf(K)forK2fC;R;Q;Zg.Ifwehaveanyhopeofgettingacomplextorusoutofthis,wewillneedc0f(Z)tobeafulllatticeinsideofc0f(R).Ofcourse,thismeansthatc0f(Z)isafreeZ-moduleofrankequaltodimRc0f(R)andthatc0f(Z)R=c0f(R).Weshowthefollowing:Proposition5.3.1.Iffisacubicorquarticform,thenc0f(Z)isafulllatticeinc0f(R)undertherepresentationofSection5.1.1.Thisfailsifdegf>4.Proof.Foracubicform,theelementsofc0f(Z)arematriceswithentriesinZ[!]where!=1+ip32isaprimitivecuberootofunity.Hencec0f(Z)livesinMN(Z[!])forsomeN,whichisafreeZ-module.SincesubmodulesoffreeZ-modulesarethemselvesfree,wehavethatc0f(Z)is58afreeZ-module.Wemustshowthattherankofc0f(Z)isthesameasthedimensionofc0f(R).ButclearlytheR-spanofc0f(Z)isc0f(R),whichshowsrankc0f(Z)dimc0f(R).Conversely,sincef1;!gisalinearlyindependentsetoverR,anyZ-linearlyindependentsetofmatricesinMN(Z[!])willbeR-linearlyindependentaswell.Thisshowsrankc0f(Z)dimc0f(R),andhencec0f(Z)isafulllatticeinc0f(R)whenfisacubicform.Whenfisaquarticform,theargumentisidentical,justreplace!withieverywhere.Forhigherdegreeforms,thelastpartoftheargumentbreaksdown.Asanexample,withfaquinticformand˘aprimitive5throotofunity,theelementsofc0f(Z)areinsideofMN(Z[˘])justasbefore.However,thesetf1;˘;˘2gisZ-linearlyindependent,butnotR-linearlyindependent.Corollary5.3.2.Letfbeacubicorquarticform.SupposethatundertherepresentationofSection5.1.1,thereisanelementJ2c0f(R)satisfyingthepropertiesin(1).Thenc0f(R)=c0f(Z)isanabelianvariety.Proof.BytheconstructionoftherepresentationinSection5.1.1,c0f(Z)forfquartic(re-spectivelycubic)consistsofmatriceswithentriesinZ[i](resp.Z[!]).ThistellsusthatQ(resp.2Q)restrictedtothelatticec0f(Z)c0f(Z)takesvaluesinZ.Proposition5.3.1showsthatc0f(Z)ˆc0f(R)isafulllattice.Thus,c0f(R)=c0f(Z)isanabelianvarietybyTheorem5.2.3.5.4ApplicationsThissectiondealswithsituationsinwhichwecananelementJsatisfyingthecondition(1).Supposef(X1;:::;Xn)isacubicformwithcotsinZoftheshapea1X31+a2X32+g(X3;:::;Xn)59Letx1;:::;xnbethecorrespondinggeneratorsofCf.Thenundertherepresentationcon-structedabovewithxi7!Ai,wehaveJ:=1p32a1a2A1A2A21A22I=2!1p3I=iIWitht=ja1a2jp3,Jclearlytherequirementsforcondition(1)above.Proposition5.3.1andTheorem5.2.3tellusc0f(R)=c0f(Z)isanabelianvariety.Thisworksinparticularwithdiagonalcubicforms(in2variables).Similarly,supposef(X1;:::;Xn)isaquarticformwithcotsinZoftheshapea1X41+a2X42+g(X3;:::;Xn)Letx1;:::;xnbethecorrespondinggeneratorsofCf.Thenundertherepresentationcon-structedabovewithxi7!Ai,wehaveJ:=1a1a2A1A2A31A32=iIThenJ(witht=ja1a2j)clearlytherequirementsforTheorem5.2.3above,andhencec0f(R)=c0f(Z)isanabelianvariety.Thisworksinparticularwithdiagonalquarticforms(in2variables).Remark5.4.1.Inthecasethatf(X;Y)=aX4+bY4isabinarydiagonalfromofdegree4,thegradedrepresentationCf!M16(C)constructedaboveisnotsurjective.However,theimageisthecyclicalgebra(a;b)4;C=Chx;y:x4=a;y4=b;iyx=xyi.Sincewehave(a;b)4;C˘=M4(C),thisgivesaminimal-dimensional,surjective,gradedrepre-sentationofCf.AccordingtoanunpublishedresultofRajeshKulkarni,sucharepresentationisunique(uptoconjugation).Hence,theabelianvarietyweconstructisuniqueinthissense.Theabovediscussionholdsaswellforbinarydiagonalformsofdegree3.605.4.1ExampleConsidertheblowupofP5atapoint.Note4HEisveryample(whereHisthestricttransformofahyperplaneinP5andEistheexceptionaldivisor).LetXbeahyperplanesectionunderthisembedding.ThenH2(X;Z)=torsionistwodimensionalwithintersectionformf=4x4y4,andwemayapplytheaboveconstructiontogetanabelianvariety.5.4.2ExampleThisexampleissimilartothevarietiesconsideredin[Lan98].LetYbeadegreedhyper-surfaceinP4(includingd=1).X=P(E)whereE=OY(n)OY(n)forn1.Letˇ:X!Ybetheprojection,hbeahyperplanesectionofY,letHdenoteˇh,andlet˘betheclassofthedivisorcorrespondingtoOX(1).Wenotethath4=0andsoH4=0.Furthermore,sinceOX(1)restrictedtoaeroveranypointofYisOP1(1)andh3=d[p](where[p]representstheclassofapointinY),weseethatH3:˘=d.WeshowthattheintersectionformonH2(X;Q)isdiagonalizable.NotethatH2(X;Q)=hH;˘iandthattheintersectiontheoryonXisbytheLeray-Hirschformula(see[Har77,AppendixAx3]and[Hat02,Theorem4D.1])˘2(ˇc1):˘+(ˇc2)=0(z)wherec1andc2aretheandsecondChernclassesofthesheafE.61SinceEisadirectsumoflinebundles,itsChernclassesarestraight-forwardtocalculate.TheChernpolynomialisct(E)=(1+(nh)t)(1(nh)t)=1(n2h2)t2Therefore,c1=0andc2=n2h2.Equation(z)thenbecomes˘2=n2H2.Intersectingthiswith:H2givesH2:˘2=n2H4=0H:˘givesH:˘3=n2H3˘=n2d˘2gives˘4=n2H2˘2=0LettingUandVcorrespondtoHand˘,respectively,theintersectionformisf(U;V)=dU3V+n2UV3ThelinearchangeofvariablesU7!nU+nVandV7!UVsendsftotheform2dn3(U4V4),whichisdiagonal.Hence,therepresentationconstructedabovegivesanabelianvariety,A.ThisabelianvarietyisuniqueinthesenseofRemark5.4.1.Furthermore,wehaveaninclusionofHodgestructuresH2(X;Q),!H1(A;Q)H1(A;Q)whichisprovedinthesamewayasProposition3.4.4.5.4.3ExampleWenowconstruct3-foldswhoseintersectionformonH2isdiagonal.Itparallelsthepreviousexample,sowewilluseanalogousnotation.LetYbeadegreedhypersurfaceinP3.ne62X=P(E)whereE=OYOY(n)anyn2Z.Thenct(E)=1+nht,soc1=nhandc2=0.TheintersectiontheoryonXisagainby(z)whichnowlookslike˘2=nH:˘Intersectingthiswith:HgivesH:˘2=nH2:˘=nd˘gives˘3=nH:˘2=n2dsinceH3=0andH2:˘=dasbefore.LettingUandVcorrespondtoHand˘,respectively,theintersectionformisf(U;V)=dU2V+nUV2+n2V3ThelinearchangeofvariablesU7!U+VandV7!nU2nVsendsftotheform3dn2(U3+V3)whichisdiagonal.Hence,therepresentationconstructedabovegivesanabelianvariety,A.ThisabelianvarietyisuniqueinthesenseofRemark5.4.1.Furthermore,wehaveaninclusionofHodgestructuresH2(X;Q),!H1(A;Q)H1(A;Q)whichisprovedinthesamewayasProposition3.4.4.5.5ProductsofSurfacesWenowlookatthecaseof4-foldswhichareproductsofsurfaces.TheKunnethisomorphismgives63H2(XY;Q)˘=Lp+q=2Hp(X;Q)Hq(Y;Q)andthecupproductisdeterminedby(ab)[(cd)=(1)degbdegc(a[c)(b[d):Ifwesupposethath1(X)=0,thenH2(XY)˘=H2(X)H2(Y)andthenegativesigninthecupproductformulaiskilled.Therefore,weseethattheintersection(quartic)formonH2(XY)isjusttheproductoftheintersection(quadratic)formsonH2(X)andH2(Y).Thisleadsustoconsiderquarticformswhichareproductsofquadraticforms.WerecallapropositionfromChilds[Chi78,Theorem8,p.274]Proposition5.5.1.Letf=ghwheregandharequadraticforms.IfCgandChhaverepresentations,thensodoesCf.Proof.Wegiveasketchoftheproof:wemaysupposethatCgandChhaverepresentationsofthesamesize(otherwisetensortherepresentationswithappropriatelysizedmatrixalgebras),sayCg!Mn(C)sendinggeneratorsxitoAiandCh!Mn(C)sendinggeneratorsyj!Bj.ThenweobtainarepresentationofCfbytheblockmatricesxi7! 0Ai0000Ai000000000!andyj7!0@00000000000BjBj0001AThisisarepresentationofsize4n.NotethatweabusivelywritexiandyjforgeneratorsofCf,correspondingtothegeneratorsofCgandCh,respectively.Ofcoursethispropositionistruefortheproductofarbitrarydegreeforms,wejustgavetheprooffortheproductofquadraticformssincethat'sallwewillneed.Wenowspecializetothecasewheref=q2foraquadraticformq.Geometrically,wearethinkingof4-foldsoftheformXXforasurfaceX.64Forthenexttheorem,wesomenotation.Recallthatifwehaveagradedrep-resentationofCf,wewritec0f(R)todenotethedegree0partoftheimageofCf(R).Iffurthermorethedegreeoffiseven,wewritec+ftodenotethesubalgebraoftheimageofCfconsistingofeven-gradedelements.Allofourprevioustheoremsarevalidwithc0freplacedbyc+f.Theorem5.5.2.LetXbeacomplexsurfacewithh1=0;h22,andwithintersectionpairingofsignatureotherthan(1;1).LetfdenotethequarticintersectionformonH2(XX).ThenthereexistsagradedrepresentationofCfforwhichc+f(R)=c+f(Z)isanabelianvariety.Proof.Bythediscussionabove,wehavethatf=q2,whereqistheintersectionformonH2(X).Wemaysupposethatq=a1W21++anW2nisdiagonalwithcotsinZ.Therefore,f=(a1X21++anX2n)(a1Y21++anY2n)Asusual,wewillwritexiandyiandwiasthegeneratorsofthealgebra,corre-spondingtoXiandYiandWi,respectively.WehavetherepresentationofCqfromSection3.1whichwebrecallhere:w17!A1=pa10B@01101CAw27!A2=pa2i0B@01101CA...65wn7!An=0B@10011CA0B@10011CApani0B@01101CAforneven.Whennisoddusetherepresentationfor(n+1)-termsandforgetwn+1.Thismapisinjective.Wemayassumethata1a2>0(duetotherestrictiononthesignatureofq).TheninCq(R),theelementJ=1pa1a2A1A2J2=1andJ=J.UndertherepresentationinProposition5.5.1,weseethat1pa1a2(x1+y1)(x2+y2)7! 00J0000JJ0000J00!CalltheimageelementJ2c2f(R).ThenJ2=I,J=,andpa1a2J2c+f(Z).Thus,left-multiplicationbyJacomplexstructureonc+f(R)satisfying(1).SinceundertherepresentationofCq,Cq(Z)isafulllatticeinsideCq(R),weimmediatelyhavethatc+f(Z)ˆc+f(R)isafulllatticebecauseitisobtainedbytakingblocksoftherepresentationofCq.Finally,QrestrictedtothelatticetakesvaluesinZ.Notethattheonlyelementsofcfwithnonzerotracemustliveinc0f.Elementsofc0fareblockdiagonalmatriceswherethematricesineachblockcomefromtherepresentationofCq.ButtheonlybasiselementsofCqwithnonzerotracearetheconstantintegermultipliesofI(seeLemma3.2.3).ThistellsusthatundertherepresentationofProposition5.5.1,QrestrictedtothelatticetakesvaluesinZ.Therefore,Theorem5.2.3saysthatthequotientc+f(R)=c+f(Z)isanabelianvariety.RecallinTheorem3.2.8,wecreatedanabelianvarietyC+q(R)=C+q(Z)fromaquadraticformqintwoormorevariablesofsignatureotherthan(1;1).66Theorem5.5.3.LetXbeacomplexsurfacewithh1=0;h22,andwithintersectionpair-ingqofsignatureotherthan(1;1).LetfdenotethequarticintersectionformonH2(XX).ThentheabelianvarietyC+q(R)=C+q(Z)isasubquotientoftheabelianvarietyc+f(R)=c+f(Z)fromTheorem5.5.2.Proof.Letf=q2foraquadraticformq=a1W21++anW2nwithintegercots.WeshowthattheimageofCfundertherepresentationfromProposition5.5.1(whichweagaindenotebycf)containsasubalgebrawhichsurjectsontoCq.LetwidenotethegeneratorsofCq,andletxiandyidenotethegeneratorsofCfeachcorrespondingtowi(asintheprevioustheorem).Themapwi7!AidiscussedintheproofoftheprevioustheoremidenCqwithasubalgebraofamatrixalgebra.Therepresentation'ofCffromProposition5.5.1has'(xi+yi)=0@0Ai0000Ai0000AiAi0001ANowthemap :0@0Ai0000Ai0000AiAi0001A7!AiasurjectivemapfromthesubalgebraB=hI;'(xi+yi)icfontoCq.Wedon'tspecifytheoverwhichthealgebraissincethisissurjectivewhenweconsiderthealgebrasoverC;R;Q;orevenZ(weincludeIinthegeneratingsetofthesubalgebra,sinceitmaybenecessaryforsurjectivitywhenconsideringthealgebraoverZ).WewriteBRtodenotethealgebraspannedbytheelementsofBwithcotsinR,andsimilarlyforC,Q,andZ.Assuminga1a2>0andthecomplexstructureonC+q(R)isbyleft-multiplicationbyJ=1pa1a2A1A2,thecomplexstructureonc+f(R)isthenbyJ2c2f(R).We67seethatJ2BRandJ7!J,sothemap iscompatiblewiththecomplexstructures.Therefore,wehaveasurjectivemapofcomplextoriBR=BZ!C+q(R)=C+q(Z).SinceBR=BZisacomplexsubtorusofanabelianvariety,itisitselfanabelianvariety.Thisthetheorem.Example5.5.4.Inparticular,whenXisaK3surface,wegetthattheKuga-SatakeabelianvarietyassociatedtoXisasubquotientofoftheabelianvarietyassociatedtoH2(XX)asinTheorem5.5.2.685.6ConstructioninFamiliesWenowexamineourconstructionforcubicandquarticformsworksinfamilies.Sup-poseVisavariationofHodgestructureofweightkonXwithamultilinearformf,ofdegreed(weassumed=3or4throughout).ThenwecanformthelocalsystemofalgebrasC(V;f)asinthequadraticcase,andwehaveitsgradedcomponents:C0(V;f);:::;Cd1(V;f).Thesetupforperformingourconstructioninfamiliesisthefollowing.SupposewehaveWalocalsystemofcomplexvectorspacesonXforwhichEnd(W)isZ=dZ-graded.ConsideragradedmorphismoflocalsystemsC(VC;fC)!End(W)andconsidertheinducedmapsonC(VR;fR)andC(VZ;fZ).Weletc(V;f)denotetheimageofC(V;f).Assumethatc0(VZ;fZ)isafulllatticeinsideofc0(VR;fR).Furthermore,supposethereisalocally-constantsectionJofc0(VR;fR)suchthatforeachx2X,leftmultiplicationbyJxacomplexstructureontheerc0(VR;fR)x.ThefollowingisarestatementofProposition4.2.1.Proposition5.6.1.Withtheaboveassumptions,c0(V;f)isaVHSonXofweight1.ThemajoryisinconditionsforwhichthisisapolarizedVHSofweight1.Theissueisbeingabletogloballyconjugatetransposeofamatrix.IfEnd(W)isaunitarybundle,weareabletogloballyconjugatetranspose.ByunitarybundlewemeanthereisacoverfUgofXwhichtrivializesthebundleEnd(W)forwhichthetransitionfunctionsaregivenbyconjugationbyaunitarymatrix.Inthiscase,wecanthe\conjugatetranspose"asamap:End(W)!End(W)ofC1-vectorbundles.Todothis,letUandUbetwoopensetsinXinthetrivializingcoverofEnd(W).WewriteUMN(C)andUMN(C)fortheopensetsaboveUandU,respectively,with69transitionfunction'.AboveU\U,considerthefollowingdiagramMN(C)'>MN(C)MN(C)_'>MN(C)_Byassumption,'isgivenbyconjugationbyaunitarymatrixS(i.e.S=S1),hencewehave'(A)=SAS1=(S1)AS=SAS1='(A)sothediagramabovecommutes.Therefore,conjugatetransposeisaglmapwhichisR-linearoners.Proposition5.6.2.AssumethehypothesesofProposition5.6.1andthatEnd(W)isauni-tarybundle.SupposetJisasectionofc0(VZ;fZ)forsomet>0andthatJcondition(1)oners.Furthermore,ifRe(tr(xvw))2Zforanyv;w2c0(VZ;x;fZ;x),thenc0(V;f)isapolarizedVHSofweight1onX.Proof.ThisisjustaglobalversionofTheorem5.2.3.Alloftheassumptionswehavemademaketheresultimmediate.70BIBLIOGRAPHY71BIBLIOGRAPHY[BHPVdV04]WolfBarth,KlausHulek,ChrisPeters,andAntoniusVandeVen.CompactComplexSurfaces.ErgebnissederMathematikundihrerGrenzgebiete[ASeriesofModernSurveysinMathematics].Springer,secondedition,2004.[BHS88]orgenBackelin,JurgenHerzog,andHerbertSanders.Matrixfactorizationsofhomogeneouspolynomials.InAlgebra|somecurrenttrends(Varna,1986),volume1352ofLectureNotesinMath.,pages1{33.Springer,Berlin,1988.[BL04]ChristinaBirkenhakeandHerbertLange.Complexabelianvarieties,volume302ofGrundlehrenderMathematischenWissenschaften[FundamentalPrinci-plesofMathematicalSciences].Springer-Verlag,Berlin,secondedition,2004.[Cat14]EduardoCattani.IntroductiontovariationsofHodgestructure.InHodgethe-ory,volume49ofMath.Notes,pages297{332.PrincetonUniv.Press,Prince-ton,NJ,2014.[CG72]C.HerbertClemensandPhillipA.TheintermediateJacobianofthecubicthreefold.Ann.ofMath.(2),95:281{356,1972.[Cha13]AdamChapman.p-CentralSubspacesofCentralSimpleAlgebras.2013.Thesis(Ph.D.){Bar-IlanUniversity.[Chi78]LindsayN.Childs.Linearizingofn-icformsandgeneralizeddalgebras.LinearandMultilinearAlgebra,5(4):267{278,1977/78.[CKM12]EmreCoskun,RajeshS.Kulkarni,andYusufMustopa.Onrepresentationsofalgebrasofternarycubicforms.InNewtrendsinnoncommuta-tivealgebra,volume562ofContemp.Math.,pages91{99.Amer.Math.Soc.,Providence,RI,2012.[Del72]PierreDeligne.LaconjecturedeWeilpourlessurfacesK3.Invent.Math.,15:206{226,1972.[Gri68]PhillipA.Periodsofintegralsonalgebraicmanifolds.I.Constructionandpropertiesofthemodularvarieties.Amer.J.Math.,90:568{626,1968.72[Gri70]PhillipA.s.Periodsofintegralsonalgebraicmanifolds:Summaryofmainresultsanddiscussionofopenproblems.Bull.Amer.Math.Soc.,76:228{296,1970.[Har77]RobinHartshorne.Algebraicgeometry.Springer-Verlag,NewYork-Heidelberg,1977.GraduateTextsinMathematics,No.52.[Hat02]AllenHatcher.Algebraictopology.CambridgeUniversityPress,Cambridge,2002.[HT88]DarrellHaileandStevenTesser.OnAzumayaalgebrasarisingfromalgebras.J.Algebra,116(2):372{384,1988.[Huy15]DanielHuybrechts.Lecturesonk3surfaces.2015.Availableathttp://www.math.uni-bonn.de/people/huybrech/K3Global.pdf.[IIPS71]I.R.ShafarevichI.I.Pyatetskii-Shapiro.Atorellitheoremforalgebraicsur-facesoftypek3.Math.USSR-Izv.,5(3):547{588,1971.[KS67]MichioKugaandIchir^oSatake.AbelianvarietiesattachedtopolarizedK3-surfaces.Math.Ann.,169:239{242,1967.[Lan98]AdrianLanger.Fano4-foldswithscrollstructure.NagoyaMath.J.,150:135{176,1998.[Mor85]DavidR.Morrison.TheKuga-Satakevarietyofanabeliansurface.J.Algebra,92(2):454{476,1985.[Rev77]Ph.Revoy.Algebresdeetalgebresexterieures.J.Algebra,46(1):268{277,1977.[Rob69]NorbertRoby.Algebresdedesformespolynomes.C.R.Acad.Sci.ParisSer.A-B,268:A484{A486,1969.[Sat66]I.Satake.algebrasandfamiliesofabelianvarieties.NagoyaMath.J.,27:435{446,1966.[Sch73]WilfriedSchmid.VariationofHodgestructure:thesingularitiesoftheperiodmapping.Invent.Math.,22:211{319,1973.73[VdB87]M.VandenBergh.Linearisationsofbinaryandternaryforms.J.Algebra,109(1):172{183,1987.[vG00]BertvanGeemen.Kuga-SatakevarietiesandtheHodgeconjecture.InThearithmeticandgeometryofalgebraiccyclesAB,1998),volume548ofNATOSci.Ser.CMath.Phys.Sci.,pages51{82.KluwerAcad.Publ.,Dordrecht,2000.[Voi05]ClaireVoisin.AgeneralizationoftheKuga-Satakeconstruction.PureAppl.Math.Q.,1(3,part2):415{439,2005.[VS02]C.VoisinandL.Schneps.HodgeTheoryandComplexAlgebraicGeometryI:.Numberv.1inCambridgeStudiesinAdvancedMathematics.CambridgeUniversityPress,2002.74