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Publié par | universitat_duisburg-essen |
Publié le | 01 janvier 2007 |
Nombre de lectures | 57 |
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CyclicandcoverCoingmpls,exCalmultabi-Yiplauicationmanifolds
DissertatDoktorionderzurNatErlaurwisngungsenscdeshaftenGrades
(Dr.rer.nat.)
vorgelegtbeim
FachbereichMathematik
derUniversit¨atDuisburg-Essen
novJanChristianRohde
inneorgebGelsnhecenkir
ZweitErstererGGutautcachhter:ter:ProPriv.-Doz.f.Dr.EDr.ckarMarttinViehMweo¨gller
Datumderm¨undlichenPr¨ufung:14.Dezember2007
2
Acknowledgments
IamverygratefultoEckartViehwegforgivingmethesubjectofthisthesisandhis
excellentguidanceandsupport,notonlyduringtheworkonthistopic,butfromthe
beginningofmymathemathicalstudies.
Moreover,IwouldliketothankespeciallyMartinM¨ollerformanyfruitfulandstimu-
latIingwhinanttstaondthankdiscusKasiongns,ZuoandandforrSteaefadingnMt¨histuller-Stahesischanfodrpotheintinghintouttotsevheeralessaymistak[48]esof.
C.familiesVoisin,of3-whicmahpronifoldsvidedandthefinmallyainofideatheoftconstrheuctionconstructioofthenofBorceamany-Voisinexamplestowofer.CMCY
MainzManyandtothanGeksbhatordUlricBo¨hckleG¨ortforzfhisorinhisterestinginstructreaivedingcoursecourseaboautboutShimShimurauravacurvrietieses.at
Moreover,IwouldliketothankBerndSiebertandClaireVoisinforstimulatingdis-
cussionsaboutmaximalityandtheVoisinmirrorfamilies.
IwishtothankIreneBouw,JuanCervino,AndreChatzistamatiou,JochenHeinloth,
carefulStefanreadingKukuliesofandtheKainytroR¨ducullingtion.formanyfruitfuldiscussions,andPoojaSinglaforthe
Finally,Iwanttothankmyfatherverymuchforhissupportduringthelastyearsand
makinguntroubledstudiespossible.
3
4
Contents
Introduction
7
1AnintroductiontoHodgestructuresandShimuravarieties11
1.1Thebasicdefinitions..............................11
1.2Jacobians,PolarizationsandRiemann’sTheorem..............14
1.3ShimuradataandSiegel’supperhalfplane..................17
1.4TheconstructionofShimuravarieties.....................21
1.5ShimuravarietiesofHodgetype........................22
2Cycliccoversoftheprojectiveline27
2.1Descriptionofacycliccoveroftheprojectiveline..............27
2.2Thelocalsystemcorrespondingtoacycliccover...............29
2.3Thecohomologyofacover...........................32
2.4Cycliccoverswithcomplexmultiplication..................33
3Somepreliminariesforfamiliesofcycliccovers37
3.1Thetheoreticalfoundations..........................37
3.2Familiesofcoversoftheprojectiveline....................38
3.3Thehomologyandthemonodromyrepresentation..............40
4TheGaloisgroupdecompositionoftheHodgestructure43
4.1TheGaloisgrouprepresentationonthefirstcohomology..........43
4.2QuotientsofcoversandHodgegroupdecomposition.............47
4.3UpperboundsfortheMumford-Tategroupsofthedirectsummands....48
4.4Acriterionforcomplexmultiplication.....................50
5ThecomputationoftheHodgegroup53
5.1Themonodromygroupofaneigenspace...................53
5.2TheHodgegroupofageneraldirectsummand................59
5.3Acriterionforthereachingoftheupperbound...............61
5.4Theexceptionalcases..............................65
5.5TheHodgegroupofauniversalfamilyofhyperellipticcurves........68
5.6ThecompletegenericHodgegroup......................72
6Examplesoffamilieswithdensesetsofcomplexmultiplicationfibers77
6.1ThenecessaryconditionSINT........................77
6.2TheapplicationofSINTforthemorecomplicatedcases..........83
6.3Thecompletelistsofexamples.........................89
6.4ThederivedvariationsofHodgestructures..................90
5
7TheconstructionofCalabi-Yaumanifoldswithcomplexmultiplication95
7.1Thebasicconstructionandcomplexmultiplication.............95
7.2TheBorcea-Voisintower............................98
7.3TheViehweg-Zuotower............................100
7.4Anewexample.................................102
8Thedegree3case105
8.1Prelude.....................................105
8.2AmodifiedversionofthemethodofViehwegandZuo............109
8.3Theresultingfamilyanditsinvolutions....................112
9Otherexamplesandvariations115
9.1Thedegree3case................................115
9.2Calabi-Yau3-manifoldsobtainedbyquotientsofdegree3..........117
9.3Thedegree4case................................121
9.4Involutionsonthequotientsofthedegree4example............122
9.5Theextendedautomorphismgroupofthedegree4example.........125
9.6Theautomorphismgroupofthedegree5examplebyViehwegandZuo..126
10ExamplesofCMCYfamiliesof3-manifoldsandtheirinvariants129
10.1ThelengthoftheYukawacoupling......................129
10.2Examplesobtainedbydegree2quotients...................130
10.3TheExampleobtainedbyadegree3quotientanditsmaximality.....131
10.4Outlookontoquotientsbycyclicgroupsofhighorder............134
11MaximalfamiliesofCMCYtype137
11.1FactsaboutinvolutionsandquotientsofK3-surfaces............137
11.2TheassociatedShimuradatumofD.....................138
11.3Theexamples..................................140
Bibliography145
814Index6
Introduction
WesearchforexamplesoffamiliesofCalabi-Yaumanifoldswithdensesetofcomplex
multiplicationfibersandforexamplesoffamiliesofcurveswithdensesetofcomplex
ers.fibtionultiplicamBystringtheoreticalconsiderations,oneisinterestedinCalabi-Yaumanifolds,since
Calabi-Yau3-manifoldsprovideconformalfieldtheories(CFT).Oneisespeciallyinter-
estedinCalabi-Yau3-manifoldswithcomplexmultiplication,sincesuchamanifoldhas
manysymmetriesandmirrorpairsofCalabi-Yau3-manifoldswithcomplexmultiplica-
tionyieldrationalconformalfieldtheories(RCFT)(see[18]).MoreoverS.Gukovand
C.Vafa[18]askfortheexistenceofinfinitelymanyCalabi-Yaumanifoldswithcomplex
multiplicationoffixeddimensionn.
ForaCalabi-YaumanifoldXofdimensionnwithn≤3,theconditionofcomplex
multiplicationisequivalenttothepropertythatforallktheHodgegroupofHk(X,C)
iscommutative.WewillcallanyfamilyofCalabi-Yaun-manifolds,whichhasadense
setoffiberssatisfyingthelatterpropertywithrespecttotheHodgegroups,aCMCY
familyofn-manifolds.Theauthorusesthisconditionfortechnicalreasonsandhopes
thatsuchaCMCYfamilyofn-manifoldsinanarbitrarydimensionmaybeinteresting
foritsmathematicalbeauty,too.HerewewillgivesomeexamplesofCMCYfamiliesof
3-manifoldsandexplainhowtoconstructCMCYfamiliesofn-manifoldsinanarbitrarily
dimension.highStartingwithafamilyofcycliccoversofP1withadensesetofCMfibers,E.Viehweg
andK.Zuo[46]haveconstructedaCMCYfamilyof3-manifolds.Thisconstructionof
E.ViehwegandK.Zuo[46]isgivenbyatowerofcycliccoverings,whichwillbeexplained
inSection7.3.InChapter8wewillgiveamodifiedversionofaViehweg-Zuotowerfor
oneofournewexamples.
HenceweareinterestedintheexamplesoffamiliesofcurveswithadensesetofCM
fibersbyoursearchforCMCYfamiliesofn-manifolds.Butthereisanothermotivation
givenbyanopenquestioninthetheoryofcurves,too.In[10]R.Colemanformulated
thefollowingconjecture:
Conjecture1.Fixanintegerg≥4.Thenthereareonlyfinitelymanycomplexalgebraic
curvesCofgenusgsuchthatJac(C)isofCMtype.
LetPndenotetheconfigurationspaceofn+3pointsinP1.Onecanendowthese
n+3pointsinP1withlocalmonodromydataandusethesedatafortheconstructionof
afamilyC→PnofcycliccoversontoP1(seeConstruction3.2.1).
TheactionofPGL2(C)onP1yieldsaquotientMn=Pn/PGL2(C).Byfixing3points
onP1,thequotientMncanalsobeconsideredasasubspaceofPn.
Remark2.In[25]J.deJongandR.Nootgavecounterexamplesforg=4andg=6to
theconjectureabove.In[46]E.ViehwegandK.Zuogaveanadditionalcounterexample
7
forg=6.ThecounterexamplesaregivenbyfamiliesC→PnofcycliccoversofP1with
infinitelymanyCMfibers.HerewewillfindadditionalfamiliesC→Pnofcyclicgenus
5andgenus7coversofP1withdensesetsofcomplexmultiplicationfibers,too.
AllnewexamplesC→PnoftheprecedingremarkhaveavariationVofHodge
structuressimilartotheexamplesofJ.deJongandR.Noot[25],andofE.Viehweg
andK.Zuo[46],whichwecallpure(1,n)−VHS.LetHg(V)denotethegenericHodge
groupofVandletKdenoteanarbitrarymaximalcompactsubgroupofHgad(V)(R).In
Section4.4weprovethatapure(1,n)−VHSinducesanopen(multivalued)periodmap
tothesymmetricdomainassociatedwithHgad(V)(R)/K,whichyieldsthedensesetsof
complexmultiplicationfibers.WeobtainthefollowingresultinChapter6:
Theorem3.Thereareexactly19familiesC→PnofcycliccoversofP1,whichhavea
pure(1,n)−VHS(includingallknownandnewexamples).
Wewillderusethefactthatthemonodr0omygroupMon0(V)isasubgroupofthederived
groupHg(V)andwewillstudyMon(V).LetψbeageneratoroftheGaloisgroup
ofC→PnandC(ψ)bethecentralizerofψinthesymplecticgroupwithrespecttothe
intersectionpairingonanarbitraryderfiberofC.In0Chapter4weobtaintheresult,which
willbeusefulforourstudyofHg(V)andMon(V):
Lemma4.ThemonodromygroupMon0(V)andthederivedHodgegroupHgder(V)are
containedinC(ψ).
UnfortunelywewillnotbeabletodetermineMon0(V)forallfamiliesC→Pnofcyclic
coversontoP1.ButweobtainforexamplethefollowingresultsinChapter5:
Proposition5.LetC→PnbeafamilyofcycliccoversofdegreemontoP1.Thenone
has:1.Ifthederdegreemisaprimenumber≥3,thealgebraicgroupsCder(ψ),Mon0(V)and
Hg(V)coincide.
2.IfC→P2g+2isafamilyofhyperellipticcurves,oneobtains
Mon0(V)=Hg(V)=∼SpQ(2g).
3.InthecaseofafamilyofcoversontoP1with4branchpoints,weneedapure
(1,1)−VHStoobtainanopenperiodmaptothesymmetricdomainassociated
withHgad(V)(R)/K.
ByournewexamplesofViehweg-Zuotowers,wewillonlyobtainCMCYfamilies
of2-manifolds.C.Voisin[48]hasdescribedamethodtoobtainCalabi-Yau3-manifolds
byusinginvolutionsonK3surfaces.C.Borcea[8]hasindependentlyarrivedatamore
generalversionofthelattermethod,whichallowstoconstructCalabi-Yaumanifoldsin
arbitrar