Cyclic coverings, Calabi-Yau manifolds and complex multiplication [Elektronische Ressource] / von Jan Christian Rohde
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Cyclic coverings, Calabi-Yau manifolds and complex multiplication [Elektronische Ressource] / von Jan Christian Rohde

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Cyclic coverings, Calabi-Yau manifoldsand Complex multiplicationDissertation zur Erlangung des GradesDoktor der Naturwissenschaften(Dr. rer. nat.)vorgelegt beimFachbereich Mathematikder Universit¨at Duisburg-EssenvonJan Christian Rohdegeboren inGelsenkirchenErster Gutachter: Prof. Dr. Eckart ViehwegZweiter Gutachter: Priv.-Doz. Dr. Martin Mo¨llerDatum der mun¨ dlichen Prufung¨ : 14. Dezember 20072AcknowledgmentsI am very grateful to Eckart Viehweg for giving me the subject of this thesis and hisexcellent guidance and support, not only during the work on this topic, but from thebeginning of my mathemathical studies.Moreover, I would like to thank especially Martin M¨oller for many fruitful and stimu-lating hints and discussions, and for reading this thesis and pointing out several mistakes.I want to thank Kang Zuo and Stefan Muller-Sta¨ ch for the hint to the essay [48] ofC. Voisin, which provided the main idea of the construction of many examples ofCMCYfamilies of 3-manifolds and finally of the construction of the Borcea-Voisin tower.Many thanks to Ulrich G¨ortz for his instructive course about Shimura varieties atMainz and to Gebhard B¨ockle for his interesting reading course about Shimura curves.Moreover, I would like to thank Bernd Siebert and Claire Voisin for stimulating dis-cussions about maximality and the Voisin mirror families.

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Publié par
Publié le 01 janvier 2007
Nombre de lectures 57

Extrait

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

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