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SINGULARINTERSECTIONSOFSUBGROUPSAND
CHARACTERVARIETIES
JULIENMARCHEANDGUILLAUMEMAURIN´
varietiesof3--manifoldwithtoric
boundaryMsatisfyingsometechnicalhypotheses,weprovethat
allbutaflnitenumberofitsDehnflllingsMp/qaregloballylo-
callyrigidinthefollowingsense:everyirreduciblerepresenta-
tionρ:π1(Mp/q)→SL2(C)isinflnitesimallyrigid,meaningthat
H1(Mp/q,Adρ)=0.
Thisquestionarosefromthestudyofasymptoticsproblemsin
topologicalquantumfleldtheorydevelopedin[2].Theproofre-
liesheavilyonrecentprogressindiophantinegeometryandraises
newquestionsofZilber-
thatagenericcurvelyinginaplanemultiplicativetorusinter-
aneffectiveresultofthisform,basedmainlyonaheightupper
boundofHabegger.
LetMbeacompactconnectedoriented3-manifoldwithoutbound-
,thequantumChern-Simonsthe-
oryassociatedtothegroupSU2andthelevelkgivesaninvariant
Zk(M)∈CcalledWitten-Reshetikhin--
antwasintroducedin[14]asapathintegral,andconstructedrigor-
ouslybyReshetikhinandTuraevusingtherepresentationtheoryofthe
arXiv:[]24Jun2014
quantumgroupUqsl2,see[12].Formally,onecanwrite
Z
Z(M)=eikCS(A)DA.
k
Inthisexpression,Aisa1-formonMwithvaluesintheLiealgebra
su2and
Z
12
CS(A)=−Tr(A∧dA+A∧[A∧A]).
4πM3
::57M25.
-Pinkconjecture,Charactervarieties,Dehnfllling,
Rigidity.
1:.
2MARCHEANDMAURIN´
ThemeasureDAisofcourseill-deflnedbutWittenunderstood
itscut-and-pastepropertiesfromwhichReshitikhinandTuraevcon-
phaseexpansiontothispathintegral,itlocalizesaroundthecritical
pointsoftheChern-Simonsfunctionalwhichcorrespondtotheflat
connections,thatis1-formsAsatisfyingdA+1[A∧A]=
2
equivalenceclassesofsuchconnectionscorrespondtoconjugacyclasses
ofrepresentationsρ:π1(M)→
followingasymptoticexpansion:
Xiπp
()Z(M)=e4m(ρ)+ikCS(ρ)T(M,ρ)+O(k−1/2).
k
ρ
Inthisformula,ρrunsovertheconjugacyclassesofirreduciblerep-
resentationsfromπ1(M)toSU2,m(ρ)isanelementofZ/8Zcalled
spectralflowandT(M,ρ)istheReidemeistertorsionofMtwistedby
therepresentationAdρofπ1(M)onsu2.
Thisformulaisprovedinveryfewcases,oneofthedifficultiesbe-
ingthattheReidemeistertorsionisdeflnedonlyforthoseirreducible
representationsρforwhichthespaceH1(M,Ad)
ρ
H1(M,Ad)canbeidentifledtotheZariskitangentspaceofthechar-
ρ
actervarietyX(M)atχρ(seeSection2).Hence,anecessarycondition
fortheWittenasymptoticformulatomakesenseisthatthecharacter
varietyisreducedofdimension0.
IfMisacompact,connectedandoriented3-manifoldwithtoric
boundary,wecallDehnsurgerytheresultofgluingbacktoMasolid
∂Mto∂D2×S1reversing
manifoldM∪D2×S1onlydependsonthehomotopyclassofthe
φ
simplecurveγ=φ−1(S1×{1})⊂∂
γ
this3-manifoldwithoutboundaryandcallittheDehnflllingofMwith
slopeγ.
In[2],CharlesandtheflrstauthorprovedthattheWittenconjec-
tureholdsforMifM=S3\V(K)isthecomplementofatubular
γ
neighborhoodoftheflgureeightknotK,thelinkingnumberlk(γ,K)
isnotdivisibleby4andthecharactervarietyX(Mγ)isreducedof
anyknotprovidedonehasastrongversionoftheAJ-conjectureand
someinformationontheReidemeistertorsion.
Theconditiononthecharactervarietytobereducedappearedasa
technicalpointwhichwashardtocheckeveninthecaseoftheflgure
,weproveinthisarticlethatforabroadclass:.
SINGULARINTERSECTIONS3
ofvarietiesM,thisconditionissatisfledforallbutaflnitenumberof
,weshow:
-
ducible3-manifoldwithtoricboundarysuchthat
(i)Themapr:X(M)→X(∂M)inducedbytheinclusion∂M⊂
Misproper.
(ii)ThecharactervarietyX(M)isreduced.
(iii)TheimagebyrofthesingularpointsofX(M)arenottorsion
pointsofX(∂M)(seeSection2).
Thenforallbutaflnitenumberofslopesγ,thevarietyX(Mγ)is
,thenumberofexceptionscanbe
effectivelybounded.
Itiswell-knownthatX(∂M)isthequotientofa2-dimensionaltorus
G2bytheinvolutionσ(x,y)=(x−1,y−1).Denotebyπ:G2→
mm
X(∂M),theva-
rietyC=π−1r(X(M))isaplanecurvedeflnedbytheso-calledA-
polynomial,see[4].Thefollowingnotionswillbecentraltotheproof
.
′
m
intersectsC′transversallyatP∈C∩C′ifthetwocurvesaresmooth
C∩C′ofCandC′asthesetofallpointsP∈C∩C′wherethe
sing
twocurvesaresmoothwithequaltangentlines.
Foranycoupleofrelativelyprimeintegers(p,q),letHp,qbethe
subtorusofG2deflnedbytheequationxpyq=
m
standardargumentincharactervarieties,wereducetheproofofthe
previoustheoremtoshowthatCintersectstransversallyHp,qforalmost
all(p,q).
Thisfactisconnectedtorecentquestionsindiophantinegeometry
surroundingtheZilber-,itfollowsfrom
the1999boundedheightpropertyofBombieri,MasserandZannier
(seeTheorem1in[1]).Effectiveversionsofthelatterwereworkedout
byHabeggerovertheyears(seeappendixB1of[5],Theorem7in[6]
and[7]),allowingustogiveanexplicitupperboundonthemaximal
sizeofacouple(p,q)suchthatChasnon-emptysingularintersection
withatranslateofHp,q.
ThisupperboundmightbeofinterestfortheapplicationsofThe-
-
putedfromanequationf(x,y)=
m:.
4MARCHEANDMAURIN´
fisinvolvedthroughitstotalandpartialdegreesanditslogarithmic
Weilheighth(f)(seesection3forthedeflnition).
(x,y)=0
foranirreduciblepolynomialf∈Q¯[X,Y].Letδ=deg(f),δx=
degx(f)andδy=degy(f).AssumeCisnotatranslateofasubtorus.
Then,foranytranslateγHp,qwithnon-emptysingularintersectionwith
C,thequantitymax(|p|,|q|)isatmost
354
δexp(+1)δmax(δxδy,h(f)).
Inparticular,theunionC{1}ofallsingularintersectionsoftheform
C∩singHp,qisaflniteset.
Wealsoproveamildstrengtheningofthelastsentencethatlooks
likeaperfectanalogueoftheZilber-Pinkconjectureinthecontextof
planesingularintersections.
Initsmultiplicativeform–thatis,whentheambientspaceisamul-
tiplicativetorusT=Gn–theZilber-Pinkconjecturepredictswhat
m
happenstoasubvaritetyXwhenintersectedtotheunionofallal-
gebraicsubgroupsofflxedcodimensionm(see[16]and[11]forthe
originalconjecturesand[15]forarecentpanoramaofthesubject).
UndertheassumptionthatXisnotcontainedinaproperalgebraic
subgroupofT,itisthestatementthatthesubsetsX[m]ofXdeflned
by
[
()X[m]=X(Q¯)∩ζH(Q¯)
codimH=m
ζtorsion
arenotZariski-denseinXform≥dimX+1,wheretheunionruns
overallsubtoriHofcodimensionmandalltorsionpointsζofT.
Notethat,intheparticularcaseofacurveClyinginG2,theas-
m
sumptiononCmeanspreciselythatCisnotatranslateofasubtorus
,
thisisweaker,yetitturnsouttobesufficientfortheflnitenessofC{1}.
Underthisassumption,wecanevenprovetheflnitenessofaslightly
()forC[1]
bychangingallintersectionsforsingularintersections.
m
subtorusbyatorsionpoint,then
[
C{1,tor}=C(Q¯)∩ζH(Q¯)
singp,q
p∧q=1
ζtorsion:.
SINGULARINTERSECTIONS5
isaflnitesubsetofC.
Itiswellknownthat,intheZilber-Pinkconjecture,thecodimension
valuem=dimX+1isoptimalforZariskinon--
creasedfurther,thenX[m]containsX[dimX]thatisdenseinXforallX.
Inthisrespect,
multiplicityofintersectioncanmakeupforacodimensiondropamong
theH’s:goingfromC[2]tothelargersubsetC[1]generatesinflnitely
manynewpoints,butrestrictingtothecaseofpositivemultiplicity
yieldsC{1,tor}andflnitenessisrecovered.
Thislineofthoughtgoesfurtherthanthecaseofplanecurvesand
makessenseinamoregeneralframework,leadingtonewconjecturesof
Zilber-
article.
Finally,thelasttopicwestudyhereistherelationbetweensubsetsof
theformC{1,tor}andC[2],showingthattheflrstcanbeseenasasubset
ofthesecondtypeforaZilber-Pink-likeproblemthattakesplaceina
slightlydifferentambientspace().
Acknowledgments:,
questionrelatedtocharactervarietiesofDehnflllingwassolvedwith
[9].
ittous.
(Γ)thealgebraic
varietyofallrepresentationsρ:Γ→SL2(Q¯).Thisvarietyisgenerally
adoptherethefleldQ¯
groupSL(Q¯)actsonR(Γ)=gρg−1:wedenotethealgebraic
2
quotientbyX(Γ)=R(Γ)//SL2(Q¯).Wereferto[10,3]forthegeneral
theoryandcollectheresomefacts.
(i)Givenarepresentationρ∈R(Γ)wedeflneitscharacterχρ:
Γ→Q¯bytheformulaχρ(γ)=Trρ(γ).Asaset,X(Γ)is
thequotientofR(Γ)bytherelationρ∼ρ′iffχ=χ′.This
ρρ
justiflesthenamecharactervariety.
(ii)Ifρ,ρ′aretwoelementsofR(Γ)withχ=χ′andρirre-
ρρ
ducible,thenρandρ′areconjugated.
(iii)ThealgebraofregularfunctionsonX(Γ)isgeneratedbythe
so-calledtracefunctionsdeflnedforanyγ∈Γbyfγ(ρ)=
Trρ(γ).:.
6MARCHEANDMAURIN´
(iv)Arepresentationρ∈R(Γ)isreducibleifandonlyifforall
α,β∈Γonehasf[α,β](ρ)=
charactersisZariski-closedinX(Γ)andisdenotedbyXred(Γ)
whereasitscomplementisdenotedbyXirr(Γ).
(v)Atanirreduciblerepresentationρ,thereisanaturalisomor-
phismTX(Γ)≃H1(Γ,Ad).
χρρ
(vi)IfΓ=Z2,weconsiderthemorphismπ:G2→X(Γ)mapping
m
(x,y)tothecharacteroftherepresentationρx,ydeflnedby
ab
xy0
ρx,y(a,b)=−a−b.
0xy
Itiswell-knownthatπinducesanisomorphismbetweenthe
quotientofG2bytheinvolutionσ(x,y)=(x−1,y−1)and
m
X(Γ).Inparticular,wewilldenotebyX(Γ)tortheimagebyπ
ofthetorsionpointsofG2.
m
(vii)Ifφ:Γ→Γ′isagrouphomomorphism,itinducesanalgebraic
morphismφ∗:X(Γ′)→X(Γ).
IfMisaconnectedcompactorientedmanifold,wesetX(M)=
X(π1(M)).IfMisasurfaceora3-,then
itisanEilenberg-Maclanespace,whichmeansthatthereisanatural
isomorphismH1(π(M),Ad)≃H1(M,Ad).Leti:∂M→Mbe
1ρρ
theinclusionmorphism:itinducesamapi∗:π1(∂M)→π1(M).We
denotebyrthemap(i)∗:X(M)→X(∂M)inducedbytheinclusion
∗
andcallittherestrictionmap.
IfMisaconnectedcompactoriented3-manifoldwithtoricboundary,
onecanunderstandrepresentationsofMγforagivenslopeγ⊂∂Min
thefollowingway:byVan-Kampentheorem,thefundamentalgroup
ofMistheamalgamatedproductπ(M)∗π(D2×S1).Moreover,
11
π1(∂M)
themapπ(∂M)→π(D2×S1)issurjectivewithkernelgeneratedby
11
γhenceonehasπ1(Mγ)=π1(M)/hγiwherehγiisthenormalclosure
ofγ.
Inparticular,arepresentationρ:π1(Mγ)→SL2(Q¯)isthesameas
arepresentationofρ:π1(M)→SL2(Q¯)suchthatρ(γ)=
ofcharactervarieties,X(Mγ)fltsinthefollowingdiagram(whichmay
notbecartesian)::.
SINGULARINTERSECTIONS7
X(Mγ)
t◆◆◆
ttt◆◆◆
tt◆◆◆
yyttt◆◆&&
X(M)X(D2×S1)
❏❏♣♣
❏❏❏rr′♣♣♣
❏❏♣♣♣
❏❏%%ww♣♣♣
X(∂M)
Theimageofr′istheprojectionofasubtorusofG2bythemapπ.
m
π−1r(X(M)).
m
-
,theneveryirreduciblecomponentofX(M)hasdimension
1.
,wedenotethelocalsys