ON THE CLASSIFICATION OF RANK TWO REPRESENTATIONS OF QUASIPROJECTIVE FUNDAMENTAL GROUPS

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ar X iv :m at h/ 07 02 28 7v 2 [m ath .A G] 2 7 F eb 20 07 ON THE CLASSIFICATION OF RANK TWO REPRESENTATIONS OF QUASIPROJECTIVE FUNDAMENTAL GROUPS KEVIN CORLETTE AND CARLOS SIMPSON Abstract. Suppose X is a smooth quasiprojective variety over C and ? : pi1(X,x) ? SL(2,C) is a Zariski-dense representation with quasiunipotent monodromy at infinity. Then ? factors through a map X ? Y with Y either a DM-curve or a Shimura modular stack. 1. Introduction Let X be a connected smooth quasiprojective variety over C with basepoint x. We look at representations ? : π1(X, x) ? SL(2,C). We assume throughout that the monodromy at infinity is quasi-unipotent. If X ? X is a normal-crossings compactification with com- plementary divisor D = ∑ Di, and if ?i are loops going around the components Di, this condition means that the ?(?i) are quasi-unipotent, in other words their eigenvalues are roots of unity. A representation ? is Zariski-dense if the Zariski-closure of its image is the whole group SL(2,C). A reductive representation of rank two is either Zariski-dense, or else it becomes reducible upon pullback to a finite unramified covering of X.

• zariski


• group action


• dense representation


• shimura modular


• deligne-mumford stack


• overlap between


• tautological representations into


• then ?


• any scheme then


• dm-curve

Sujets

• Chicago
• Group action

Informations

• Publié par: pefav
• Nombre de lectures: 6
• Langue: English

Extrait

ONTHECLQAUSASSIIFPICRAOTJIEOCNTIOVFERFUANNDKATMWEONTRAELPRGERSOEUNPTSATIONSOF

KEVINCORLETTEANDCARLOSSIMPSON

Abstract.
Suppose
X
isasmoothquasiprojectivevarietyover
C
and
ρ
:
π
1
(
X,x
)
→
SL
(2
,
C
)isaZariski-denserepresentationwithquasiunipotentmonodromyatinfinity.Then
ρ
factorsthroughamap
X
→
Y
with
Y
eitheraDM-curveoraShimuramodularstack.

1.
Introduction
Let
X
beaconnectedsmoothquasiprojectivevarietyover
C
withbasepoint
x
.Welook
atrepresentations
ρ
:
π
1
(
X,x
)
→
SL
(2
,
C
).Weassumethroughoutthat
themonodromy
plementarydivisor
D
=
D
i
,andif
γ
i
areloopsgoingaroundthecomponents
D
i
,this
atinfinityisquasi-unipote
P
nt
.If
X
⊂
X
isanormal-crossingscompactificationwithcom-
conditionmeansthatthe
ρ
(
γ
i
)arequasi-unipotent,inotherwordstheireigenvaluesare
rootsofunity.
Arepresentation
ρ
is
Zariski-dense
iftheZariski-closureofitsimageisthewholegroup
SL
(2
,
C
).AreductiverepresentationofranktwoiseitherZariski-dense,orelseitbecomes
reducibleuponpullbacktoafiniteunramifiedcoveringof
X
.Wewillclassifyrepresentations
ρ
whichareZariski-denseandhavequasi-unipotentmonodromyatinfinity.See[9][5][36]
[39][47]forasimilarclassificationinthereduciblecase.
Thegeometryofthefundamentalgroupofanalgebraicvarietyhasbeenstudiedfrommany
differentangles[2][4][5][6][19][22][41][53][59][62][71][97][104][105][126][128].The
methodswewilluseherearebasedonthetheoryofharmonicmappings,bothtosymmetric
spacesandcombinatorialcomplexes[26][28][35][42][46][54][60][61][65][66][69][86][91]
[113][117][118][132].
Ourclassificationisobtainedbylookingattheinterplaybetweendifferentpropertiesof
ρ
.Themainpropertyis
factorization
:wesaythat
ρ
factorsthroughamap
f
:
X
→
Y
ifitisisomorphictothepullbackofarepresentationof
π
1
(
Y,f
(
x
)).Thisnotioncanbe
extendedinacoupleofways,forexample
ρ
projectivelyfactors
through
f
iftheprojected
representationinto
PSL
(2
,
C
)factorsthrough
f
.Theotherextensionisthatitisconvenient
(andbasically–almostessential—tolookatthenotionoffactorizationthroughmapsto
Deligne-Mumfordstacks
Y
ratherthanjustvarieties.Inacertainsensethistakestheplace
ofcomplicatedstatementsinvolvingcoveringsof
X
.Itevensubsumesthenotionofprojective
factorization,becauseprojectivefactorizationisequivalenttofactorizationthroughanew
DM-stackobtainedbyputtingastackstructurewithgroup
Z
/
2(thecenterof
SL
(2
,
C
))
overthegenericpointof
Y
.
Keywordsandphrases.
Fundamentalgroup,Representation,Harmonicmap,Tree,Deligne-Mumford
stack,Shimuravariety.
1

2K.CORLETTEANDC.SIMPSON
Oneofthemaincasesoffactorizationweshallbeconcernedwithisfactorizationthrough
acurve.Asmoothone-dimensionalDM-stackwillbecalleda
DM-curve
.Recallthatan
orbicurve
isaDM-curvewhosegenericstabilizeristrivial.Anorbicurveisgivenbythe
dataofasmoothcurvetogetherwithacollectionofmarkedpointsassignedinteger(
≥
2)
weights.FactorizationthroughaDM-curveisequivalenttoprojectivefactorizationthrough
anorbicurve(Corollary3.3).
Theothercaseweneedtoconsiderariseswhentherepresentationismotivic,infactcomes
fromafamilyofabelianvarieties.Thefamiliesofabelianvarietieswhosemonodromyrep-
resentationsbreakupintoranktwopiecesaregivenbymapstocertainShimuravarieties
orstacks.TheseShimuravarietiesarecloselyanalogoustoHilbertmodularvarieties.How-
ever,Hilbertmodularvarietiesparametrizeabelianvarietieswithrealmultiplication,while
ingeneralweneedtolookatabelianvarietieswithmultiplicationbyatotallyimaginary
extensionofatotallyrealfield.Theconditionthatthetautologicalrepresentationgoesinto
SL
(2)basicallysaysthattheuniversalcoveringoftheShimuravarietyisapolydiski.e.
aproductofone-dimensionaldisks.Weworkwithoutlevelstructureandcallthesethings
polydiskShimuramodularDM-stacks
.AclassicalexampleisthecaseofShimuracurves.The
preciseconstructionwillbereviewedin
§
9below.If
H
isapolydiskShimuraDM-stackthen
π
1
(
H
)hasatautologicalrepresentationinto
SL
(2
,L
)foratotallyimaginaryextension
L
of
atotallyrealfield,andthisgivesacollectionoftautologicalrepresentationsinto
SL
(2
,
C
)
indexedbytheembeddings
σ
:
L
→
C
.
ClassifyingourranktwoZariski-denserepresentationsquasi-unipotentatinfinity,will
consistthenofshowingthatanysuchrepresentationfactorsthroughamap
f
:
X
→
Y
,
with
Y
beingeitheraDM-curve,orelseapolydiskShimuramodularDM-stack.Weconsider
as“known”therepresentationsonthesetargetstacks
Y
.Theremaybesomeoverlapbetween
thesetwocases,butoneofourbasictasksistohavepropertieswhichdeterminewhichcase
oftheclassificationwewillwanttoproveforagivenrepresentation.
Sincewearelookingatrepresentationsonquasiprojectivevarieties,wedefine
rigidity
in
awaywhichtakesintoaccountthemonodromyatinfinity.Fixanormalcrossingscom-
pactificationof
X
.Foreachcomponent
D
i
ofthedivisoratinfinity,wehaveawell-defined
conjugacyclassofelementsof
π
1
(
X,x
)correspondingtoaloop
γ
i
goingaroundthatcompo-
nent.Thusforagivenrepresentation
ρ
thisgivesaconjugacyclass
C
i
inthetargetgroup.We
areassumingthatthesemonodromyelementsarequasi-unipotent,so
C
i
isaquasi-unipotent
conjugacyclass.Wecandefineanaffinevariety
R
(
X,x,SL
(2)
,
{
C
i
}
)ofrepresentationssuch
thatthemonodromies
ρ
(
γ
i
)arecontainedintheclosuresofthe
C
i
.Let
M
(
X,SL
(2)
,
{
C
i
}
)
denoteitsuniversalcategoricalquotientbytheconjugationaction.Wesaythat
ρ
is
rigid
if
itrepresentsanisolatedpointinthemodulispace
M
(
X,SL
(2)
,
{
C
i
}
)obtainedbylookingat
itsownconjugacyclasses.InthecaseofaZariski-denserepresentation,thisformofrigidity
meansthatthereisnonon-isotrivialfamilyofrepresentationsallhavingthesameconjugacy
classesatinfinity,goingthrough
ρ
(Lemma6.5).
Apropertywhichplaysasimilarrolebutwhichiseasiertostateis
integrality
.Say
thatarepresentation
ρ
is
integral
ifitisconjugate,in
SL
(2
,
C
),toarepresentation
ρ
:
π
1
(
X,x
)
→
SL
(2
,A
)for
A
aringofalgebraicintegers.ForZariski-denserepresentations,
thisisequivalenttoaskingthatthetraces
Tr
(
ρ
(
γ
))bealgebraicintegersforall
γ
∈
π
1
(
X,x
).

RANKTWOREPRESENTATIONS

3

Saythat
ρ
comesfromacomplexvariationofHodgestructure
ifthereisastructureof
complexvariationofHodgestructureonthecorrespondinglocalsystem
V
.
Themainrelationshipbetweenallofthesenotionsisthefollowingfirstresult.
Theorem1.
Suppose
ρ
:
π
1
(
X,x
)
→
SL
(2
,
C
)
isarepresentationwithquasi-unipotent
monodromyatinfinity,suchthat
ρ
doesnotprojectivelyfactorthroughanorbicurve,or
equivalently
ρ
doesn’tfactorthroughamaptoaDM-curve.Then
ρ
isrigidandintegral.
Rigidityimpliesthat
ρ
comesfromacomplexvariationofHodgestructure.
Thisisalreadyknowninthecasewhen
X
isprojectivefrom[117]forrigidity,andfor
integralityGromov-Schoen[54],and[115],thelatterofwhichwasdesignedtosupportthe
originaldormantversionofthispaper.ThevariationofHodgestructurefollowsfrom[26],
see[113].Inthepresent,weextendtheresulttothequasi-projectivecaseforrepresentations
withquasi-unipotentmonodromyatinfinity.ThevariousstatementsinTheorem1appear
asTheorems6.8,7.3and8.1below.
TheunderlyingargumentforbothrigidityandintegralitycomesfromTheorem5.13about
harmonicmapstoBruhat-Titstrees.Thisstrategyisperhapsworthcommentingon.Ithas
itsoriginsintheworkofBassandSerre[8][108][109],Culler-Shalen[29]andGromov-Schoen
.]45[Itwouldcertainlyhavebeenpossibletotreattherigidityquestionusingharmonicmaps
tosymmetricspaces[26][38][42][44][86].Forintegrality,though,itisnecessarytousethe
theoryofharmonicmapstoBruhat-Titstrees[54].Furthermore,thereisasortofanalogy
betweenthetwonotions:integralitymeansthatarepresentationinto
SL
(2
,
Q
p
)goesinto
acompactsubgroup,whereasrigiditymaybethoughtofassayingthatarepresentation
into
SL
(2
,
C
(
t
))goesintoacompactsubgroup,muchasin[29].So,wethoughtitwou