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ON THE CLASSIFICATION OF RANK TWO REPRESENTATIONS OF QUASIPROJECTIVE FUNDAMENTAL GROUPS

67 pages
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


Voir plus Voir moins

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,wethoughtitwould
beinterestingtouseharmonicmapstotreestotreatbothcasesatonce.Thisreducesthe
volumeofmaterialaboutharmonicmaps.Ontheotherhanditintroducesanadditional
difficulty,becausetheBruhat-Titstreefor
SL
(2
,
C
(
t
))isnotlocallycompact.Weovercome
thisbymakingareductiontothecaseofrepresentationsin
SL
(2
,
F
q
(
t
))where
F
q
isafinite
field.Thisreductionisfairlystandardbutitrequiresafinitenesstheoremforthenumber
ofpossiblemapstoahyperbolicDM-curve(Proposition2.8),seeTheorem6.8.Delzant
appearstohaveindependentlyfoundthisproofofthefactthatnonfactorizationimplies
rigidity.Somepiecesofhisproof,includingthefinitenessstatement,appearin[36]but
thatpaperisorientedtowardsprovingasimilarstatementforrepresentationsinasolvable
group.Theclassificationforsolvablerepresentationsin[9]and[36]complementsthepresent
paperbecausewerestrictheretoZariskidenserepresentations,whichareirreducibleover
anyfinitecovering.
Supposenowthat
ρ
doesnotfactorthroughacurve.Fromtheaboveresultsweobtain
that
ρ
isrigid,integral,andcomesfromacomplexvariationofHodgestructure.Let
A
be
aringofalgebraicintegerssuchthat
ρ
isdefinedover
A
.Let
σ
:
A

C
beanembedding
(notnecessarilytheidentityone).Let
ρ
σ
bethecomposedrepresentation
π
1
(
X,x
)

ρ
A

SL
(2
,A
)

σ

SL
(2
,
C
)
.
Then
ρ
σ
isrigidtoo.Hence
ρ
σ
comesfromacomplexvariationofHodgestructurefor
every
σ
.Wewillusethisdatatoconstructafactorizationof
ρ
throughapolydiskShimura

4

m

oudalrD-Mtsakc∗H.sAbfeK.CORLETTEANDC.SIMPSON

orethismeansthatthereisamap
f
:
X

H
suchthat
ρ
isthepullback
f
ofoneofthetautologicalranktwolocalsystemson
H
.Themainidea
isthatbecausetherankis2,theHodgetypescanbechosentobe(1
,
0)and(0
,
1);then,
byintegralitywegetafamilyofabelianvarieties.Theconstructionofthemapto
H
is
straightforwardbutithastotakeintoaccountthenotionofpolarizationandthespecial
structureofabelianvarietieswhoseHodgestructuressplitintoranktwopiecesoveratotally
imaginaryfield.Thisgivesthestatementofourmainclassificationresult,see
§
11:
Theorem2.
Suppose
X
isasmoothquasiprojectivevarietyand
ρ
:
π
1
(
X,x
)

SL
(2
,
C
)
is
aZariski-denserepresentation.Supposethatthemonodromytransformationsaroundcom-
ponentsofthedivisoratinfinityarequasi-unipotent.Theneither
ρ
comesfromamap
f
:
X

Y
toaDM-curve
Y
,orelse
ρ
comesfrompullbackofoneofthetautological
representationsbyamap
f
:
X

H
toapolydiskShimuramodularDM-stack
H
.
Thetwocasesdescribedinthistheoremcanoverlap:therecanberigidlocalsystemson
anorbicurve.ForanyranksuchthingsareclassifiedbyKatz’salgorithm[64],butinthe
ranktwocasetheycanbeseenexplicitlyashypergeometricsystems
§
7.1.
Thispaperisaprojectwhichwehavebeenentertainingsincearound1990.Itwasmoti-
vatedbyGromov’spaper[53],andspurredonbyalecturebyR.SchoeninChicagoabout
[54].Intheprojectivecasethepaper[115]wasdoneinthecontextofthisprogram,in
ordertoobtaintheproofofTheorem1andthusTheorem2;inparticular[115]shouldbe
consideredasanintegralpartofthepresentproject.
AnimportantelementwasM.Larsen’sexplanationofhowtogetinformationonthefield
ofdefinitionoftherepresentation.Hisargument,firstreportedin[113,Lemma4.8],playsa
crucialroleinstartingoff
§
10below.
Aswehavetakensuchalongtimetowriteuptheclassificationresult,theambientstate
oftechnologyhasevolvedinthemeantime[14][61][86][97]whichmakesitreasonableto
givestatementsforquasiprojectivevarieties.Anotherrecentadvanceisthatthenotionof
DM-stackhasbecomestandardandwell-understood,forexampleBehrend-Noohi[10]have
classifiedDM-curvesinawaywhichisveryusefulforourconsiderations.
Oneofthereasonsforgettingbacktothisprojectisthatrecentlytherehavebeensome
explicitconstructionsofranktwolocalsystemsonquasiprojectivevarieties,whichinsome
casescanbeseenascomingfromprojectivevarietiesbypassingtoafiniteramifiedcover.
TheexamplesweknowofarethoseofBoalch[15]andPanov[97].Itwouldbeinteresting
explicitlytodeterminethefactorizationsfortheseexamples,butwedon’ttreatthatquestion
here.ThatquestionseemstobeansweredinsomecasesbyarecentpaperofBenHamedand
Gavrilov[12]whichgivesanexplicitgeometricoriginforsolutionsofPainleve´VIequations.
Kontsevichhasanumberofconjecturesaboutlocalsystemsoncurves[73].
2.
LocalsystemsonDeligne-Mumfordstacks
Inordertoobtainoptimalstatementsofthetypeweshallconsiderthroughoutthispaper,
itisconvenienttoconsiderthenotionoflocalsystemoveraDeligne-Mumfordstack.
Recallthata
Deligne-Mumfordstack
isa1-stack
X
onthesiteofschemesover
C
with
theetaletopology,suchthatthereexistsasurjectivemorphismofstacks
f
:
Z

X
from
ascheme
Z
to
X
,with
f
“representableandetale”.Theseconditionsmeanthatif
Y
is

RANKTWOREPRESENTATIONS

5

anyschemethen
Z
×
X
Y

Y
isanetalemorphismofschemes.Wereferthereaderto
[1][10][20][34][79][95]forgeneralreferencesaboutthisnotion.Intuitively,aDeligne-
Mumfordstackisanobject
X
whichlookslikethequotientofanalgebraicvariety
Z
byan
“equivalencerelation”
R

Z
×
Z
suchthattheprojectionsfrom
R
tothetwofactorsare
etale(inparticularquasi-finite).Technicallyspeaking
R
:=
Z
×
X
Z
isalsoprovidedwith
amultiplicationmakingitintothemorphism-objectforagroupoidwhoseobject-objectis
Z
.Inpractice,inthecasesweshallconsiderinthispaper,
X
willoftenbethequotientof
asmoothvariety
Z
byanactionofafinitegroup
G
.Thisisthecasewhen
X
isaShimura
modularstack,andalmostalwaysthecasewhen
X
isanorbicurveorDM-curve.Inthe
Shimuramodularcase,
X
hasmanyGaloiscoveringsbyShimuramodularvarietiesobtained
byimposingsomelevelstructure.Intheonedimensionalcase,asidefromasmallnumber
ofdegeneratesituations,anorbicurvehasaGaloiscoveringwhichisasmoothcurve(see
Lemmas2.3and2.6below).
If
X
isaDM-stackthenweobtainits
coarsemodulispace
X
coarse
whichistheuniversal
algebraicspacewithamapfrom
X
.Thisexists,andthemap
X

X
coarse
isfinite,by
Keel-Mori[68].
Theorem2.1.
If
X
isaDM-stack,thereisaZariskiopencoveringof
X
coarse
suchthatthe
pullbacksoftheopensetsarequotientstacksbyfinitegroupactions.
Proof:
Thisisstatedin[125]Proposition1.17,withforproofareferenceto[129]2.8which
shouldbetakeninlightof[68].Asketchofproofisgivenin[1]Lemma2.2.3.

Becauseofthistheorem,thereadermaywithoutdangerimaginethatthewords“DM-
stack”andsoforth,basicallymeanvarietiesmodulofinitegroupactions.The
irreducible
components
ofaDM-stackwillbydefinitionbetheirreduciblecomponentsofthecoarse
modulispace,whichpullbacktoclosedsubstacksof
X
.
Localpropertiesof
X
aredefinedbyrequiringthesamepropertiesfortheetalecovering
scheme
Z

X
occuringintheabovedefinition.Inparticular,aDM-stack
X
is
smooth
if
forone(orequivalentlyany)surjectiveetalemapfromascheme
Z

X
,thescheme
Z
is
smooth.WerestrictourattentioninthispapertosmoothDM-stacks.A
point
isamorphism
x
:
Spec
(
C
)

X
,whichwedenoteabusivelyby
x

X
.Apointcanhave“automorphisms”,
whichisthephenomenonnewtostacks.
ThenotionofDeligne-Mumfordstackisageneralizationandtransfertothealgebraiccat-
egory,oftheclassicalnotionof“orbifold”or
V
-manifold[67][106][107].Thisnotionadapts
tothealgebraiccategory:an
orbifold
isasmoothDM-stacksuchthatthegeneralpointof
anyirreduciblecomponenthastrivialautomorphismgroup.Theassociatedcomplexanalytic
stackinthiscaseisexactlyacomplexanalytic
V
-manifold.ThenthereisadenseZariski
opensubsetofthecoarsemodulispace
X
coarse
overwhichtheprojectionisanisomorphism,
thusanorbifoldhasadenseopensubstackwhichisanalgebraicspace.
If
X
=
Z/G
isaquotientstackbytheactionofafinitegroup(withthequotientbeing
irreducible,say),then
X
isanorbifoldifandonlyif
G
actsfaithfully.Thecoarsemoduli
spaceistheclassicalquotientspaceoftheaction.Orthogonaltothiscaseistheimportant
exampleofthequotientofasinglepointbyatrivialactionofafinitegroup
G
:thisisa
DM-stackdenoted
BG
.Allofitspointsareisomorphicandtheyallhaveautomorphism
groupisomorphicto
G
.

6

K.CORLETTEANDC.SIMPSON

ThesetwoexamplespermitustoformallpossiblesmoothDM-stacks,asshownbythe
followingstructureresult.
Proposition2.2.
Suppose
X
isasmoothDM-stack.Thereisauniversalorbifold
X
orb
(i.e.asmoothDMstackwithtrivialgenericstabilizers)withamap
φ
:
X

X
orb
suchthat
locallyintheetaletopologyoverthebase,themap
φ
isisomorphictotheprojectionofa
productofthebaseorbifoldwitha
BG
(thisisusuallycalleda
gerb
).
Proof:
(See[10]).UsingTheorem2.1,overanopensetwherethestackhastheform
Z/G
forasmoothirreduciblevariety
G
,let
H
bethekerneloftheaction.Itisanormalsubgroup
(sinceitisthekernelofthemapfrom
G
totheautomorphismsof
Z
).Inthiscase,
X
orb
isthequotientstackof
Z
by
G/H
,and
X
isagerbover
X
orb
withfiber
BH
(use
Z
itself
fortheetaleneighborhoodinquestion:wehave
X
×
X
orb
Z
=

Z
×
BH
).Thissatisfiesa
universalpropertysotheselocalconstructionsgluetogethertogive
X
orb
intheglobalcase.
Remark:
Thecoarsemodulispacesof
X
and
X
orb
coincide.
2.1.
Fundamentalgroupandlocalsystems.
If
X
isaDM-stackand
x

X
isapoint
thenweobtainthe
fundamentalgroup
π
1
(
X,x
).SeeNoohi[95]forageneraldiscussion.
Inthequotientcase
X
=
Z/G
,thefundamentalgroupmaybeviewedmoresimplyasan
extension
1

π
1
(
Z,z
)

π
1
(
X,x
)

G

1
.
Thegroupinthemiddlemaybedefinedasthesetofpathsin
Z
startingat
z
andgoingto
anypreimageof
x
.Pathsarecomposedbyfirsttranslatingbyanappropriateelementof
g
,
thenjuxtaposingpaths.
Wealsohavethenotionof
localsystem
over
X
,whichisbynowastandardnotion,see
[76]forexample,or[125]forthecaseof
D
-modules.Thecaseoflocalsystemsonthemoduli
stackofhyperellipticcurveswasmentionnedinatalkbyR.Hain,see[55,p.12].Alocal
systemisacollection
L
Z,f
oflocalsystemsoverschemes
Z
foreverysection
f
:
Z

X
,
togetherwithfunctorialitymaps:if
f
:
Z

X,f

:
Z


X
aretwomaps,andif
g
:
Z


Z
isamaptogetherwithanaturaltransformation
η
from
f

g
to
f

thenwegetamapfrom
g

(
L
Z,f
)to
L
Z

,f

.Thesesatisfysomenaturalaxioms.
Weobtainatensorcategoryoflocalsystemsover
X
.
If
L
isalocalsystemofrank
r
over
X
andif
x

X
isabasepointthenthefiber
L
x
isa
C
-vectorspaceofdimension
r
,andthemonodromyisanactionof
π
1
(
X,x
)onit.If
X
is
smoothandirreduciblewithfixedbasepoint
x
thenthecategoryoflocalsystemsover
X
is
equivalenttothecategoryofrepresentationsof
π
1
(
X,x
).
If
X
=
Z/G
isaquotientstackofavariety
Z
,alocalsystemon
X
maybeseenasa
G
-equivariantlocalsystemon
Z
,i.e.apair(
V
Z

)where
V
Z
isalocalsystemon
Z
and
α
isanactionassociatingtoeach
g

G
anisomorphism
α
(
g
):
g

V
Z
=

V
Z
.Theactionis
requiredtosatisfyacocyclecondition:if
g,h

G
then
α
(
g
)

g

α
(
h
)=
α
(
gh
).
ThenotionoflocalsystemonaDMstackbeamsbackdowntotheworldofvarietiesin
thefollowingway:if
Y
isavarietyand
f
:
Y

X
isamaptoaDeligne-Mumfordstack
X
thenforanylocalsystem
V
on
X
thepullback
f

V
isalocalsystemon
Y
.

RANKTWOREPRESENTATIONS7
Somethingnewhereisthatamap
f
canhave
automorphisms
.Roughlyspeakingan
automorphismof
f
isasectionofthesheafofstabilizergroupspulledbackover
f
.More
generallyweshouldspeakof
isomorphisms
betweenmaps
f,g
:
Y

X
.Wehavethefollow-
ingfunctoriality:if
a
:
f

g
isanisomorphismfrom
f
to
g
thenweobtainanisomorphism
ofpullbacklocalsystems
a

V
:
f

V
=

g

V
.Thissatisfiessomeusualfunctorialityand
associativityidentities.Thisphenomenonwillappearinourstatementswhenwewantto
saythatamapisunique.
2.2.
DM-curvesandorbicurves.
Weareparticularlyinterestedinthecaseofobjectsof
dimension1.A
DM-curve
isasmoothDM-stackofdimension1.An
orbicurve
isanorbifold
ofdimension1,soanorbifoldisaDM-curvewithtrivialgenericstabilizer.Inthiscase
Proposition2.2saysthataDM-curveisalwaysagerboveracanonicalorbicurve.If
X
isan
orbicurvethenitscoarsemodulispace
X
coarse
isagainasmoothcurve(thisisspecialtothe
caseoforbicurves:ingeneralthecoarsemodulispaceofasmoothorbifoldwillhavefinite
quotientsingularities).
Anetalecoveringisafiniteetalemap;overthecomplexnumbersthisisthesamething
asafinitetopologicalcoveringspace.ForDM-curves,etalenessismeasuredintermsofa
localchartforthestack.Inpracticaltermsthiscomesdowntosayingthattheramification
index,takingcorrectlyintoaccounttheorbifoldindices,is1.
Thedataofanorbicurveisdeterminedbythesmoothcurve
X
coarse
,togetherwithafinite
setofpoints
P
j
andaninteger
n
j

2attachedtoeachpoint.Anonconstantmorphism
fromaconnectedsmoothcurveto
X
isanynonconstantmap
f
:
Z

X
coarse
suchthatfor
any
z

Z
lyingoversome
P
j
theramificationorderof
f
at
z
isdivisibleby
n
j
.Themapis
etalewhentheramificationat
P
j
isexactly
n
j
.
Wedenoteanorbicurvebythenotation(
Y,n
1
,...,n
k
)with
Y
=
X
coarse
asmoothcurve
and
n
i
thesequenceofintegersorganizedindecreasingorder(leftoutofthisnotationisthe
choiceofpoints
P
1
,...,P
k

X
).
ThepaperofBehrendandNoohi[10]isaverycompletedescriptionofthepossibilities
forDM-curvesandorbicurves,andthereaderisreferredthere.Recallherethegeneral
outlinesoftheirclassificationwhichgeneralizestheclassificationofcurves.Anorbicurveis
either
spherical
,
elliptic
,or
hyperbolic
.Thesphericalorbicurvesare
P
1
,the
drops
(
P
1
,a
)and
footballs
(
P
1
,a,b
)aswellasthefinitelistofcases
(
P
1
,
2
,
2
,
2)
,
(
P
1
,
2
,
3
,
3)
,
(
P
1
,
2
,
3
,
4)
,
(
P
1
,
2
,
3
,
5)
.
Theuniversalcoveringsoftheseorbicurvesareeither
P
1
,drops,orfootballswithrelatively
primeindices.
Lemma2.3.
Anorbicurvewhichisnotsphericalaslistedabove,hasanetalecoveringwhich
isaregularcurvedifferentfrom
P
1
.Inparticulartheassociatedcomplex-analyticorbicurve
hasacontractibleuniversalcovering.
Proof:
See[10].

Nowwecandefinethe
elliptic
orbicurvestobethosewhichhaveacoveringbyeither
A
1
or
G
m
oranellipticcurve(thus,theyarethosewhoseuniversalcoveringis
C
);andthe
hyperbolic
orbicurvesaretheremainingones—thosewhoseuniversalcoveringisadisc.

8

K.CORLETTEANDC.SIMPSON

Notealsothatanellipticorhyperbolicorbicurvehasinfinitefundamentalgroupexceptfor
theellipticcasewherethecoveringis
A
1
.Inallellipticandhyperboliccases,thehomotopy
typeisa
K
(
π
1
,
1).
WesaythataDM-curve
X
isspherical,ellipticorhyperbolicaccordingtothetypeofthe
associatedorbicurve
X
orb
.
Lemma2.4.
Suppose
X
isaDM-curveand
ρ
:
π
1
(
X,x
)

SL
(2
,K
)
isaZariski-dense
representationforanyinfinitefield
K
.Then
X
ishyperbolic.
Proof:
Sphericalorbicurveshavefinitefundamentalgroups,andellipticorbicurveshave
fundamentalgroupswhicharevirtuallyabelian.Thusthesameholdsforsphericaland
ellipticDM-curves.NeitherofthesetypesofgroupscanhaveZariski-denserepresentations
to
SL
(2
,K
)foraninfinitefield
K
.

Inviewofthislemma,forfactorizationquestionswewillmainlybelookingathyperbolic
orbicurves.
Lemma2.5.
Suppose
X
isaDM-curvewithbasepoint
x
.Supposetheunderlyingorbifold
Y
:=
X
orb
iseitherellipticorhyperbolic.Then
π
2
(
Y
)=0
andthegerb
p
:
X

Y
(locally
aproductwith
BG
)iscompletelydeterminedbytheinducedextension
1

G

π
1
(
X,x
)

π
1
(
Y,p
(
x
))

1
.
Thuswemaythinkof
X
asbeinggivenbythedataofanorbicurve
Y
plusanextensionof
thefundamentalgroupbyagroup
G
.If
Z
isaschemethenasubmersivemapfrom
Z
to
X
isthesamethingasasubmersivemapfrom
Z
to
Y
plusaliftingofthemaponfundamental
groupsfrom
π
1
(
Z,z
)
to
π
1
(
X,x
)
.
Proof:
ThisisbasicallythesameasBehrend-Noohi[10]Proposition4.7.
If
X
isalreadyanorbicurvethenthegenericstabilizergroup
G
istrivial,
BG
=Spec(
C
)
and
Y
=
X
.
Ingeneral,let
Z
bethecoarsemodulispacefor
X
.Itisanormal(hencesmooth)curve.
Let
Y
betheorbicurveobtainedbysetting,foreachpoint
z

Z
,theorbifoldstructureat
z
tobetheramificationdegreeofthemap
X

Z
over
z
(thisisdifferentfrom1foronly
afinitenumberofpoints).Thereisauniquefactorizationtoamap
p
:
X

Y
(notethat
thereisnoquestionofnaturaltransformationsherebecause
p
issubmersiveandthegeneric
stabilizerson
Y
aretrivial).Themap
p
isetalebythechoiceoforbifoldindices.Anetale
coveringisafibrationintheetaletopology,andthefiberoverapoint(sayageneralpoint)
isaone-pointDM-stack,henceoftheform
BG
forafinitegroup
G
.Weobtainalongexact
sequenceinhomotopy.If
X
isellipticorhyperbolic,thenbydefinitionthesameistrueof
Y
.Thustheuniversalcoveringof
Y
is
C
oradisk[10],so
π
2
(
Y
)=0.Inparticular,thelong
exacthomotopysequenceofhomotopygroupsgivesanextensionasinthestatementofthe
lemma.Thefactthatthisextensionuniquelydeterminesthefibration
p
:
X

Y
,plusthe
statementaboutmapsto
X
,willbeleftasexercisesinnonabeliancohomology[18].

WewillseebelowthatthenotionofDM-curvegivesahandywayofdealingwiththe
differencebetween
SL
(2)and
PSL
(2)forfactorizationstatements.Inparticular,sincewe
aremainlyusingthenotionofDM-curveforthatpurpose,itmostlysufficestoconsiderthe
casewherethegroup
G
inLemma2.5isofordertwo.
Puttingtogethertheaboveinformationweobtainthefollowing.

RANKTWOREPRESENTATIONS

9

Lemma2.6.
If
X
isanellipticorhyperbolicDM-curve,thenthereisanetaleGalois
covering
Z

X
with
Z
asmoothcurve;inparticular
X
isaquotientof
Z
bytheGalois
group
G
—notethattheactionof
G
on
Z
mightnotbefaithful;thisishowwegetnontrivial
genericstabilizers.Asdescribedabovewehaveanextension
1

π
1
(
Z
)

π
1
(
X
)

G

1
.
Proof:
UseLemma2.5toexpress
X
asagerboveranorbicurve
Y
.Fromthelongexact
sequenceweseethat
Y
hasinfinitefundamentalgroup,sowearenotinthedegeneratecases
ofLemma2.3.Thusthereisanetalecoveringmap
V

Y
fromasmoothcurveto
Y
.
Pullingback,wegetagerb
U
:=
X
×
Y
V
over
V
.Thegerbcorrespondstoanextension
1

H

π
1
(
U
)

π
1
(
V
)

1
.
Let
U


U
denotethecoveringcorrespondingtothemap
π
1
(
U
)

Aut(H).Thisisagain
agerbcorrespondingtoacentralextension
1

Z
(
H
)

π
1
(
U

)

π
1
(
V

)

1
.
Thecentralextensionisclassifiedbyanelementof
H
2
(
V

,Z
(
H
)).Thereisafiniteetale
coveringspace
V
′′

V

suchthatthepullbackoftheelementiszero,thatistheextension
splits.Thisgivesanetalecoveringmorphismfrom
V
′′
backdownto
X
.Galoiscompletion
yieldsthecurve
Z
.

Duetothislemma,andthefactthattheotherstacksweconsider(Shimuramodularstacks)
arebyconstructionanalyticquotientsandareseentobealgebraicquotientsbylookingat
modulispacesforobjectswithlevelstructure,itispossibletorestrictourattentionto
Deligne-Mumfordstackswhicharequotients.
2.3.
Finiteness.
Weproveafinitenessstatementwhichisthekeytoourmethodofproofof
theimplication“nonfactoringimpliesrigid”in
§
6below.Ithasalsoindependantlyappeared
inDelzant’spreprint[36]withamoreconceptualproof.
Lemma2.7.
Given
g
and
b
,thereisaninteger
N
givingthefollowingbound.Forany
hyperbolicorbicurve
X
,andanyregularcurve
Y
withanonconstantmap
f
:
Y

X
,
assumethattheprojectivecompactification
Y
of
Y
hasgenus

g
,andthenumberofpoints
in
Y

Y
is

b
.Thenforanypoint
P
intheimageof
f
,theorbifoldindexof
X
at
P
is
.N≤Proof:
Wemayassumethat
X
iscompact.Indeed,ifnotthenwecouldaddinorbifold
structuresatthepointsatinfinityof
X

X
coarse
.Notethatwearenotaskinginthe
hypothesisthatthemap
f
besurjective,onlythatthepoint
P
inquestionbeintheimage.
Ifwefixanorbifoldindexof

7attheseadditionalpointsthentheneworbifoldwillalso
behyperbolic(see[10]),sowemayassume
X
iscompact.
Fix
f
:
Y

X
,andpoints
Q

Y
and
P
=
f
(
Q
)

X
.Let
n
betheorbifoldindexof
X
at
P
.Denoteby
f
theextensionof
f
to
Y

X
(where
X
denotestheusualcurvewhichis
theprojectivecompletionof
X
coarse
).
Theramificationdegreeof
f
at
Q
isdivisibleby
n
,sothecontributiontotheHurwitz
formulafrom
Q
isatleast
n

1.If
g
(
X
)

1wehave
2
g
(
Y
)

2

d
(2
g
(
X

2)+(
n

1)

n

1

10K.CORLETTEANDC.SIMPSON
whichgivestheboundinquestion.
Theremainingproblemistotreatthecase
X
=
P
1
.Since
X
ishyperbolic,thereare
atleasttwootherorbifoldpoints
P
1
and
P
2
besides
P
0
:=
P
.Let
n
1
and
n
2
denotetheir
1−orbifoldindices.Let
d
denotethedegreeofthemap
f
.If
Q

isapointin
f
(
P
i
)theneither
Q


Y
,inwhichcasetheramificationdegree
r
Q

isdivisibleby
n
i
(if
i
=0wearedenoting
n
0
:=
n
);orelse
Q


Y

Y
.Thereareatmost
d/n
i
pointsinthefirstcase,andatmost
b
pointsinthelastcase.
TheHurwitzcontributionforthefiberover
P
i
isbounded:
1−X(
r
Q


1)=
d

#(
f
(
P
i
))

d

b

d/n
i
.
f
(
Q

)=
P
i
ThereforetheHurwitzformulagives
1112
g
(
Y
)

2
≥−
2
d
+3
d

3
b

d
(++)
.
n
0
n
1
n
2
For
n
0
=
n
thisgivesabound
1

2

2
g
(
Y
)

3
b
+(1

1

1)
.
ndn
1
n
2
Inthecasethatatleastoneof
n
1
or
n
2
are

3weobtain
111(1
−−
)

6nn21os12

2
g
(
Y
)

3
b
1
n

d
+6
.
If
n

6thenwehaveourboundsowecanassumethat
n

7.Thus
3
b
+2
g
(
Y
)

21
,≥24drod

42(3
b
+2
g
(
Y
)

2)
.
Thisgivesaboundfor
d
hencefor
n

d
.
Assumenowthat
n
1
=
n
2
=2.Inthiscasetheremustbeafourthorbifoldpoint,because
theorbifoldswithindices(2
,
2
,a
)aresphericalforany
a
.Addinginthefourthpointtothe
abovecalculationstheboundbecomes
1

2

2
g
(
Y
)

4
b
+(2

1

1

1)
.
ndn
1
n
2
n
3
ereH1111(2
−−−
)

n
1
n
2
n
3
2
os12

2
g
(
Y
)

4
b
1
n

d
+2
.

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