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PRYM VARIETIES OF SPECTRAL COVERS

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English

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PRYM VARIETIES OF SPECTRAL COVERS TAMAS HAUSEL AND CHRISTIAN PAULY Abstract. Given a possibly reducible and non-reduced spectral cover pi : X ? C over a smooth projective complex curve C we determine the group of connected components of the Prym variety Prym(X/C). As an immediate application we show that the finite group of n- torsion points of the Jacobian of C acts trivially on the cohomology of the twisted SLn-Higgs moduli space up to the degree which is predicted by topological mirror symmetry. In particular this yields a new proof of a result of Harder–Narasimhan, showing that this finite group acts trivially on the cohomology of the twisted SLn stable bundle moduli space. 1. Introduction Recently there has been renewed interest in the topology of the Hitchin fibration. The Hitchin fibration is an integrable system associated to a complex reductive group G and a smooth complex projective curve C. It was introduced by Hitchin [Hi] in 1987, originating in his study of a 2-dimensional reduction of the Yang-Mills equations. In 2006, Kapustin and Witten [KW] highlighted the importance of the Hitchin fibration for S-duality and the Geometric Langlands program. While the work of Ngo [N2] in 2008 showed that the topology of the Hitchin fibration is responsible for the fundamental lemma in the Langlands program. In Ngo's work and later in the work of Frenkel and Witten [FW] a certain symmetry of the Hitchin fibration plays an important role.

  • group pic0

  • spectral cover

  • space up

  • line bundles over

  • bundle moduli

  • cover pi

  • sj ?

  • ?1 ?


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PRYMVARIETIESOFSPECTRALCOVERS

TAMA´SHAUSELANDCHRISTIANPAULY

Abstract.
Givenapossiblyreducibleandnon-reducedspectralcover
π
:
X

C
overa
smoothprojectivecomplexcurve
C
wedeterminethegroupofconnectedcomponentsofthe
PrymvarietyPrym(
X/C
).Asanimmediateapplicationweshowthatthefinitegroupof
n
-
torsionpointsoftheJacobianof
C
actstriviallyonthecohomologyofthetwistedSL
n
-Higgs
modulispaceuptothedegreewhichispredictedbytopologicalmirrorsymmetry.Inparticular
thisyieldsanewproofofaresultofHarder–Narasimhan,showingthatthisfinitegroupacts
triviallyonthecohomologyofthetwistedSL
n
stablebundlemodulispace.

1.
Introduction
RecentlytherehasbeenrenewedinterestinthetopologyoftheHitchinfibration.TheHitchin
fibrationisanintegrablesystemassociatedtoacomplexreductivegroupGandasmooth
complexprojectivecurve
C
.ItwasintroducedbyHitchin[Hi]in1987,originatinginhisstudy
ofa2-dimensionalreductionoftheYang-Millsequations.In2006,KapustinandWitten[KW]
highlightedtheimportanceoftheHitchinfibrationfor
S
-dualityandtheGeometricLanglands
program.WhiletheworkofNgoˆ[N2]in2008showedthatthetopologyoftheHitchinfibration
isresponsibleforthefundamentallemmaintheLanglandsprogram.InNgoˆ’sworkandlater
intheworkofFrenkelandWitten[FW]acertainsymmetryoftheHitchinfibrationplaysan
importantrole.
InthispaperwefocusontheHitchinfibrationforthegroupG=SL
n
andforalinebundle
M
over
C
,i.e.,themorphism
nM(1)
h
:
M−→A
n
0
=
H
0
(
C,M
j
)
.
2=jHere
M
denotesthequasi-projectivemodulispaceofsemi-stableHiggsbundles(
E,φ
)over
C
ofrank
n
,fixeddeterminantΔandwithtrace-freeHiggsfield
φ

H
0
(
C,
End
0
(
E
)

M
).Inthe
caseofSL
n
theabovementionedsymmetrygroupoftheHitchinfibrationisthePrymvarietyof
aspectralcover.Forthetopologicalapplicationsthedeterminationofitsgroupofcomponents
isthefirststep.Ngoˆworkswithintegral,thatisirreducibleandreduced,spectralcurves;but
itisinterestingtoextendhisresultstonon-integralcurves.Forreduciblebutreducedspectral
curvesitwasachievedbyChaudouardandLaumon[CL],whoprovedtheweightedfundamental
lemmabygeneralizingNgoˆ’sresultstoreducedspectralcurves.Inthispaperwedeterminethe
groupofconnectedcomponentsofthePrymvarietyfornon-reducedspectralcurvesaswell.
Inordertostatethemaintheoremweneedtointroducesom
S
enotation.Weassociatetoany
spectralcover
π
:
X

C
afinitegroup
K
asfollows:let
X
=
i

I
X
i
beitsdecompositioninto
irreduciblecomponents
X
i
,let
X
ired
betheunderlyingreducedcurveof
X
i
,
m
i
themultiplicity
2000
MathematicsSubjectClassification.
Primary14K30,14H40,14H60.
1

2TAMA´SHAUSELANDCHRISTIANPAULY
of
X
ired
in
X
i
and
X
e
ired
thenormalizationof
X
i
.Wedenoteby
π
e
i
:
X
e
ired

C
theprojection
onto
C
andintroducethefinitesubgroups
K
i
=ker
π
e
i

:Pic
0
(
C
)
−→
Pic
0
(
X
e
ired
)

Pic
0
(
C
)
,
aswellasthesubgroups(
K
i
)
m
i
=[
m
i
]

1
(
K
i
),where[
m
i
]denotesmultiplicationby
m
i
inthe
PicardvarietyPic
0
(
C
)parameterizingdegree0linebundlesover
C
.Finally,weput
\(2)
K
=(
K
i
)
m
i

Pic
0
(
C
)
.
I∈iWedenoteby
C
n
themultiplecurvewithtrivialnilpotentstructureoforder
n
havingunderlying
reducedcurve
C
.
WeconsiderthenormmapNm
X/C
:Pic
0
(
X
)

Pic
0
(
C
)betweentheconnectedcomponents
oftheidentityelementsofthePicardschemesofthecurves
X
and
C
anddefinethePrym
variety
Prym(
X/C
):=ker(Nm
X/C
)
.
Ourmainresultisthefollowing

Theorem1.1.
Let
π
:
X

C
beaspectralcoverofdegree
n

2
.Withthenotationabovewe
havethefollowingresults:
(1)
Thegroupofconnectedcomponents
π
0
(Prym(
X/C
))
ofthePrymvariety
Prym(
X/C
)
equals
π
0
(Prym(
X/C
))=
K
b
,
where
K
b
=Hom(
K,
C

)
isthegroupofcharactersof
K
.
(2)
Thenaturalhomomorphismfromthegroupof
n
-torsionlinebundles
Pic
0
(
C
)[
n
]
to
π
0
(Prym(
X/C
))
givenby
Φ:Pic
0
(
C
)[
n
]
−→
π
0
(Prym(
X/C
))

7→
[
π

γ
]
,
where
[
π

γ
]
denotestheclassof
π

γ

Pic
0
(
X
)
in
π
0
(Prym(
X/C
))
issurjective.In
particular,weobtainanupperboundfortheorder
|
π
0
(Prym(
X/C
))
|≤
n
2
g
,
where
g
isthegenusofthecurve
C
.
(3)
Themap
Φ
isanisomorphismifandonlyif
X
equalsthenon-reducedcurve
C
n
with
trivialnilpotentstructureoforder
n
.
Similardescriptionsof
π
0
(Prym(
X/C
))weregivenin[N1]inthecaseofintegralspectral
curvesandby[CL]inthecaseofreduciblebutreducedspectralcurves.Also[dCHM]use
specialcasesforSL
2
.
Foracharacteristic
a
∈A
n
0
wedenoteby
π
:
X
a

C
theassociatedspectralcoverofdegree
n
(seesection2.2)andby
K
a
thesubgroupofPic
0
(
C
)definedin(2)andcorrespondingtothe
cover
X
a
.LetΓ

Pic
0
(
C
)[
n
]beacyclicsubgroupoforder
d
ofthefinitegroupPic
0
(
C
)[
n
]of
n
-torsionlinebundlesover
C
andlet
A

⊂A
n
0
denotetheendoscopicsublocusofcharacteristics
na
suchthattheassociateddegree
n
spectralcover
π
:
X
a

C
comesfromadegree
d
spectral
coveroverthee´taleGaloiscoverof
C
withGaloisgroupΓ(fortheprecisedefinitionseesection
5.1).Withthisnotationwehavethefollowing

Theorem1.2.
Wehaveanequivalence
Γ

K
a
⇐⇒
a
∈A

.

3

Thisgivesadescriptionofthelocusofcharacteristics
a
∈A
n
0
suchthatthePrymvariety
00Prym(
X
a
/C
)isnon-connected,becauseclearly
A
Γ
2
⊂A
Γ
1
ifΓ
1

Γ
2
.
Corollary1.3.
Thesublocusofcharacteristics
a
∈A
n
0
suchthatthePrymvariety
Prym(
X
a
/C
)
isnotconnectedequalstheunion
[0(3)
A
endo
:=
A
Γ
,
where
Γ
variesoverallcyclicsubgroupsofprimeorderof
Pic
0
(
C
)[
n
]
.
Calculatingthedimensionsoftheendoscopicloci
A

willleadtoanimmediatetopological
application.RecallthatPic
0
(
C
)[
n
]actson
M
bytensorization,andthiswillinduceanaction
ontherationalcohomology
H

(
M
;
Q
).Wethenhave
Theorem1.4.
Let
n>
1
and
p
n
bethesmallestprimedivisorof
n
.Assumethat
M
=
K
C
,the
canonicalbundleof
C
,andthat
(
n,
deg(Δ))=1
.Thentheactionof
Pic
0
(
C
)[
n
]
on
H
k
(
M
;
Q
)
istrivial,providedthat
k

2
n
2
(1

1
/p
n
)(
g

1)
.
Infactthisresultshouldbesharp,asthetopologicalmirrorsymmetryconjecture[HT,
Conjecture5.1]predictsthatthesmallestdegreewherePic
0
(
C
)[
n
]actsnon-triviallyis
k
=
n
2
(1

1
/p
n
)(2
g

2)+1
.
ThisresultshintsatthecloseconnectionbetweenNgoˆ’sstrategyin[N1,N2]forstudyingthe
symmetriesoftheHitchinfibrationandthetopologicalmirrorsymmetryconjecturesin[HT].
Morediscussiononthisconnectioncanbefoundin[Hau2].
Finallylet
N
denotethemodulispaceofstablevectorbundlesofrank
n
andfixeddeterminant
Δover
C
.AgainthefinitegroupPic
0
(
C
)[
n
]actson
N
bytensorizationandthuson
H

(
N
;
Q
).
Asthecohomology
H

(
N
;
Q
)isasummandinthecohomologyof
H

(
M
;
Q
)wecandeduce
Corollary1.5.
Thefinitegroup
Pic
0
(
C
)[
n
]
actstriviallyon
H

(
N
;
Q
)
.
ThiswasthemainapplicationofHarder–Narasimhanin[HN,Theorem1].Ourproofhere
canbeconsideredasanexampleofboththeabelianizationphilosophyofAtiyah–Hitchin[At,
§
6.3]andNgoˆ’sstrategy[N1,N2]ofstudyingthesymmetriesoftheHitchinfibration.
Thepaperisorganizedasfollows.Insections2and3werecallbasicresultsonspectral
coversandonthenormmapNm
X/C
.Insections4and5weprovethetwomaintheorems.In
section6wedescribetheactionofthePrymvarietyPrym(
X
a
/C
)onthefiberover
a
∈A
n
0
oftheSL
n
-Hitchinfibration.Finallyinsection7weapplytheresultsinthispapertoprove
Theorem1.4anditsCorollary1.5.
Notation:
Givenasheaf
F
overascheme
X
andasubset
U

X
wedenoteby
F
(
U
)orby
Γ(
U,
F
)thespaceofsectionsof
F
over
U
.
Acknowledgements:
Duringthepreparationofthispaperthefirstauthorwassupportedbya
RoyalSocietyUniversityResearchFellowshipwhilethesecondauthorwassupportedbyaMarie
CurieIntra-EuropeanFellowship(PIEF-GA-2009-235098)fromtheEuropeanCommission.We

4TAMA´SHAUSELANDCHRISTIANPAULY
wouldliketothankJean-MarcDre´zet,NigelHitchin,M.S.NarasimhanandDanielSchaubfor
helpfuldiscussions.

2.
Preliminaries
2.1.
Twolemmasonabelianvarieties.
Givenanabelianvariety
A
andapositiveinteger
n
wedenoteby[
n
]:
A

A
themultiplicationby
n
,by
A
[
n
]=ker[
n
]itssubgroupof
n
-torsion
0pointsandby
A
ˆ=Pic(
A
)itsdualabelianvariety.Weconsider
f
:
A
−→
B
ahomomorphismbetweenabelianvarietieswithkernel
K
=ker(
f
)whichweassumetobe
finite.Welet
f
ˆ:
B
ˆ

A
ˆdenotethedualmapinducedby
f
.Weintroducethequotientabelian
variety
A
0
=
A/K
,sothatwecanwritethehomomorphism
f
asacompositemap
jµf
=
j

µ
:
A
−→
A
0
−→
B,
where
µ
isanisogenywithkernel
K
and
j
isinjective.
Lemma2.1.
Thegroupofconnectedcomponentsoftheabeliansubvariety
ker(
f
ˆ)

B
ˆ
equals
π
0
(ker(
f
ˆ))=
K
b
,
where
K
b
=Hom(
K,
C

)
isthegroupofcharactersof
K
.
Proof.
Weconsiderthedualmap
ˆµˆjf
ˆ:
B
ˆ
−→
A
ˆ
0
−→
A
ˆ
,
andobservethat
µ
ˆ:
A
ˆ
0

A
ˆisanisogenywithkernel
K
b
(seee.g.[BL]Proposition2.4.3)and
j
ˆhasconnectedfibers(seee.g.[BL]Proposition2.4.2).Thelemmathenfollows.

Wealsosupposethat
A
and
B
areprincipallypolarizedabelianvarieties,i.e.thepolarizations
induceisomorphisms
A
=

A
ˆand
B
=

B
ˆ.
Lemma2.2.
Weassumethatthereexistsahomomorphism
g
:
B

A
suchthat
g

f
=[
n
]
forsomeinteger
n
.Thenthedualofthecanonicalinclusion
i
:
K,

A
[
n
]
isasurjectivemap
i
ˆ:
A
[
n
]=
A
ˆ[
n
]
−→
K
b
,
whichcoincideswiththerestrictionto
A
[
n
]
ofthecompositemap
j
ˆ

g
ˆ:
A

B

A
ˆ
0
.
Proof.
Itsufficestoobservethattheisogeny
f
ˆ

g
ˆ=[
n
c
]=[
n
]factorizesas
ˆ[
n
]:
A

j


g
ˆ
A
ˆ
0

µ
ˆ

A,
that
K
b
=ker(
µ
ˆ),andthat
j
ˆ

g
ˆissurjective.Henceacanonicalsurjection
A
[
n
]

K
b
,whichis
dualtotheinclusion
i
:
K,

A
[
n
],since[
n
c
]=[
n
].

52.2.
Spectralcovers.
Inthissectionwereviewsomeelementaryfactsonspectralcovers.
Let
C
beacomplexsmoothprojectivecurveandlet
M
bealinebundleover
C
withdeg
M>
0.Wedenoteby
|
M
|
thetotalspaceof
M
andby
π
:
|
M
|−→
C
theprojectiononto
C
.Thereisacanonicalcoordinate
t

H
0
(
|
M
|


M
)onthetotalspace
|
M
|
.Thedirectimagedecomposesasfollows
∞Mπ

O
|
M
|
=Sym

(
M

1
)=
M

i
.
0=iDefinition2.3.
Aspectralcover
X
ofdegree
n
overthecurve
C
andassociatedtotheline
bundle
M
isthezerodivisorin
|
M
|
ofanon-zerosection
s

H
0
(
|
M
|


M
n
)
.
Sinceaspectralcover
X
isasubschemeof
|
M
|
,itisnaturallyequippedwithaprojection
onto
C
,whichwealsodenoteby
π
.Thedecompositionofthesection
s
accordingtothedirect
mus∞MH
0
(
|
M
|


M
n
)=
H
0
(
C,M
n

M

i
)(projectionformula)
0=i=
H
0
(
C,M
n
)
⊕∙∙∙⊕
H
0
(
C,M
)

H
0
(
C,
O
C
)
givesanexpression
s
=
s
0
+
ts
1
+
∙∙∙
+
t
n

1
s
n

1
+
t
n
s
n
with
s
j

H
0
(
C,M
n

j
).Herewe
∼alsodenoteby
s
j
itspull-backto
|
M
|
.Wenotethatthereisanisomorphism
π

:Pic(
C
)
−→
Pic(
|
M
|
),henceanylinebundleover
|
M
|
isoftheform
π

L
forsomelinebundle
L

Pic(
C
).
Moregenerally,wehaveadecomposition
H
0
(
|
M
|


L
)=
H
0
(
C,L
)

H
0
(
C,LM

1
)
⊕∙∙∙⊕
H
0
(
C,LM

d
)forsomeinteger
d
andanysection
s

H
0
(
|
M
|


L
)canbewrittenintheform
(4)
s
=
s
0
+
ts
1
+
∙∙∙
+
t
d

1
s
d

1
+
t
d
s
d
,s
j

H
0
(
C,LM

j
)
.
Lemma2.4.
Let
π
:
X

C
beaspectralcover.Thentheunderlyingreducedcurveofeach
irreduciblecomponentof
X
isagainaspectralcoverassociatedtothelinebundle
M
.
Proof.
Itsufficestoshowthatifthesection
s

H
0
(
|
M
|


M
n
)decomposesas
s
=
s
(1)

s
(2)
with
s
(
i
)

H
0
(
|
M
|


L
i
)for
i
=1
,
2and
L
1
L
2
=
M
n
,then
L
i
=
M
n
i
and
n
1
+
n
2
=
n
.By(4)
thesection
s
(
i
)
canbewrittenas
(5)
s
(
i
)
=
s
(
i
)
+
ts
(
i
)
+
∙∙∙
+
t
n
i
s
(
i
)
,
n10iwith
s
j
(
i
)

H
0
(
C,L
i
M

j
)and
s
(
ni
i
)
6
=0.Moreover
n
i
=deg(
X
(
i
)
/C
)with
X
(
i
)
=Zeros(
s
(
i
)
).By
consideringthehighestordertermsof(5)weobtaintherelations
n
1
+
n
2
=
n
and
s
(
n
1
1
)

s
(
n
2
2
)
=
s
n

H
0
(
C,
O
C
).Since
s
n
isanon-zeroconstantsection,weconcludethat
L
i
=
M
n
i
.

WeintroducetheSL
n
-andGL
n
-Hitchinspaceforthelinebundle
M
overthecurve
C
nnMMA
n
0
(
C,M
)=
H
0
(
C,M
j
)and
A
n
(
C,M
)=
H
0
(
C,M
j
)
.
j
=2
j
=1
Ifnoconfusionarises,wesimplydenotethesevectorspacesby
A
n
0
and
A
n
.Notethat
A
n
0
⊂A
n
.
Givenanelement
a
=(
a
1
,...,a
n
)
∈A
n
with
a
j

H
0
(
C,M
j
),calledacharacteristic,we
associateto
a
aspectralcoverofdegree
n
π
a
:
X
a
−→
C,X
a
=Zeros(
s
a
)
⊂|
M
|
,

TAMA´SHAUSELANDCHRISTIANPAULY

6htiw(6)
s
a
=
t
n
+
a
1
t
n

1
+
∙∙∙
a
n

1
t
+
a
n

H
0
(
|
M
|


M
n
)
.
Remark2.5.
Given
a
∈A
n
weobservethatthepull-backofthespectralcover
X
a
⊂|
M
|
by
theautomorphismof
|
M
|
givenbytranslationwiththesection

an
1
,i.e.(
x,y
)
7→
(
x,y

n
1
a
1
(
x
)),
equalsthespectralcover
X
a
0
forsome
a
0
∈A
n
0
;equivalentlydothechangeofvariables
t
7→
t

an
1
.
Hence
X
a
=

X
a
0
.Itthereforesufficestorestrictourstudytospectralcovers
X
a
for
a
∈A
n
0
.

2.3.
Non-reducedcurves.
Let
X
beanirreduciblecurvecontainedinasmoothsurfaceand
let
X
red
denoteitsunderlyingreducedcurve.Thenthereexistsaglobalsection
s
ofaline
bundlesuchthat
X
red
=Zeros(
s
)andaninteger
k
suchthat
X
=Zeros(
s
k
).Weintroduce
thesubschemes
X
i
=Zeros(
s
i
)for
i
=1
,...,k
,sothatwehaveafiltrationof
X
byclosed
subschemes
X
red
=
X
1

X
2
⊂∙∙∙⊂
X
k
=
X.
Inthatcasewesaythat
X
hasanilpotentstructureoforder
k
.Foranyinteger
i
wedenoteby
O
X
i
thestructuresheafofthesubscheme
X
i

X
.Notethat
O
X
i
isnaturallya
O
X
-module.
Weneedtorecallaresultonthelocalstructureofcoherentsheavesonnon-reducedcurves.
Theorem2.6
([D2]The´ore`me3.4.1)
.
Let
X
beacurvewithnilpotentstructureoforder
k
and
let
E
beacoherentsheafover
X
.Thenthereexistsanopensubset
V

X
dependingon
E
and
integers
m
i
suchthat

M
k

m
i
E
|
V
−→O
X
i
|
V
.
1=iThesheafontherightiscalledaquasi-freesheaf.

3.
Thenormmap
Inthissectionwerecallthedefinitionofthenormmapandprovesomeofitsproperties.The
standardreferencesare[G1]section6.5and[G2]section21.5.
3.1.
Definition.
Let
C
beasmoothprojectivecurveandlet
π
:
X
−→
C
beanyfinitedegree
n
coveringof
C
.The
O
C
-algebra
π

O
C
willbedenoted
B
andthegroupof
invertibleelementsin
B
by
B

.Notethat
B
isalocallyfreesheafofrank
n
.Let
U

C
bean
opensubsetandlet
s

Γ(
U,
B
)=Γ(
π

1
(
U
)
,
O
X
)bealocalsection.Onedefines([G1]section
6.5.1)
N
X/C
(
s
):=det(
µ
s
)

Γ(
U,
O
C
)
where
µ
s
:
B
|
U
→B
|
U
isthemultiplicationwiththesection
s
.Moreover
s
isinvertibleinΓ(
U,
B
)
ifandonlyif
N
X/C
(
s
)isinvertibleinΓ(
U,
O
X
).Wehavethefollowingobviousrelations
(7)
N
X/C
(
s

s
0
)=
N
X/C
(
s
)

N
X/C
(
s
0
)
,N
X/C
(
λs
)=
λ
n
N
X/C
(
s
)
foranylocalsections
s
and
s
0
of
B
andanylocalsection
λ
of
O
C
.

7Let
L
beaninvertible
B
-module.Wecanchooseacovering
{
U
λ
}
λ

Λ
of
C
byopensubsets
∼andtrivializations
η
λ
:
L
|
U
λ
−→B
|
U
λ
.Then(
ω
λ,µ
)
λ,µ

Λ
with
ω
λ,µ
=
η
λ

η
µ
−|
1
U
λ

U
µ

Γ(
U
λ

U
µ
,
B
)
∗∗isa1-cocyclewithvaluesin
B
and(
N
X/C
(
ω
λ,µ
))
λ,µ

Λ
isa1-cocyclewithvaluesin
O
C
,the
sheafofinvertibleelementsof
O
C
.This1-cocycledeterminesaninvertiblesheafover
C
,which
wedenotebyNm
X/C
(
L
).Thefollowingpropertieseasilyfollowfrom(7)
(8)Nm
X/C
(
L⊗L
0
)=Nm
X/C
(
L
)

Nm
X/C
(
L
0
)
,
Nm
X/C
(
π

M
)=
M

n
,
foranytwoinvertiblesheaves
L
and
L
0
over
X
andforanyinvertiblesheaf
M
over
C
.We
thereforeobtainagrouphomomorphismbetweenthePicardgroupsofthecurves
X
and
C
calledthenormmap
Nm
X/C
:Pic(
X
)
−→
Pic(
C
)
,
L7→
Nm
X/C
(
L
)
.

3.2.
Properties.
Inthecase
X
issmooth,thenormmapNm
X/C
hasamoreexplicitdescrip-
tionintermsofdivisorsassociatedtolinebundles.
Proposition3.1
([G2]section21.5)
.
Assumethat
X
isasmoothcurve.Thenormmap,as
definedabove,coincideswiththemap
XXL
=
O
X
(
n
i
p
i
)
7→
Nm
X/C
(
L
)=
O
C
(
n
i
π
(
p
i
))
,
i

Ii

I
where
n
i

Z
and
p
i

X
.Notethatthism
P
apiswell-defined,i.e.
Nm
X/C
(
L
)
onlydependson
thelinearequivalenceclassofthedivisor
i

I
n
i
p
i
.
Fromnowonthecurve
X
isagainanarbitrarycoverof
C
.
Lemma3.2.
Let
0
→E→F→T→
0
beanexactsequenceof
O
X
-modules.Weassume
that
E
and
F
aretorsion-freeandthat
T
isatorsionsheaf.Let
ϕ

bealocalmorphismover
π

1
(
U
)
forsomeopensubset
U

C
betweenexactsequences
0
−→E−→F−→T−→
0
(9)

ϕ
E

ϕ
F

ϕ
T
0
−→E−→F−→T−→
0
.
Weconsiderthe
O
C
-linearmapsinducedby
ϕ
E
and
ϕ
F
inthedirectimagesheaves
π

E
and
π

F
.Thenwehavetheequality
det(
ϕ
E
)=det(
ϕ
F
)

Γ(
U,
O
U
)
.
Proof.
Itisenoughtoshowthatthetwolocalsectionsdet(
ϕ
E
)anddet(
ϕ
F
)coincideinthe
localrings
O
C,p
foreverypoint
p

U
.Weput
A
=
O
C,p
and
K
=
Fr
(
A
)anddenoteby
E
,
F
and
T
thecorresponding
A
-modulesofsheaves
E
,
F
and
T
.Then
E
and
F
arefree
A
-modules,
hencewehaveinjections
E,

E

A
K
and
F,

F

A
K
.Since
T
isatorsionmodule,we
have
T

A
K
=0.Thenafterlocalizing(9)at
p

C
andtakingtensorproductwith
K
,we
obtainthecommutativediagram
∼E

A
K
−−
=
−→
F

A
K
y
ϕ
E

id
y
ϕ
F

id
∼E

A
K
−−
=
−→
F

A
K,

8TAMA´SHAUSELANDCHRISTIANPAULY
wherethehorizontalmapsareisomorphisms.So
ϕ
E

id
and
ϕ
F

id
areconjugate,hence
det(
ϕ
E

id
)=det(
ϕ
F

id
)

K
.Ontheotherhanddet(
ϕ
E

id
)anddet(
ϕ
F

id
)are
elementsin
A

K
,henceweobtainthedesiredequality.

Inthesequelwewillusethefollowingpropertiesofthenormmap:

Corollary3.3.
Let
E
and
F
betwotorsion-free
O
X
-modulessuchthat
0
−→E−→F−→T−→
0
,
where
T
isatorsion
O
X
-module.Let
s

Γ(
U,
B
)=Γ(
π

1
(
U
)
,
O
X
)
bealocalsectionof
B
over
theopensubset
U

C
.Weconsiderthemapsinducedbythemultiplicationwiththesection
s
inthedirectimagesheaves
π

E
and
π

F
,whichwedenoteby
µ
s
E

Hom
O
C
(
U
)
(
π

E
(
U
)


E
(
U
))
Fand
µ
s

Hom
O
C
(
U
)
(
π

F
(
U
)


F
(
U
))
.Thenwehavetheequality
det(
µ
s
E
)=det(
µ
s
F
)

Γ(
U,
O
C
)
.
Lemma3.4.
Let
p
:
X
e

X
beacoveringsuchthatthecokernelofthecanonicalinclusion
O
X
,

p

O
X
e
isatorsion
O
X
-module.Then,foranyinvertiblesheaf
L
over
X
wehave
∗Nm
X
e
/C
(
p
L
)=Nm
X/C
(
L
)
.
Proof.
Weconsidertheexactsequence
(10)0
−→O
X
−→
p

O
X
e
−→T−→
0
,
where
T
isatorsion
O
X
-module.Notethatthedirectimage
p

O
X
e
istorsion-free.Wedenotethe
O
C
-alg

ebra
π

p

O
X
e
by
B
e
.Notethat
B
e
isa
B
-module.Let
L
beaninvertible
O
X
-module,
η
λ
:
L
|
U
λ
−→B
|
U
λ
beasetoftrivializationsof
L
as
B
-module,and(
ω
λ,µ
)
λ,µ

Λ
bethecorresponding
1-cocyclewithvaluesin
B

.Thenthepull-back
p

L
correspondstoa1-cocycle(
p

ω
λ,µ
)
λ,µ

Λ
withvaluesin
B
e

obtainedfrom(
ω
λ,µ
)
λ,µ

Λ
underthecanonicalinclusion
B
,
→B
e
.Wenow
applyCorollary3.3totheexactsequence(10)andconcludethat
N
X
e
/C
(
p

ω
λ,µ
)=
N
X/C
(
ω
λ,µ
)

Γ(
U
λ

U
µ
,
O
C
).Thisprovesthelemma.

rSLemma3.5.
Let
X
=
i
=1
X
i
bethedecompositionof
X
intoirreduciblecomponents
X
i
.For
aninvertiblesheaf
L
,wedenoteby
L
i
=
L⊗
O
X
O
X
i
itsrestrictionto
X
i
.Then,wehavethe
equality
rONm
X/C
(
L
)=Nm
X
i
/C
(
L
i
)
.
1=irFProof.
Weapplythepreviouslemmatothecovering
p
:
X
e
=
i
=1
X
i

X
givenbythedisjoint
unionofthecurves
X
i
.

Lemma3.6.
Let
X
beanirreduciblecurveandlet
j
:
X
red
,

X
beitsunderlyingreduced
curve.Let
m
bethemultiplicityof
X
red
in
X
.Then,foranyinvertiblesheaf
L
over
X
wehave
Nm
X/C
(
L
)=Nm
X
red
/C
(
j

L
)

m
.
Proof.
The
O
C
-algebra
B
=
π

O
X
comesequippedwithanilpotentidealsheaf
J⊂B
such
that
B
red
=
B
/
J
=
π

O
X
red
.Wechooseacovering
{
U
λ
}
λ

Λ
of
C
byop

ensubsetswhich
trivializetheinvertiblesheaf
L
,i.e.thereexistsisomorphisms
η
λ
:
L
|
U
λ
−→B
|
U
λ
andsuch
that
J
|
U
λ
isgeneratedbyanelement
t
∈B
|
U
λ
.Thenmultiplicationwiththeinvertibleelement
ω
λ,µ
=
η
λ

η
µ
−|
1
U
λ

U
µ
preservesthefiltration
t
m

1
B
|
U
λ
⊂∙∙∙⊂
t
B
|
U
λ
⊂B
|
U
λ
andactsonthe

9quotientsasmultiplicationwith
ω
λr,eµd
∈B
|
rUe
λ
d

U
µ
.Itfollowsthat
N
X/C
(
ω
λ,µ
)=
N
X
red
/C
(
ω
λr,eµd
)
m
,
whichprovesthelemma.

3.3.
ThePrymvariety
Prym(
X/C
)
.
Givenaspectralcover
π
:
X

C
wedenotebyPic
0
(
X
)
theconnectedcomponentoftheidentityelementofthePicardgroupof
X
(seee.g.[Kl]).We
thendefinethePrymvarietyPrym(
X/C
)tobethekerneloftheNormmapNm
X/C
Prym(
X/C
):=kerNm
X/C
:Pic
0
(
X
)
−→
Pic
0
(
C
)
.
Werecallthat
n
denotesthedegreeofthecover
π
:
X

C
.Wechooseanamplelinebundle
O
C
(1)over
C
anddenoteby
O
X
(1)=
π

O
C
(1)itspull-backto
X
andby
δ
=deg
O
C
(1).
Definition3.7.
Let
E
beacoherent
O
X
-module.Therankanddegreeof
E
withrespectto
thepolarization
O
X
(1)
aretherationalnumbers
rk(
E
)
and
deg(
E
)
determinedbytheHilbert
polynomial
χ
(
X,
E⊗O
X
(
l
))=
nδl
rk(
E
)+deg(
E
)+rk(
E
)
χ
(
O
X
)
.
Theslopeof
E
isdefinedby
µ
(
E
)=
drekg((
EE
))
.Thesheaf
E
isstable(resp.semi-stable)if
E
is
torsion-freeandforanypropersubsheaf
E
0
⊂E
wehavetheequality
µ
(
E
0
)

(
E
)
(resp.

).
Remark3.8.
Thedefinitionsofrankanddegreeofacoherentsheaf
E
over
X
abovecoincide
withtheclassicaloneswhenthecurve
X
isintegral.The(semi-)stabilityconditionabove
coincideswiththe(semi-)stabilityconditionintroducedin[S1].
Remark3.9.
Usingtheequality
χ
(
X,
E⊗O
X
(
l
))=
χ
(
C,π

E⊗O
C
(
l
))weobtainthefollowing
formulae
(11)
n
rk(
E
)=rk(
π

E
)anddeg(
E
)+rk(
E
)
χ
(
O
X
)=deg(
π

E
)+rk(
π

E
)
χ
(
O
C
)
.
Proposition3.10.
Let
E
beatorsion-free
O
X
-moduleofintegralrank
r
=rk(
E
)
andlet
L
be
aninvertible
O
X
-module.Thenwehavetherelation
det(
π

(
E⊗L
))=det(
π

E
)

Nm
X/C
(
L
)

r
.
Proof.
Weshallusethenotationofsection3.1.Since
E
istorsion-free,thedirectimage
π

E
is
alocallyfree
O
C
-module.Wechooseacovering
{
U
λ
}
λ

Λ
of
C
forwhichboth
L
and
π

E
are
trivialized,i.e.,suchthatthereexistslocalisomorphisms
α
λ
:
π

E
|
U
λ


→O
U

λ
rn

λ
:
L
|
U
λ


→B
U
λ
.
Since
L
istrivialon
U
λ
wehaveanisomorphism
id
E

τ
λ
:
E⊗L
|
U
λ
−→E⊗B
|
U
λ
,
whichwecanconsiderasanisomorphismbetween
O
C
-modules
id
E

τ
λ
:
π

(
E⊗L
)
|
U
λ
−→
π

E
|
U
λ
.
Wecomposewith
α
λ
toobtainatrivializationof
π

(
E⊗L
)
|
U
λ
β
λ
:
π

(
E⊗L
)
|
U
λ
id

E


τ
λ
π

E
|
U
λ

α
λ
→O
U

λ
rn
.
Given
λ,µ

Λwecannowwritethetransitionfunctions
f
λ,µ
=
β
λ

β
µ

1
ofthevectorbundle
π

(
E⊗L
)as
1−f
λ,µ
:
O
U

rn

α
µ

(
π

E
)
|
U
λ,µ
id
E



ω
λ,µ
(
π

E
)
|
U
λ,µ

α
λ
→O
U

rn
,
λλ

10TAMA´SHAUSELANDCHRISTIANPAULY
wherewedenoteby
ω
λ,µ
=
τ
λ

τ
µ

1
the
B

-valuedtransitionfunctionsofthelinebundle
L
.We
deducefromthisexpressiontherelation
det(
f
λ,µ
)=det(
g
λ,µ
)

det(id
E

ω
λ,µ
)
,
where
g
λ,µ
=
α
λ

α
µ

1
denotesthetransitionfunctionsofthevectorbundle
π

E
.Hencethe
propositionfollowsifweshowtherelationdet(id
E

ω
λ,µ
)=det(
ω
λ,µ
)
r
,whichisprovedinthe
nextLemma.

Lemma3.11.
Let
E
beatorsion-free
O
X
-moduleandlet
s

Γ(
U,
B
)=Γ(
π

1
(
U
)
,
O
X
)
bea
localsectionof
B
overtheopensubset
U

C
.Wedenoteby
µ
s
E

Hom
O
C
(
U
)
(
π

E
(
U
)


E
(
U
))
themapinducedbymultiplicationwiththesection
s
.Thenwehaveanequality
det(
µ
s
E
)=det(
µ
s
)
r

Γ(
U,
O
C
)
.
Proof.
ByLemma2.6thereexistsanopensubset
j
:
V,

X
suchthat
j

E
isisomorphicto
m⊕j

Q
where
Q
isaquasi-freesheafoftheform

ik
=1
O
X
ii
.WethenapplyCorollary3.3tothe
twoexactsequences
0
−→E−→
j

j

E−→T
1
−→
0
,
and0
−→Q−→
j

j

Q−→T
2
−→
0
,
where
T
i
aretorsionsheaves.Thisleadstotheequalitydet(
µ
s
E
)=det(
µ
s
Q
).Itthereforesuffices
tocomputedet(
µ
s
Q
)intermsofdet(
µ
s
).Weput
n
=
k

l
with
l
=deg(
X
red
/C
).Thenwehave
1
X
k
1
X
k
1
X
k
r
=rk(
E
)=rk(
Q
)=
m
i
rk(
π

O
X
i
)=
m
i
il
=
m
i
i.
n
i
=1
n
i
=1
k
i
=1
Let
A
=
O
C,p
denotethelocalringatthepoint
p

C
andlet
B
denotethelocalizationof
π

O
X
atthepoint
p

C
.Thus
B
isaprojective
A
-moduleofrank
n
equippedwithafiltration
t
k

1
B
⊂∙∙∙⊂
tB

B,t

B
with
t
k
=0
.
Weput
B
1
=
B/tB
,thelocalizationof
π

O
X
red
atthepoint
p

C
.Since
B
isprojectivewe
canchooseasplitting
B
=
B
1

tB
1
⊕∙∙∙⊕
t
k

1
B
1
.
Usingthisdecompositionwecanwriteasection
s

B
as
s
=
s
0
+
ts
1
+
∙∙∙
+
t
k

1
s
k

1
with
s
j

B
1
.Moreover,thelocalizationof
π

O
X
i
atthepoint
p

C
isgivenby
B
i
:=
B
1

tB
1
⊕∙∙∙⊕
t
i

1
B
1
andthematrixofthemultiplicationwith
s
in
B
i
iswithrespecttothis
decompositionlowerblock-triangularandhasdeterminantdet(
µ
sB
i
)=det(
µ
sB
01
)
i
.Therefore
kPYdet(
µ
s
Q
)=det(
µ
sB
i
)
m
i
=det(
µ
sB
1
)
ik
=1
im
i
.
01=iOntheotherhanddet(
µ
s
)=det(
µ
s
O
X
)=det(
µ
sB
k
)=det(
µ
sB
01
)
k
,whichleadstothedesired
equality.

Takingthetrivialsheaf
E
=
O
X
inProposition3.10weobtainthefollowingdescriptionof
thenormmap:
Corollary3.12.
Foranyinvertible
O
X
-module,wehave
Nm
X/C
(
L
)=det(
π

L
)

det(
π

O
X
)

1
.