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REGULATORS OF CANONICAL EXTENSIONS ARE TORSION: THE SMOOTH DIVISOR CASE

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Niveau: Supérieur, Doctorat, Bac+8
ar X iv :0 70 7. 03 72 v2 [ ma th. AG ] 4 J ul 20 07 REGULATORS OF CANONICAL EXTENSIONS ARE TORSION: THE SMOOTH DIVISOR CASE JAYA NN IYER AND CARLOS T SIMPSON Abstract. In this paper, we prove a generalization of Reznikov's theorem which says that the Chern-Simons classes and in particular the Deligne Chern classes (in degrees > 1) are torsion, of a flat vector bundle on a smooth complex projective variety. We consider the case of a smooth quasi–projective variety with an irreducible smooth divisor at infinity. We define the Chern-Simons classes of the Deligne's canonical extension of a flat vector bundle with unipotent monodromy at infinity, which lift the Deligne Chern classes and prove that these classes are torsion. Contents 1. Introduction 1 2. Idea for the construction of secondary classes 4 3. The C∞-trivialization of canonical extensions 10 4. Patched connection on the canonical extension 13 5. Compatibility with the Deligne Chern class 20 6. Rigidity of the secondary classes 22 7. A deformational variant of the patching construction in K-theory 25 8. Hermitian K-theory and variations of Hodge structure 31 9. The generalization of Reznikov's theorem 37 References 39 1. Introduction Chern and Simons [Chn-Sm] and Cheeger [Ch-Sm] introduced a theory of differential cohomology on smooth manifolds.

  • has unipotent

  • let

  • once around

  • projective variety

  • b?

  • deligne chern

  • characters has

  • unipotent monodromy around


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REGULATORSOFCANONICALEXTENSIONSARETORSION:THE
SMOOTHDIVISORCASE
JAYANNIYERANDCARLOSTSIMPSON

Abstract.
Inthispaper,weproveageneralizationofReznikov’stheoremwhichsays
thattheChern-SimonsclassesandinparticulartheDeligneChernclasses(indegrees
>
1)aretorsion,ofaflatvectorbundleonasmoothcomplexprojectivevariety.We
considerthecaseofasmoothquasi–projectivevarietywithanirreduciblesmoothdivisor
atinfinity.WedefinetheChern-SimonsclassesoftheDeligne’s
canonicalextension
of
aflatvectorbundlewithunipotentmonodromyatinfinity,whichlifttheDeligneChern
classesandprovethattheseclassesaretorsion.

Contents

1.Introduction
2.Ideafortheconstructionofsecondaryclasses
3.The
C

-trivializationofcanonicalextensions
4.Patchedconnectiononthecanonicalextension
5.CompatibilitywiththeDeligneChernclass
6.Rigidityofthesecondaryclasses
7.Adeformationalvariantofthepatchingconstructionin
K
-theory
8.Hermitian
K
-theoryandvariationsofHodgestructure
9.ThegeneralizationofReznikov’stheorem
References

140131022252137393

1.
Introduction
ChernandSimons[Chn-Sm]andCheeger[Ch-Sm]introducedatheoryofdifferential
cohomologyonsmoothmanifolds.Forvectorbundleswithconnection,theydefined
classesorthesecondaryinvariantsintheringofdifferentialcharacters.Theseclasses
lifttheclosedformdefinedbythecurvatureformofthegivenconnection.Inparticular
whentheconnectionisflat,thesecondaryinvariantsyieldclassesinthecohomologywith
R
/
Z
-coefficients.ThesearetheChern-Simonsclassesofflatconnections.
0
MathematicsClassificationNumber:14C25,14D05,14D20,14D21
0
Keywords:LogarithmicConnections,Delignecohomology,Secondaryclasses.
1

2

J.N.IYERANDC.T.SIMPSON

Thefollowingquestionwasraisedin[Ch-Sm,p.70-71](seealso[Bl,p.104])byCheeger
andSimons:
Question1.1.
Suppose
X
isasmoothmanifoldand
(
E,

)
isaflatconnectionon
X
.
AretheChern-Simonsclasses
c
b
i
(
E,

)
of
(
E,

)
torsionin
H
2
i

1
(
X,
R
/
Z
)
,for
i

2
?
Suppose
X
isasmoothprojectivevarietydefinedoverthecomplexnumbers.Let
(
E,

)beavectorbundlewithaflatconnection

.S.Bloch[Bl]showedthatfora
unitaryconnectiontheChern-SimonsclassesaremappedtotheChernclassesof
E
inthe
Delignecohomology.TheaboveQuestion1.1togetherwithhisobservationledhimto
conjecturethattheChernclassesofflatbundlesaretorsionintheDelignecohomologyof
X
,indegreesatleasttwo.
A.BeilinsondefineduniversalsecondaryclassesandH.Esnault[Es]constructedsec-
ondaryclassesusingamodifiedsplittingprincipleinthe
C
/
Z
-cohomology.Theseclasses
areshowntobeliftingsoftheChernclassesintheDelignecohomology.Theseclasses
alsohaveaninterpretationintermsofdifferentialcharacters,andtheoriginal
R
/
Z
classes
ofChern-Simonsareobtainedbytheprojection
C
/
Z

R
/
Z
.Theimaginarypartsof
the
C
/
Z
classesareBorel’svolumeregulators
Vol
2
p

1
(
E,

)

H
2
p

1
(
X,
R
).Allthecon-
structionsgivethesameclassinodddegrees,calledasthesecondaryclassesonX(see
[DHZ],[Es2]foradiscussiononthis).
Reznikov[Re],[Re2]showedthatthesecondaryclassesof(
E,

)aretorsionintheco-
homology
H
2
i

1
(
X,
C
/
Z
)of
X
,when
i

2.Inparticular,heprovedtheabovementioned
conjectureofBloch.
Ouraimhereistoextendthisresultwhen
X
issmoothandquasi–projectivewith
anirreduciblesmoothdivisor
D
atinfinity.Weconsideraflatbundleon
X
whichhas
unipotentmonodromyatinfinity.Wedefinesecondaryclasseson
X
(extendingtheclasses
on
X

D
oftheflatconnection)andwhichlifttheDeligneChernclasses,andshowthat
theseclassesaretorsion.
Ourmaintheoremis
Theorem1.2.
Suppose
X
isasmoothquasi–projectivevarietydefinedover
C
.Let
(
E,

)
beaflatconnectionon
U
:=
X

D
associatedtoarepresentation
ρ
:
π
1
(
U
)

GL
r
(
C
)
.
Assumethat
D
isasmoothandirreducibledivisorand
(
E,

)
betheDelignecanonical
extensionon
X
withunipotentmonodromyaround
D
.Thenthesecondaryclasses
c
b
p
(
ρ/X
)

H
2
p

1
(
X,
C
/
Z
)
of
(
E,

)
aretorsion,for
p>
1
.If,furthermore,
X
isprojectivethentheChernclasses
of
E
aretorsionintheDelignecohomologyof
X
,indegrees
>
1
.

Whatwedoherecaneasilybegeneralizedtothecasewhen
D
issmoothandhas
severaldisjointirreduciblecomponents.Ontheotherhand,thegeneralizationtoanormal

REGULATORSOFCANONICALEXTENSIONSARETORSION:THESMOOTHDIVISORCASE3
crossingsdivisorpresentssignificantnewdifficultieswhichwedon’tyetknowhowto
handle,sothiswillbeleftforthefuture.
Themainconstructionsinthispaperareasfollows.Wewillconsiderthefollowing
situation.Suppose
X
isasmoothmanifold,and
D

X
isaconnectedsmoothclosed
subsetofrealcodimension2.Let
U
:=
X

D
andsupposewecanchooseareasonable
tubularneighborhood
B
of
D
.Let
B

:=
B

U
=
B

D
.Itfollowsthat
π
1
(
B

)

π
1
(
B
)
issurjective.Thediagram
∗BB→)1(↓↓XU→isahomotopypushoutdiagram.Notealsothat
B
retractsto
D
,and
B

hasatubular
structure:
B

=

S
×
(0
,
1)
where
S
=

∂B
isacirclebundleover
D
.
Wesaythat(
X,D
)is
complexalgebraic
if
X
isasmoothcomplexquasiprojective
varietyand
D
anirreduciblesmoothdivisor.
Supposewearegivenarepresentation
ρ
:
π
1
(
U
)

GL
r
(
C
),correspondingtoalocal
system
L
over
U
,orequivalentlytoavectorbundlewithflatconnection(
E,

).Let
γ
bealoopgoingoutfromthebasepointtoapointnear
D
,oncearound,andback.
Then
π
1
(
B
)isobtainedfrom
π
1
(
B

)byaddingtherelation
γ

1.Weassumethat
the
monodromyof
ρ
atinfinityisunipotent
,bywhichwemeanthat
ρ
(
γ
)shouldbeunipotent.
Thelogarithmisanilpotenttransformation
11N
:=log
ρ
(
γ
):=(
ρ
(
γ
)

I
)

(
ρ
(
γ
)

I
)
2
+(
ρ
(
γ
)

I
)
3

...,
32wheretheseriesstopsafterafinitenumberofterms.
Inthissituation,thereisacanonicalandnaturalwaytoextendthebundle
E
toa
bundle
E
over
X
,knownasthe
Delignecanonicalextension
[De].Theconnection

extendstoaconnection

whosesingulartermsinvolvedlooklocallylike
Ndθ
where
θ
istheangularcoordinatearound
D
.Inanappropriateframethesingularitiesof

areonlyinthestrictuppertriangularregionoftheconnectionmatrix.Inthecomplex
algebraiccase,(
E,

)areholomorphic,andindeedalgebraicwithalgebraicstructure
uniquelydeterminedbytherequirementthat

haveregularsingularities.Theextended
bundle
E
isalgebraicon
X
and

becomesalogarithmicconnection[De].
Wewilldefine
extendedregulatorclasses
c
b
p
(
ρ/X
)

H
2
p

1
(
X,
C
/
Z
)
whichrestricttotheusualregulatorclasseson
U
.Theirimaginarypartsdefine
extended
volumeregulators
whichwewriteas
Vol
2
p

1
(
ρ/X
)

H
2
p

1
(
X,
R
).

4

J.N.IYERANDC.T.SIMPSON

Thetechniquefordefiningtheextendedregulatorclassesistoconstructa
patched
connection

#
over
X
.Thiswillbeasmoothconnection,howeveritisnotflat.Still,
thecurvaturecomesfromthesingularitiesof

whichhavebeensmoothedout,sothe
curvatureisupper-triangular.Inparticular,theChernformsfor

#
arestillidentically
zero.TheCheeger-Simonstheoryofdifferentialcharactersprovidesaclassof

#
inthe
groupofdifferentialcharacters,mappingtothegroupofclosedforms.Sincetheimage,
whichistheChernform,vanishes,thedifferentialcharacterliesinthekernelofthis
mapwhichisexactly
H
2
p

1
(
X,
C
/
Z
)[Ch-Sm,Cor.2.4].Thisistheconstructionofthe
regulatorclass.
TheproofofDupont-Hain-ZuckerthattheregulatorclassliftstheDeligneChernclass,
goesthroughwordforwordheretoshowthatthisextendedregulatorclassliftsthe
DeligneChernclassofthecanonicalextension
E
inthecomplexalgebraiccase.Forthis
part,weneed
X
projective.
Wealsogiveadifferentconstructionoftheregulatorclasses,usingthedeformationthe-
oremin
K
-theory.Thefiltrationwhichwewillusetodefinethepatchedconnection,also
leadstoapolynomialdeformationon
B

betweentherepresentation
ρ
anditsassociated-
graded.Then,usingthefactthat
BGL
(
F
[
t
])
+
ishomotopy-equivalentto
BGL
(
F
)
+
and
thefactthatthesquare(1)isahomotopypushout,thisallowsustoconstructamapfrom
X
to
BGL
(
F
)
+
andhencepullbacktheuniversalregulatorclasses.Corollary7.5below
saysthatthesearethesameastheextendedregulatorsdefinedbythepatchedconnec-
tion.Ontheotherhand,thecounterpartofthedeformationconstructioninhermitian
K
-theoryallowsustoconcludethattheextendedvolumeregulatoriszerowhenever
ρ
underliesacomplexvariationofHodgestructureinthecomplexalgebraiccase.
Arigiditystatementforthepatchedconnectionsisdiscussedandprovedinmoregen-
eralityin
§
6.AlloftheingredientsofReznikov’soriginalproof[Re2]arenowpresentfor
theextendedclasses,includingMochizuki’stheoremthatanyrepresentationcanbede-
formedtoacomplexvariationofHodgestructure[Mo].Thusweshowthegeneralization
ofReznikov’sresult.
Acknowledgements
:WethankP.Deligneforhavingusefuldiscussions.Hissuggestiontoconsider
aglueingconstructionofthesecondaryclasses(see
§
2.3)andhisletter[De3],motivatedsomeofthe
mainconstructions,andwearethankfultohim.WealsothankH.Esnaultforexplainingsomeofher
constructionsin[Es].ThefirstnamedauthorissupportedbytheNationalScienceFoundation(NSF)
underagreementNo.DMS-0111298.

2.
Ideafortheconstructionofsecondaryclasses

WebeginbyrecallingthedifferentialcohomologyintroducedbyChern,Cheegerand
Simons[Ch-Sm],[Chn-Sm].Sincewewanttolookatlogarithmicconnections,weconsider
thesecohomologiesoncomplexanalyticvarietiesandontheirsmoothcompactifications.
Ouraimistodefinesecondaryclassesinthe
C
/
Z
-cohomologyforlogarithmicconnections

REGULATORSOFCANONICALEXTENSIONSARETORSION:THESMOOTHDIVISORCASE5

whichhaveunipotentmonodromyalongasmoothboundarydivisor.Aglueingconstruc-
tionwassuggestedbyDeligne,whichusesglueingofsecondaryclassesontheopenvariety
andonatubularneighbourhoodoftheboundarydivisor.In
§
4thiswillbemadeprecise
usingapatchedconnection.
Let
X
beanonsingularvarietydefinedoverthecomplexnumbers.Inthefollowing
discussionwewillinterchangeablyusethenotation
X
forthealgebraicvarietyorthe
underlyingcomplexanalyticspace.

2.1.
Analyticdifferentialcharacterson
X
[Ch-Sm]
.
Let
S
k
(
X
)denotethegroupof
k
-dimensionalsmoothsingularchainson
X
,withintegercoefficients.Let
Z
k
(
X
)denote
thesubgroupofcycles.Letusdenote
S

(
X,
Z
):=Hom
Z
(
S

(
X
)
,
Z
)
thecomplexof
Z
-valuedsmoothsingularcochains,whoseboundaryoperatorisdenoted
by
δ
.Thegroupofsmoothdiffferential
k
-formson
X
withcomplexcoefficientsisdenoted
by
A
k
(
X
)andthesubgroupofclosedformsby
A
ckl
(
X
).Then
A

(
X
)iscanonicallyem-
beddedin
S

(
X
),byintegratingformsagainstthesmoothsingularchains.Infact,we
haveanembedding
i
Z
:
A

(
X
)
֒

S

(
X,
C
/
Z
)
.
Thegroupofdifferentialcharactersofdegree
k
isdefinedas
H
c
k
(
X,
C
/
Z
):=
{
(
f,α
)

Hom
Z
(
Z
k

1
(
X
)
,
C
/
Z
)

A
k
(
X
):
δ
(
f
)=
i
Z
(
α
)and

=0
}
.
Thereisacanonicalandfunctorialexactsequence:
(2)0
−→
H
k

1
(
X,
C
/
Z
)
−→
H
c
k
(
X,
C
/
Z
)
−→
A
ckl
(
X,
Z
)
−→
0
.
Here
A
ckl
(
X,
Z
):=ker(
A
ckl
(
X
)
−→
H
k
(
X,
C
/
Z
)).Similarly,onedefinesthegroupof
differentialcharacters
H
c
k
(
X,
R
/
Z
)with
R
/
Z
-coefficients.
Forthestudyofinfinitesimalvariationsofdifferentialcharacters,wehavethefollowing
remarkaboutthetangentspace.
Lemma2.1.
Thegroupofdifferentialcharactershasthestructureofinfinitedimensional
abelianLiegroup.Itstangentspaceattheorigin(orbytranslation,atanypoint)is
naturallyidentifiedas
c
k

A
k

1
(
X,
C
)
T
0
H
(
X,
C
/
Z
)=
dA
k

2
(
X,
C
)
.
Proof.
Atangentvectorcorrespondstoapath(
f
t

t
).Anelement
β

A
k

1
(
X,
C
)maps
tothepathgivenby
f
t
(
z
):=
t
z
β
and
α
t
:=
td
(
β
).Lookingattheaboveexactsequence
R(2),weseethatthismapinducesanisomorphismfrom
A
k

1
(
X,
C
)
/dA
k

2
(
X,
C
)tothe
tangentspaceof
H
c
k
(
X,
C
/
Z
).

6

J.N.IYERANDC.T.SIMPSON

2.2.
SecondaryclassesandtheCheeger-Chern-Simonsclasses.
Suppose(
E,

)
isavectorbundlewithaconnectionon
X
.ThentheChernforms
k2c
k
(
E,

)

A
cl
(
X,
Z
)
for0

k

rank(
E
),aredefinedusingtheuniversalWeilhomomorphism[Chn-Sm].
Thereisaninvariantandsymmetricpolynomial
P
ofdegree
k
in
k
variablesonthe
Liealgebra
gl
r
suchthatifΩisthecurvatureof

then
c
k
(
E,

)=(

1)
k
P

,...,
Ω).
When
X
i
=
X
foreach
i
,then
P
(
X,...,X
)=trace(

k
X
)(see[Gri-Ha,p.403]),however
thewedgeproducthereistakeninthevariable
C
r
,notthewedgeofformsonthebase.If
PX
isadiagonalmatrixwitheigenvalues
λ
1
,...,λ
r
then
P
(
X,...,X
)=
I
λ
i
1

λ
i
k
.We
canalsoexpress
P
intermsofthetracesofproductsofmatrices.Inthisexpression,the
highestordertermof
P
isthesymmetrizationof
Tr
(
X
1

X
k
)multipliedbyaconstant,
thelowerordertermsaresymmetrizationsof
Tr
(
X
1

X
i
1
)
Tr
(

)

Tr
(
X
i
a
+1

X
k
),
withsuitableconstantcoefficients.
Thecharacteristicclasses
c
b
k
(
E,

)

H
d
2
k
(
X,
C
/
Z
)
aredefinedin[Ch-Sm]usingafactorizationoftheuniversalWeilhomomorphismandlook-
ingattheuniversalconnections[Na-Ra].Theseclassesarefunctorialliftingsof
c
k
(
E,

).
Oneofthekeypropertiesoftheseclassesisthevariationalformulaincaseofafamily
ofconnections.If
{∇
t
}
isa
C

familyofconnectionson
E
,then—referingtoLemma2.1
forthetangentspaceofthespaceofdifferentialcharacters—wehavetheformula
dd(3)
dtc
b
k
(
E,

t
)=
k
P
(
dt

t
,
Ω
t
,...,
Ω
t
)
,
see[Ch-Sm,Proposition2.9].
If
E
istopologicallytrivial,thenanyconnectionisconnectedbyapathtothetrivial
connectionforwhichthecharacteristicclassisdefinedtobezero.Thevariationalformula
thusservestocharacterize
c
b
k
(
E,

t
)forall
t
.
Remark2.2.
Iftheform
c
k
(
E,

)
iszero,thentheclass
c
b
k
(
E,

)
liesin
H
2
k

1
(
X,
C
/
Z
)
.
If
(
E,

)
isaflatbundle,then
c
k
(
E,

)=0
andtheclasses
c
b
k
(
E,

)
arecalledthe
secondaryclasses
or
regulators
of
(
E,

)
.Noticethattheclassdependsonthechoiceof

.WewillalsorefertotheseclassesastheChern-Simonsclassesin
C
/
Z
-cohomology.
Inthecaseofaflatbundle,aftergoingtoafinitecoverthebundleistopologically
trivialbytheresultofDeligne-Sullivanwhichwillbediscussedin
§
3below.Thus,at
leastthepullbacktothefinitecoverof
c
b
k
(
E,

)canbeunderstoodusingthevariational
methodsdescribedabove.
Beilinson’stheoryofuniversalsecondaryclassesyieldclassesforaflatconnection
(
E,

),
)4(

c
b
k
(
E,

)

H
2
k

1
(
X,
C
/
Z
)
,k

1

REGULATORSOFCANONICALEXTENSIONSARETORSION:THESMOOTHDIVISORCASE7

whicharefunctorialandadditiveoverexactsequences.Furthermore,Esnault[Es]usinga
modifiedsplittingprinciple,Karoubi[Ka2]using
K
-theoryhavedefinedsecondaryclasses.
Theseclassesarefunctorialandadditive.Theseclassesthenagreewiththeuniversally
definedclassin(4)(see[Es,p.323]).
When
X
isasmoothprojectivevariety,Dupont-Hain-Zucker[Zu],[DHZ]andBrylinski
[Br]haveshownthattheChern–SimonsclassesareliftingsoftheDeligneChernclass
c
k
D
(
E
)underthemapobtainedbydividingoutbytheHodgefilteredpiece
F
k
,
H
2
k

1
(
X,
C
/
Z
)
−→
H
D
2
k
(
X,
Z
(
k
))
.
Byfunctorialityandadditiveproperties,theclassesin(4)lifttheChern-Simonsclasses
definedaboveusingdifferentialcharacters,viatheprojection
C
/
Z

R
/
Z
.
Infact,Cheeger-Simonsexplicitlytooktherealpartintheirformulaatthestartof
[Ch-Sm,
§
4].Seealso[Bl]forunitaryconnections,[So],[Gi-So]when
X
issmoothand
projective;foradiscussiononthissee[Es2].

2.3.
Secondaryclassesoflogarithmicconnections.
Suppose
X
isanonsingular
varietyand
D

X
anirreduciblesmoothdivisor.Let
U
:=
X

D
.Chooseatubular
neighborhood
B
of
D
andlet
B

:=
B

U
=
B

D
.
Let(
E,

)beacomplexanalyticvectorbundleon
U
withaconnection

.Consider
alogarithmicextension(
E,

)(see[De])on
X
oftheconnection(
E,

).Assumingthat
theresiduesarenilpotent,wewanttoshowthattheclasses
c
b
k
(
E,

)

H
2
i

1
(
U,
C
/
Z
)
extendon
X
togiveclassesinthecohomologywith
C
/
Z
-coefficientswhichmaptothe
DeligneChernclassof
E
.
WewanttousetheMayer-Vietorissequence(asuggestionfromDeligne)tomotivate
aconstructionofsecondaryclassesinthissituation.Thepreciseconstructionwillbe
carriedoutin
§
4.
Considertheresiduetransformation
η
:
E
−→
E

Ω
X
(log
D
)

re

s
E
⊗O
D
.
Byassumption
η
isnilpotentandlet
r
betheorderof
η
.
Considerthe
Kernelfiltration
of
E
D
inducedbythekernelsoftheoperator
η
:
0=
W
0
,D

W
1
,D

W
2
,D

...

W
r,D
=
E
D
.
ereHW
j,D
:=kernel(
η

j
:
E
D
−→
E
D
)
.
Denotethegradedpieces
Gr
j
(
E
D
):=
W
j,D
/W
j
+1
,D
andtheassociatedgraded
Gr
(
E
D
):=

rj

=10
Gr
j
(
E
D
)
.

8J.N.IYERANDC.T.SIMPSON
Lemma2.3.
Eachgradedpiece
Gr
j
(
E
D
)
(for
0

j<r
)isendowedwithaflatcon-
nectionalong
D
.Furthermore,thefiltrationof
E
D
by
W
j,D
extendstoafiltrationof
E
byholomorphicsubbundles
W
r
definedinatubularneighborhood
B
ofthedivisor
D
.On
B

thesesubbundlesarepreservedbytheconnection

,and

inducesoneachgraded
piece
Gr
j
(
E
B

)
aconnectionwhichextendstoaflatconnectionover
B
,andinducesthe
connectionmentionedinthefirstphrase,on
Gr
j
(
E
D
)
.

Proof.
Suppose
n
isthedimensionofthevariety
X
.Consideraproductof
n
-opendisks
Δ
n
withcoordinates(
t
1
,t
2
,...,t
n
)aroundapointofthedivisor
D
sothat
D
islocally
definedby
t
1
=0.Let
γ
bethegeneratorofthefundamentalgroupofthepunctureddisk
Δ
n
−{
t
1
=0
}
.Then
γ
isthemonodromyoperatoractingonafibre
E
t
,for
t

Δ
n
−{
t
1
=
0
}
.Theoperator
11N
=log
γ
=(
γ

I
)

(
γ

I
)
2
+(
γ

I
)
3

...
32isnilpotentsincebyassumptionthelocalmonodromy
γ
isunipotent.Further,theorder
ofunipotencyof
γ
coincideswiththeorderofnilpotencyof
N
.Considerthefiltrationon
thefibre
E
t
inducedbytheoperator
N
:
0=
W
0
(
t
)

W
1
(
t
)

...

W
r
(
t
)=
E
t
.
suchthat
W
j
(
t
):=kernel(
N
j
:
E
t
−→
E
t
)
.
Denotethegradedpieces
gr
tj
:=
W
j
(
t
)
/W
j
+1
(
t
)
.
Thenwenoticethattheoperator
N
actstriviallyonthegradedpieces
gr
tj
.Thismeans
that
γ
actsasidentityon
gr
tj
.Inotherwords,
gr
tj
(for
t

Δ
n
)formsalocalsystemon
Δ
n
andextendsasalocalsystem
gr
j
inatubularneighbourhood
B
of
D
in
X
.
Theoperationof
γ
around
D
canbeextendedtotheboundary(see[De]or[Es-Vi,
c)Proposition]).Moreprecisely,theoperation
γ
(resp.
N
)extendstothesheaf
E
and
definesanendomorphism
γ
(resp.
N
)of
E
D
suchthat
exp(

2
πi.η
)=
γ
D
.
Thisimpliesthatthekernelsdefinedbytheresiduetransformation
η
and
N
arethesame
over
D
.Thegradedpiece
Gr
j
isthebundleassociatedtothelocalsystem
gr
j
inatubular
neighbourhood
B
of
D
in
X
.

Corollary2.4.
If
(
E
B
,

B
)
denotestherestrictionof
(
E,

)
onthetubularneighbourhood
B
,theninthe
K
0
-group
K
an
(
B
)
ofanalyticvectorbundles,wehavetheequality
E
B
=
Gr
(
E
B
)=

j
Gr
j
(
E
B
)
.

REGULATORSOFCANONICALEXTENSIONSARETORSION:THESMOOTHDIVISORCASE9
Corollary2.5.
Wecandefinethesecondaryclassesoftherestriction
(
E
B
,

B
)
tobe
c
b
i
(
E
B
,

B
):=
c
b
i
(
Gr
(
E
B
))
in
H
2
i

1
(
B,
C
/
Z
)
.
Fortheaboveconstruction,wecouldhavereplacedthekernelfiltrationbyDeligne’s
monodromyweightfiltration
0=
W

r

1

...

W
r
=
E
orindeedbyanyfiltrationoftheflatbundle(
E
B

,

B

)satisfyingthefollowingcondition:
wesaythat
W

is
graded-extendable
ifitisafiltrationbyflatsubbundlesorequivalentlyby
sub-localsystems,andifeachassociated-gradedpiece
Gr
jW
correspondstoalocalsystem
whichextendsfrom
B

to
B
.
Consideratubularneighbourhood
B
of
D
,asobtainedinLemma2.3,and
B

:=
B

U
=
B

D
.AssociatetheMayer-Vietorissequenceforthepair(
U,B
):
H
2
i

2
(
B

,
C
/
Z
)

H
2
i

1
(
X,
C
/
Z
)

H
2
i

1
(
B,
C
/
Z
)

H
2
i

1
(
U,
C
/
Z
)

H
2
i

1
(
B

,
C
/
Z
)

.
Considertherestrictions(
E
B
,

B
)on
B
and(
E,

)on
U
.Thenwehavethesecondary
classes,definedinCorollary2.5,
(5)
c
b
i
(
E
B
,

B
)

H
2
i

1
(
B,
C
/
Z
)
dna(6)
c
b
i
(
E,

)

H
2
i

1
(
U,
C
/
Z
)
suchthat
c
b
i
(
E
B
,

B
)
|
B

=
c
b
i
(
E,

)
|
B


H
2
i

1
(
B

,
C
/
Z
)
.
TheaboveMayer-Vietorissequenceyieldsaclass
(7)
c
b
i
(
E,

)

H
2
i

1
(
X,
C
/
Z
)
whichisobtainedbyglueingtheclassesin(5)and(6).
Assuch,theMayer-Vietorissequencedoesn’tuniquelydeterminetheclass:thereis
apossibleindeterminacybytheimageof
H
2
i

2
(
B

,
C
/
Z
)undertheconnectingmap.
Nonetheless,wewillshowin
§
4,usingapatchedconnection,thatthereisacanonically
determinedclass
c
b
i
(
E,

)asabovewhichisfunctorialandadditive(
§
6)andmoreoverit
liftstheDeligneChernclass(
§
5).

10J.N.IYERANDC.T.SIMPSON
3.
The
C

-trivializationofcanonicalextensions
Tofurthermotivatetheconstructionofregulatorclasses,wedigressforamomentto
giveageneralizationoftheresultofDeligneandSullivanontopologicaltrivialityofflat
bundles,tothecaseofthecanonicalextension.Thetopologicalmodelofthecanonical
extensionweobtaininthissection,onanideacommunicatedtousbyDeligne[De3],
motivatestheconstructionofafiltrationtriplein
§
7.3whichisrequiredtodefineregulator
classesusing
K
-theory.
Suppose
X
isaproper
C

-manifoldofdimension
d
.Let
E
beacomplexvectorbundleof
rank
n
.Itiswell-knownthatif
N

2
d
,thentheGrassmanianmanifoldGrass(
n,
C
n
+
N
)of
n
-dimensionalsubspacesof
C
n
+
N
,classifiescomplexvectorbundlesofrank
n
onmanifolds
ofdimension

d
.Inotherwords,givenacomplexvectorbundle
E
on
X
,thereexistsa
morphism
f
:
X
−→
Grass(
n,
C
n
+
N
)
suchthatthepullback
f

U
ofthetautologicalbundle
U
onGrass(
n,
C
n
+
N
)is
E
.Ifthe
morphism
f
ishomotopictoaconstantmapthen
E
istrivialasa
C

-bundle.This
observationisusedtoobtainanupperboundfortheorderoftorsionofBettiChern
classesofflatbundles.

3.1.
C

-trivializationofflatbundles.
Suppose
E
isequippedwithaflatconnection

.ThentheChern-WeiltheoryimpliesthattheBettiChernclasses
c
iB
(
E
)

H
2
i
(
X,
Z
)
aretorsion.AnupperboundfortheorderoftorsionwasgivenbyGrothendieck[Gk].An
explanationofthetorsion-propertyisgivenbythefollowingtheoremduetoDeligneand
Sullivan:
Theorem3.1.
[De-Su]
Let
V
beacomplexlocalsystemofdimension
n
onacompact
polyhedron
X
and
V
=
V
⊗O
X
bethecorrespondingflatvectorbundle.Thereexistsa
finitesurjectivecovering
π
:
X
˜
−→
X
of
X
suchthatthepullbackvectorbundle
π

V
is
trivialasa
C

-bundle.
Anupperboundfortheorderoftorsionisalsoprescribedintheirproofwhichdepends
onthefieldofdefinitionofthemonodromyrepresentation.

3.2.
C

-trivializationofcanonicalextensions.
Suppose
X
isacomplexanalyticva-
riety
D

X
asmoothirreducibledivisor,andput
U
:=
X

D
.Consideraflatvector
bundle(
E,

)on
U
anditscanonicalextension(
E,

)on
X
.Assumethattheresidues
of

arenilpotent.ThenacomputationofthedeRhamChernclassesbyEsnault[Es-Vi,
AppendixB]showsthattheseclassesarezero.ThisimpliesthattheBettiChernclasses
of
E
aretorsion.WewanttoextendtheDeligne-Sullivantheoreminthiscase,reflecting
thetorsionpropertyoftheBettiChernclasses.