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

Sujets

• Once Around
• Algebraic variety

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 9
• Langue: English

Extrait

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,theChern