A SHORT COURSE ON GEOMETRIC MOTIVIC INTEGRATION

Description

Niveau: Supérieur, Doctorat, Bac+8
ar X iv :m at h/ 05 07 40 4v 1 [m ath .A G] 2 0 J ul 20 05 A SHORT COURSE ON GEOMETRIC MOTIVIC INTEGRATION MANUEL BLICKLE Abstract. These notes grew out of the authors e?ort to understand the theory of motivic integration. They give a short but thorough introduction to the flavor of mo- tivic integration which nowadays goes by the name of geometric motivic integration. Motivic integration was introduced by Kontsevich and the foundations were worked out by Denef, Loeser, Batyrev and Looijenga. We focus on the smooth complex case and present the theory as self contained as possible. As an illustration we give some applications to birational geometry which originated in the work of Mustat¸aˇ. Contents 1. The invention of motivic integration. 2 2. Geometric motivic integration 4 2.1. The value ring of the motivic measure 5 2.2. The arc space J∞(X) 7 2.3. An algebra of measurable sets 9 2.4. The measurable function associated to a subscheme 10 2.5. Definition and computation of the motivic integral 12 3. The transformation rule 14 3.1. Images of cylinders under birational maps. 16 3.2. Proof of transformation rule using Weak Factorization 19 4. Brief outline of a formal setup for the motivic measure. 20 4.1. Properties of the motivic measure 21 4.2.

• kontsevich's result


• kxi ?


• ky ?


• hodge numbers


• bb bb


• birationally equivalent


• jj jj


• aa aa


• kxi ? π?i


• smooth varieties

Sujets

• Gordon
• Salt Lake City
• Bayreuth
• Birational geometry

Informations

• Publié par: profil-feym-2012
• Nombre de lectures: 37
• Langue: English

Extrait

ASHORTCOURSEONGEOMETRICMOTIVICINTEGRATION
MANUELBLICKLE
Abstract.
Thesenotesgrewoutoftheauthorsefforttounderstandthetheoryof
motivicintegration
.Theygiveashortbutthoroughintroductiontotheflavorofmo-
tivicintegrationwhichnowadaysgoesbythenameof
geometricmotivicintegration
.
MotivicintegrationwasintroducedbyKontsevichandthefoundationswereworked
outbyDenef,Loeser,BatyrevandLooijenga.Wefocusonthesmoothcomplexcase
andpresentthetheoryasselfcontainedaspossible.Asanillustrationwegivesome
applicationstobirationalgeometrywhichoriginatedintheworkofMusta¸taˇ.

Contents
1.Theinventionofmotivicintegration.
2.Geometricmotivicintegration
2.1.Thevalueringofthemotivicmeasure
2.2.Thearcspace
J
∞
(
X
)
2.3.Analgebraofmeasurablesets
2.4.Themeasurablefunctionassociatedtoasubscheme
2.5.Definitionandcomputationofthemotivicintegral
3.Thetransformationrule
3.1.Imagesofcylindersunderbirationalmaps.
3.2.ProofoftransformationruleusingWeakFactorization
4.Briefoutlineofaformalsetupforthemotivicmeasure.
4.1.Propertiesofthemotivicmeasure
4.2.Motivicintegrationonsingularvarieties
5.Birationalinvariantsviamotivicintegration
5.1.Notationfrombirationalgeometry
5.2.Proofofthresholdformula
5.3.Boundsforthelogcanonicalthreshold
5.4.Inversionofadjunction
5.5.Geometryofarcspaceswithoutexplicitmotivicintegration.
AppendixA.AnelementaryproofoftheTransformationrule.
A.1.Therelativecanonicaldivisoranddifferentials
A.2.ProofofTheorem3.3
References

Date
:28.July,2005.

1

24579012141619102124252527213234373738314

2MANUELBLICKLE
1.
Theinventionofmotivicintegration.
MotivicintegrationwasintroducedbyKontsevich[31]toprovethefollowingresult
conjecturedbyBatyrev:Let

X
1
BB|
X
2
BBBB|||
|π
1
BB


~
~
|||
π
2
Xbetwocrepantresolutionsofthesingularitiesof
X
,whichitselfisacomplexprojective
Calabi-Yau
1
varietywithatworstcanonicalGorensteinsingularities.Crepant(asin
nondiscrepant
)meansthatthepullbackofthecanonicaldivisorclasson
X
isthe
canonicaldivisorclasson
X
i
,
i.e.
thediscrepancydivisor
E
i
=
K
X
i
−
π
i
∗
K
X
isnumer-
icallyequivalenttozero.InthissituationBatyrevshowed,using
p
-adicintegration,
that
X
1
and
X
2
havethesamebettinumbers
h
i
=dim
H
i
(
,
C
).ThisleadKontse-
vichtoinvent
motivicintegration
toshowthat
X
1
and
X
2
evenhavethesameHodge
numbers
h
i,j
=dim
H
i
(
,
Ω
j
).
Thisproblemwasmotivatedbythe
topologicalmirrorsymmetrytest
ofstringtheory
whichassertsthatif
X
and
X
∗
areamirrorpair
2
ofsmoothCalabi-Yauvarietiesthen
theyhavemirroredHodgenumbers
h
i,j
(
X
)=
h
n
−
i,j
(
X
∗
)
.
AsthemirrorofasmoothCalabi-Yaumightbesingular,onecannotrestricttothe
smoothcaseandtheequalityofHodgenumbersactuallyfailsinthiscase.Therefore
Batyrevsuggested,inspiredbystringtheory,thatoneshouldlookinsteadattheHodge
numbersofacrepantresolution,ifsuchexists
3
.Theindependenceofthesenumbers
fromthechosencrepantresolutionisKontsevich’sresult.Thismakesthe
stringyHodge
numbers
h
is,tj
(
X
)of
X
,definedas
h
i,j
(
X
′
)foracrepantresolution
X
′
of
X
,welldefined.
Thisleadstoamodifiedmirrorsymmetryconjecture,assertingthatthestringyHodge
numbersofamirrorpairareequal[3].
Batyrev’sconjectureisnowKontsevich’stheoremandthesimplestformtophrase
itmightbe:

1
Usually,anormalprojectivevariety
X
ofdimension
n
iscalledCalabi-Yauifthecanonicaldivisor
K
X
istrivialand
H
i
(
X,
O
X
)=0for0
<i<n
.Thislastconditiononthecohomologyvanishing
isnotnecessaryforthestatementsbelow.Inthecontextofmirrorsymmetryitseemscustomaryto
dropthislastconditionandcall
X
Calabi-Yauassoonas
K
X
=0(andthesingularitiesaremild),
see[2].
2
Toexplainwhatamirrorpairisinausefulmannerliesbeyondmyabilities.Forourpurposeone
canthinkofamirrorpair(somewhattautologically)asbeingapairthatpassesthetopologicalmirror
symmetrytest.AnotherachievementofBatyrev[3]wastoexplicitlyconstructthemirrortoamildly
singular(toric)Calabi-Yauvariety.
3
Calabi-Yauvarietiesdonotalwayshavecrepantresolutions.IthinkoneofBatyrev’spapers
discussesthis.

ASHORTCOURSEONGEOMETRICMOTIVICINTEGRATION3
Theorem1.1
(Kontsevich)
.
BirationallyequivalentsmoothCalabi-Yauvarietieshave
thesameHodgenumbers.
4
Proof.
Theideanowistoassigntoanyvarietya
volume
inasuitablering
M
ˆ
k
such
thattheinformationabouttheHodgenumbersisretained.Thefollowingdiagram
illustratestheconstructionof
M
ˆ
k
:
Var
k
/
/
K
0
(Var
k
)
/
/
K
0
(Var
k
)[
L
−
1
]
/
/
M
ˆ
k
JJJJJJJE
JJJ
$
$
Z
[
u,v
]

/
/
Z
[
u,v,
(
uv
)
−
1
]

/
/
Z
[
u,v,
(
uv
)
−
1
]
∧
jsmoothprojectivevariety
X
isgivenby
E
(
X
)=(
−
1)
i
dim
H
i
(
X,
Ω
X
)
u
i
v
j
.Ingen-
Thediagonalmapisthe(compactlysupported)
P
Hodgecharacteristic,whichona
eralitisdefinedviamixedHodgestructures
5
[8,9,10],satisfies
E
(
X
×
Y
)=
E
(
X
)
E
(
Y
)
forallvarieties
X,Y
andhasthepropertythatfor
Y
⊆
X
aclosed
k
-subvarietyone
has
E
(
X
)=
E
(
Y
)+
E
(
X
−
Y
).ThereforetheHodgecharacteristicfactorsthrough
the
naiveGrothendieckring
K
0
(
Var
k
)whichistheuniversalobjectwiththelatter
6property.Thisexplainsthelefttriangleofthediagram.
Thebottomrowofthediagramisthecompositionofalocalization(inverting
uv
)
andacompletionwithrespecttonegativedegree.
M
k
isconstructedanalogously,
byfirstinverting
L
−
1
=[
A
k
1
](apre-imageof
uv
)andthencompletingappropriately
(negativedimension).Whereasthebottommapsareinjective(easyexercise),themap
K
0
(Var
k
)
−→M
ˆ
k
ismostlikelynotinjective.Theneedtoworkwith
M
ˆ
k
insteadof
K
0
(Var
k
)arisesinthesetupoftheintegrationtheoryanwillbecomeclearlater.
7
Clearly,byconstructionitisnowenoughtoshowthatbirationallyequivalentCalabi-
Yauvarietieshavethesame
volume
,i.e.thesameclassin
M
ˆ
k
.Thisisachievedviathe
allimportant
birationaltransformationrule
ofmotivicintegration.Roughlyitasserts
thatforaproperbirationalmap
π
:
Y
−→
X
theclass[
X
]
∈M
ˆ
k
isan
expression
in
Y
and
K
Y/X
only:
[
X
]=
L
−
ord
KY/X
ZY4
ThereisnowaproofbyIto[28]ofthisresultusing
p
-adicintegration,thuscontinuingtheideasof
BatyrevwhoprovedtheresultforBettinumbersusingthistechnique.Furthermoretherecentweak
factorizationtheoremofWl odarczyk[1]allowsforaproofavoidingintegrationofanysort.
5
Recently,Bittner[4]gaveanalternativeconstructionofthecompactlysupportedHodgecharac-
teristic.SheusestheweakfactorizationtheoremofWl odarczyk[1]toreducethedefinitionof
E
to
thecaseof
X
smoothandprojective,whereitisasgivenabove.
6
K
0
(Var
k
)isthefreeabeliangroupontheisomorphismclasses[
X
]of
k
-varietiessubjecttothe
relations[
X
]=[
X
−
Y
]+[
Y
]for
Y
aclosedsubvarietyof
X
.Theproductisgivenby[
X
][
Y
]=[
X
×
k
Y
].
Thesymbol
L
denotestheclassoftheaffineline[
A
k
1
].
7
Infact,recentresultsofF.LoeserandR.Cluckers[6],andJ.Sebag[41]indicatethatthefull
completionmaynotbenecessary,andallthevolumesofmeasurablesetsarecontainedinasubring
of
M
ˆ
k
thatcanbeconstructedexplicitly.

4MANUELBLICKLE
Tofinishofftheprooflet
X
1
and
X
2
bebirationallyequivalentCalabi-Yauvarieties.
WeresolvethebirationalmaptoaHironakahut:
YAπ
1
}}}}}AAAAA
π
2
~
~
}}}AA


X
1
_______
/
/
X
2
BytheCalabi-Yauassumptionwehave
K
X
i
≡
0andtherefore
K
Y/X
i<