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# A SHORT COURSE ON GEOMETRIC MOTIVIC INTEGRATION

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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

##### Birational geometry

Informations

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

Extrait

ASHORTCOURSEONGEOMETRICMOTIVICINTEGRATION
MANUELBLICKLE
Abstract.
Thesenotesgrewoutoftheauthorseﬀorttounderstandthetheoryof
motivicintegration
.Theygiveashortbutthoroughintroductiontotheﬂavorofmo-
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
5.5.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|||

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
that
X
1
and
X
2
havethesamebettinumbers
h
i
=dim
H
i
(
,
C
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
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.
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
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
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
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<

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