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# 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

Subjects

##### Birational geometry

Informations

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

K
Y

π
i

K
X
i

K
Y
.Hencethedivisors
K
Y/X
1
and
K
Y/X
2
arenumericallyequivalent.This
numericalequivalenceimpliesinfactanequalityofdivisors
K
X/X
1
=
K
X/X
2
since,
againbytheCalabi-Yauassumption,dim
H
0
(
X,K
Y
)=dim
H
0
(
X
i
,
O
X
i
)=1.
8
Bythe
transformationrule,[
X
1
]isanexpressiondependingonlyon
Y
and
K
X/X
1
=
K
X/X
2
.
Thesameistruefor[
X
2
]andthuswehave[
X
1
]=[
X
2
]asdesired.

ThesenoteswerestartedduringaworkingseminaratMSRIduringtheyearof2003
andtookshapeinthecourseofthepast2yearswhileIwasgivingintroductorylectures
onthesubject.Theyhavetakenmewaytoomuchtimetoﬁnishandwouldnothave
beenﬁnishedatallifitweren’tfortheencouragementofmanypeople:Thanksgoes
toalltheparticipantsoftheseminaronmotivicintegrationatMSRI(2002/2003),
oftheSchwerpunktJuniorenTagunginBayreuth(2003)andthepatientlistenersof
themini-coursesatKTH,Stockholm(2003),theUniversityofHelsinki(2004)andthe
2.
Geometricmotivicintegration
Wenowassumethat
k
isalgebraicallyclosedandofcharacteristiczero.Infact,
thereisonepoint(seesection4.1)wherewewillassumethat
k
=
C
inordertoavoid
choosetoreplace
k
by
C
wheneveritiscomforting.Westressthattherearesigniﬁcant
(thoughmanageable)obstaclesonehastoovercomeifonewantsto(a)workwith
singularspacesor(b)withvarietiesdeﬁnedoverﬁeldswhicharenotuncountableor
notalgebraicallyclosed.Orputdiﬀerently:Thetheorydevelopsnaturally(foran
algebraicgeometer),andeasily,inthesmoothcaseover
C
,aswehopetodemonstrate
below.Inordertotransferthisintuitiontoanyothersituation,nontivialresultsand
extracareisnecessary.
AlltheresultsinthesenotesappearedinthepapersofDenefandLoeser,Batyrev
andLooijenga.OurexpositionisparticularlyinﬂuencedbyLooijenga[33]andBatyrev
8
Ingeneral,theconditionthat
X
1
and
X
2
haveacommonresolution
Y
suchthat
K
Y/X
1
isnumer-
icallyequivalent
K
Y/X
1
iscalled
K
–equivalence.WeshowedabovethattwobirationalCalabi–Yau
varietiesare
K
–equivalent.Formildlysingular
X
i
(saycanonical)itcanbederivedfromtheneg-
ativitylemma[30,Lemma3.39]that
K
–equivalenceimpliesactualequalityofdivisors
K
Y/X
1
and
K
Y/X
1
.HencetheCalabi–Yauassumptionwasnotessentialtoconcludethis(butprovidesasimple
argument).

ASHORTCOURSEONGEOMETRICMOTIVICINTEGRATION5
ofHales[25]andVeys[44],bothexplaintheconnectionto
p
whichwedonotdiscussinthesenotesatall.Theabovementionedreferencesarealso
Wewilldiscussnoneofthemexceptforcertainapplicationstobirationalgeometry.
Wewillnowintroducethebuildingblocksofthetheory.Theseare:
(1)Thevalueringofthemeasure:
M
ˆ
k
,alocalizedandcompletedGrothendieck
J

(
X
)
,thespaceofformalarcsover
X
.
(3)Analgebraofmeasurablesetsof
J

(
X
)andameasure:
cylinders/stablesets
andthevirtualeulercharacteristic.
(4)Aninterestingclassofmeasurable/integrablefunctions:
Contactorderofan
(5)Achangeofvariablesformula:
Kontsevich’sbirationaltransformationrule.
Thesebasicingredientsappearwithvariationsinallversionsofmotivicintegration
(onecouldargue:ofanytheoryofintegration).

2.1.
Thevalueringofthemotivicmeasure.
anyprevious(classical?)theoryofintegrationsincethevaluesofourmeasuredonot
liein
R
varietiesbyaprocessoflocalizationandcompletion.Thisingeniouschoiceisakey
featureofthetheory.
WestartwiththenaiveGrothendieckringofthecategoryofvarietiesover
k
.
9
This
isthering
K
0
(Var
k
)generatedbytheisomorphismclassesofallﬁnitetype
k
–varieties
andwithrelation[
X
]=[
Y
]+[
X

Y
]foraclosed
k
-subvariety
Y

X
,thatissuch
thattheinclusion
Y

X
isdeﬁnedover
k
.Thesquarebracketsdenotetheimage
of
X
in
K
0
(Var
/k
).Theproductstructureisgivenbytheﬁberproduct,[
X
]

[
Y
]=
[
X
×
k
Y
](=[(
X
×
k
Y
)
red
]).Thesymbol
L
isreservedfortheclassoftheaﬃneline
[
A
k
1
]and1=1
k
denotesSpec
k
.Thus,forexample,[
P
n
]=
L
n
+
L
n

1
+
...
+
L
+1.
Roughlyspeakingthemap
X
7→
[
X
]isrobustwithrespecttochoppingup
X
intoa
disjointunionoflocallyclosedsubvarieties.
10
Byusingastratiﬁcationof
X
bysmooth
subvarieties,thisshowsthat
K
0
(Var
k
)isgeneratedbytheclassesofsmoothvarieties.
11
Inasimilarfashiononecanassigntoeveryconstructiblesubset
C
of
X
aclass[
C
]by
expressing
C
asacombinationofsubvarieties.

9
Alternatively,theGrothendieckringofﬁnitetypeschemesover
k
X

X
red
=

.AsBjornPoonenpointsout,theﬁnitetypeassumptioniscrucialhere.Ifonewould
allownonﬁnitetypeschemes,
K
0
(Var
k
)wouldbezero.Forthislet
Y
beany
k
–schemeandlet
X
be
aninﬁnitedisjointunionofcopiesof
Y
.Then[
X
]+[
Y
]=[
X
]andtherefore[
Y
]=0.
10
ThisiselegantlyillustratedinthearticleofHales[25]whichemphasizespreciselythispointof
K
0
(Var
k
)beinga
scissorgroup
.
11
In[4],Bittnershowsthat
K
0
(Var
k
)istheabeliangroupgeneratedbysmoothprojectivevarieties
subjecttoaclassofrelationswhicharisefromblowingupatasmoothcenter:If
Z
isasmooth
subvarietyof
X
,thentherelationis[
X
]

[
Z
]=[Bl
Z
X
]

[
E
],where
E
istheexceptionaldivisorof
theblowup.

6MANUELBLICKLE
Exercise2.1.
Verifytheclaiminthelastsentence.Thatis:showthatthemap
Y
7→
[
Y
]
for
Y
aclosedsubvarietyof
X
naturallyextendstothealgebraofconstructible
subsetsof
X
.
meansonecanwrite
X
=
X
i
asaﬁnitedisjointunionoflocallyclosedsubsets
X
i
Exercise2.2.
Let
Y
−→
X
F
beapiecewisetrivialﬁbrationwithconstantﬁber
Z
.This
suchthatovereach
X
i
onehas
f

1
X
i
=

X
i
×
Z
and
f
isgivenbytheprojectiononto
X
i
.Showthatin
K
0
(Var
k
)
onehas
[
Y
]=[
X
]

[
Z
]
.
Thereisanaturalnotionofdimensionofanelementof
K
0
(Var
k
).Wesaythat
τ

K
0
(Var
k
)is
d
–dimensional
ifthereisanexpressionin
K
0
(Var
k
)
τ
=
a
i
[
X
i
]
Xwith
a
i

Z
and
k
-varieties
X
i
ofdimension

d
,andifthereisnoexpressionlikethis
withalldim
X
i

d

1.Thedimensionoftheclassoftheemptyvarietyissettobe
−∞
.Itiseasytoverify(exercise!)thatthemap
dim:
K
0
(Var
k
)
−→
Z
∪{−∞}
satisﬁesdim(
τ

τ

)

dim
τ
+dim
τ

anddim(
τ
+
τ

)

max
{
dim
τ,
dim
τ

}
withequality
inthelatterifdim
τ
6
=dim
τ

.
Thedimensioncanbeextendedtothelocalization
M
k
d
=
ef
K
0
(Var
k
)[
L

1
]simplyby
demandingthat
L

1
hasdimension

1.Toobtainthering
M
ˆ
k
inwhichthedesired
measurewilltakevalueswefurthercomplete
M
k
withrespecttotheﬁltrationinduced
bythedimension.
12
The
n
thﬁlteredsubgroupis
F
n
(
M
k
)=
{
τ
∈M
k
|
dim
τ

n
}
.
Thisgivesusthefollowingmapswhichwillbethebasisforconstructingthesought
aftermotivicmeasure:
∧Var
k
−→
K
0
(Var
k
)

i

nv

er

t

L
→M
k

−→M
ˆ
k
.
12
InLooijenga[33],thisiscalledthevirtualdimension.AsdescribedbyBatyrev,composingthe
dimensiondim:
M
k
−→
Z
∪{−∞}
withtheexponential
Z

R

ex

p

(
−−
)

R
+
andbyfurtherdeﬁning
∅7→
0wegetamap
δ
k
:
M
k
−→
R
+
,
0
whichisa
non-archimediannorm
.Thatmeansthefollowingpropertieshold:
(1)
δ
k
(
A
)=0iﬀ
A
=0=[

]in
M
k
.
(2)
δ
k
(
A
+
B
)

max
{
δ
k
(
A
)

k
(
B
)
}
(3)
δ
k
(
A

B
)

δ
k
(
A
)

δ
k
(
B
)
Thering
M
ˆ
k
isthenthecompletionwithrespecttothisnorm,andtherefore
M
ˆ
k
iscompletein
thesensethatallCauchysequencesuniquelyconverge.Thecondition(2)isstrongerthantheone
usedinthedeﬁnitionofanarchimediannorm.Thisnon-archimedianingredientmakesthenotionof
convergenceofsumsconvenientlysimple;asumconvergesifandonlyifthesequenceofsummands
convergestozero.
Iftherewasanequalityincondition(3)thenormwouldbecalled
multiplicative
.Itisunknown
whether
δ
ismultiplicativeon
M
k
.However,Poonen[39]showsthat
K
0
(Var
k
)containszerodivisors,
thus
δ
restrictedto
K
0
(Var
k
)is
not
multiplicativeon
K
0
(Var
k
).

ASHORTCOURSEONGEOMETRICMOTIVICINTEGRATION7
Wewillsomewhatambiguouslydenotetheimageof
X

Var
k
inanyoftherings
completionmap

isinjective,
i.e.
whetheritskernel,
F
d
(
K
0
(Var
/k
)[
L

1
]),iszero.
totherightby[
X
].Itisimportanttopointoutheret
T
hatitisunknownwhetherthe
Itisalsounknownwhetherthelocalizationisinjective.
13
Exercise2.3.
Convergenceofseriesin
M
ˆ
k
israthereasy.Forthisobservethata
dim
τ
i
tendto
−∞
as
i
approaches

.Showthatasum
i

=0
τ
i
convergesifandonly
sequenceofelements
τ
i
∈M
k
convergestozeroin
M
ˆ
k
if
P
andonlyifthedimensions
thesequenceofsummandsconvergestozero.
Exercise2.4.
Showthatin
M
ˆ
k
theequality
i

=0
L

ki
=
1

L
1

k
holds.
P2.2.
Thearcspace
J

(
X
)
.
ArcspaceswereﬁrststudiedseriouslybyNash[37]who
conjecturedatightrelationshipbetweenthegeometryofthearcspaceandthesingu-
laritiesof
X
,seeIshiiandKollar[27]forarecentexpositionofNash’sideasinmodern
language.RecentworkofMusta¸taˇ[35]supportsthesepredictionsbyshowingthatthe
X
canbedetectedbytheirreducibilityofthejetschemesforcompleteintersections.
birationalgeometry,suchasthelogcanonicalthresholdofapair,forexample,canbe
andSection5wherewewilldiscusssomeoftheseresultsindetail.
Let
X
bea(smooth)schemeofﬁnitetypeover
k
ofdimension
n
.An
m
-jetof
X
is
anorder
m
inﬁnitesimalcurvein
X
,
i.e.
itisamorphism
ϑ
:Spec
k
[
t
]
/t
m
+1
−→
X.
Thesetofall
m
-jetscarriesthestructureofascheme
J
m
(
X
),calledthe
m
th
jetscheme
,
functor
×
Spec
k
[
t
]
/t
m
+1
.Inotherwords,
Hom(
Z
×
Spec
k
[
t
]
/t
m
+1
,X
)=Hom(
Z,
J
m
(
X
))
forall
k
-schemes
Z
,
i.e.
J
m
(
X
)istheschemewhichrepresentsthecontravariantfunctor
Hom(
×
Spec
k
[
t
]
/t
m
+1
,X
).
14
Inparticularthismeansthatthe
k
–valuedpointsof
J
m
(
X
)arepreciselythe
k
[
t
]
/t
m
+1
–valuedpointsof
X
.ThesocalledWeilrestriction
ofscalars,
i.e.
thenaturalmap
k
[
t
]
/t
m
+1
−→
k
[
t
]
/t
m
,inducesamap
π
mm

1
:
J
m
(
X
)
−→
J
m

1
(
X
)andcompositiongivesamap
π
m
:
J
m
(
X
)
−→J
0
(
X
)=
X
.Asupperindices
areoftencumbersomewedeﬁne
η
m
=
π
m
and
ϕ
m
=
π
mm

1
.

13
In[39]Poonenshowsthat
K
0
(Var
/k
thoughthatthelocalizationmapisnotinjectiveandthat
M
k
map
M
k
−→M
ˆ
k
isinjective.ButrecentlyNaumann[38]foundinhisdissertationzero-divisorsin
K
0
(Var
k
)for
k
aﬁniteﬁeldandthesearenon-zeroevenafterlocalizingat
L
–thusforaﬁniteﬁeld
M
k
e.g.
k
algebraicallyclosed)theabovequestionsremainopen.
14
RepresentabilityofthisfunctorwasprovedbyGreenberg[19,20];anotherreferenceforthisfact
is[5].

8MANUELBLICKLE
Takingtheinverselimit
15
oftheresultingsystemyieldsthedeﬁnitionofthe
inﬁnite
jetscheme
,orthe
arcspace
J

(
X
)=li
←−
m
J
m
(
X
)
.
Its
k
-pointsarethelimitofthe
k
-valuedpointsHom(Spec
k
[
t
]
/t
m
+1
,X
)ofthejet
spaces
J
m
(
X
).Thereforetheycorrespondtotheformalcurves(orarcs)in
X
,that
istomapsSpec
k
J
t
K
−→
X
.
16
Therearealsomaps
π
m
:
J

(
X
)
−→J
m
(
X
)again
inducedbythetruncationmap
k
J
t
K
−→
k
J
t
K
/t
m
+1
.Ifthereisdangerofconfusionwe
sometimesdecoratetheprojections
π
withthespace.Thefollowingpictureshould
helptorememberthenotation.
a(1)
J

(
X
)
π
a
/
/
J
a
(
X
)
π
b
/
/
J
b
(
X
)
π
b
/
/
X
ηbarethemapsinducedbythenaturalsurjections
k
J
t
K
/
/
k
J
t
K
/t
a
+1
/
/
k
J
t
K
/t
b
+1
/
/
k.
Example
2.1
.
Let
X
=Spec
k
[
x
1
,...,x
n
]=
A
n
.Then,onthelevelof
k
-pointsonehas
J
m
(
X
)=
{
ϑ
:
k
[
x
1
,...,x
n
]
−→
k
J
t
K
/t
m
+1
}
coeﬃcientsof
ϑ
(
x
i
)=
j
=0
ϑ
i
t
j
.Conversely,anychoiceofcoeﬃcients
ϑ
i
determines
Suchamap
ϑ
isdeter
P
minedbyitsvaluesonthe
x
i
’s,
i.e.
itisdeterminedbythe
m
(
j
)(
j
)
apointin
J
m
(
A
n
).Choosingcoordinates
x
i
(
j
)
of
J
m
(
X
)with
x
i
(
j
)
(
ϑ
)=
ϑ
i
(
j
)
weseethat
J
m
(
X
)=

Spec
k
[
x
1(0)
,...,x
(
n
0)
,......,x
1(
m
)
,...,x
(
nm
)
]=

A
n
(
m
+1)
.
Furthermoreobservethat,somewhatintuitively,thetruncationmap
π
m
:
J
m
(
X
)
−→
X
isinducedbytheinclusion
k
[
x
1
,...,x
n
]
֒

k
[
x
1(0)
,...,x
(
n
0)
,......,x
(1
m
)
,...,x
(
nm
)
]
sending
x
i
to
x
i
(0)
.
Exercise2.5.
Let
Y

A
n
beahypersurfacegivenbythevanishingofoneequation
f
=0
.Showthat
J
m
(
Y
)
⊆J
m
(
A
n
)
isgivenbythevanishingof
m
+1
equations
and
f
(1)
=
∂∂x
i
f
(
x
(0)
)
x
i
(1)
).Showthat
f
(0)
,...,f
(
m
P
)
inthecoordinatesof
J
m
(
A
n
)
describedabove.(Observethat
f
(0)
=
f
(
x
(0)
)
(1)
J
m
(
Y
)
ispuredimensionalifandonlyif
dim
J
m
(
Y
)=(
m
+1)(
n

1)
,inwhich
case
J
m
(
Y
)
isacompleteintersection.
(2)
J
m
(
Y
)
isirreducibleifandonlyif
dim(
π
Ym
)

1
(
Y
Sing
)
<
(
m
+1)(
n

1)
.
Similarstatementsholdif
Y
islocallyacompleteintersection.
Theexistenceofthejetschemesingeneral(thatistoshowtherepresentabilityof
thefunctordeﬁnedabove)isproved,forexample,in[5].Fromtheverydeﬁnitionone
caneasilyderivethefollowinge´taleinvarianceofjetschemes,which,togetherwith
15
Forthistobedeﬁnedonecruciallyusesthattherestrictionmapsare
aﬃne
morphisms.
16
Forthisobservethatli
←−
mHom(
R,k
[
t
]
/t
m
+1
)=

Hom(
R,
li
←−
m
k
[
t
]
/t
m
+1
)=Hom(
R,k
J
t
K
).

ASHORTCOURSEONGEOMETRICMOTIVICINTEGRATION9
theexampleof
A
n
abovegivesusaprettygoodunderstandingofthejetschemesofa
smoothvariety.
Proposition2.2.
Let
X
−→
Y
bee´tale,then
J
m
(
X
)=
∼J
m
(
Y
)
×
Y
X
.
Proof.
Weshowtheequalityonthelevelofthecorrespondingfunctorsofpoints
KtJkHom(
,
J
m
(
X
))=

Hom(
×
k
Spec
t
m
+1
,X
)
dnaHom(
,
J
m
(
Y
)
×
Y
X
)=Hom(
,
J
m
(
Y
))
×
Hom(
,X
)
KtJk=Hom(
×
k
Spec
m
+1
,Y
)
×
Hom(
,X
)
.
tForthislet
Z
bea
k
–schemeandconsiderthediagram
X
O
O
h
h
PP
/
/
Y
O
O
Pp
PP
τ
PP
ϑ
Z
×
Spec
k
/
/
Z
×
Spec
k
m
J
+
t
K
1
ttoseethata
Z
-valued
m
-jet
τ

Hom(
Z
×
k
Spec
tk
m
J
+
t
K
1
,X
)of
X
inducesa
Z
-valued
m
-jet
ϑ

Hom(
Z
×
k
Spec
tk
m
J
+
t
K
1
,Y
)andamap
p

Hom(
Z,X
).Virtuallybydeﬁnition
formallye´taleness[23,Deﬁnition(17.1.1)]forthemapfrom
X
to
Y
,theconverseholds
also,
i.e.
ϑ
and
p
togetherinduceauniquemap
τ
asindicated.

Usingthise´taleinvarianceofjetschemesthecomputationcarriedoutfor
A
n
above
holdslocallyonanysmooth
X
.Thusweobtain:
Proposition2.3.
Let
X
beasmooth
k
-schemeofdimension
n
.Then
J
m
(
X
)
islocally
an
A
nm
–bundleover
X
.Inparticular
J
m
(
X
)
issmoothofdimension
n
(
m
+1)
.Inthe
sameway,
J
m
+1
(
X
)
islocallyan
A
n
–bundleover
J
m
(
X
)
.
Notethatthisisnottrueforasingular
X
tangentbundle
TX
=
J
1
(
X
)whichiswell-knowntobeabundleifandonlyif
X
is
smooth.Infact,overasingular
X
thejetschemesneednotevenbeirreduciblenor
2.3.
Analgebraofmeasurablesets.
Theprototypeofameasurablesubsetof
J

(
X
)isa
stableset
volumein
M
k
.
Deﬁnition2.4.
Asubset
A
⊆J

(
X
)iscalled
stable
ifforall
m

0,
A
m
d
=
ef
π
m
(
A
)
isaconstructiblesubset
17
of
J
m
(
X
),
A
=
π
m

1
(
A
m
)andthemap
(2)
π
mm
+1
:
A
m
+1
−→
A
m
isalocallytrivial
A
n
–bundle.
17
Theconstructiblesubsetsofascheme
Y
arethesmallestalgebraofsetscontainingtheclosed
setsinZariskitopology.

10MANUELBLICKLE
Forany
m

0wedeﬁnethe
volume
ofthestableset
A
by

X
(
A
)=[
A
m
]

L

nm
∈M
k
.
Thatthisisindependentof
m
isensuredbycondition(2)whichimpliesthat[
A
m
+1
]=
[
A
m
]

L
n
.
18
Assumingthat
X
issmoothoneusesProposition2.3toshowthatthecollection
ofstablesetsformsanalgebraofsets,whichmeansthat
J

(
X
)isstableandwith
A
and
A

stablethesets
J

(
X
)

A
and
A

A

arealsostable.Thesmoothness
of
X
furthermorewarrantsthatsocalled
cylindersets
arestable(acylinderisaset
A
=
π
m

1
B
forsomeconstructible
B
⊆J
m
(
X
)).Thusinthesmoothcasecondition
(2)issuperﬂuouswhereasingeneralitisabsolutelycrucial.Infact,amaintechnical
pointindeﬁningthemotivicmeasureonsingularvarietiesistoshowthattheclass
ofstablesetscanbeenlargedtoanalgebraof
measurable
setswhichcontainsthe
cylinders.Inparticular
J

(
X
)isthenmeasurable.Thisisachievedasonewould
expectbydeclaringasetmeasurableifitisapproximatedinasuitablesensebystable
sets.Thisisessentiallycarriedoutin[33],thoughtherearesomeinaccuracies;but
everythingshouldbeﬁneifoneworksover
C
[2,Appendix].
19
Toavoidthesetechnicalitiesweassumeuntiltheendofthissection
that
X
issmoothoverthecomplexnumbers
C
.
2.4.
Themeasurablefunctionassociatedtoasubscheme.
Fromanalgebraof
measurablesetstherearisesnaturallyanotionofmeasurablefunction.Sincewedid
notcarefullydeﬁnethemeasurablesets—wemerelydescribedtheprototypes—we
willfornowonlydiscussanimportantclassofmeasurablefunctions.
Let
Y

X
beasubschemeof
X
deﬁnedbythesheafofideals
I
Y
.To
Y
one
associatesthefunction
ord
Y
:
J

(
X
)
−→
N
∪{∞}
sendinganarc
ϑ
:
O
X
−→
k
J
t
K
totheorderofvanishingof
ϑ
along
Y
,
i.e.
tothesupre-
mumofall
e
suchthatideal
ϑ
(
I
Y
)of
k
J
t
K
iscontainedintheideal(
t
e
).Equivalently,
ord
Y
(
ϑ
)isthesupremumofall
e
suchthatthemap
ϑO
X

k
J
t
K
−→
k
J
t
K
/t
e
sends
I
Y
tozero.Notethatthismapisnothingbutthetruncation
π
e

1
(
ϑ
)
∈J
e

1
(
X
)of
ϑ
.
20
Fora(
e

1)-jet
γ
∈J
e

1
(
X
)tosend
I
Y
tozeromeanspreciselythat
γ
∈J
e

1
(
Y
).
18
Reid[40],Batyrev[2]andLooijenga[33]usethisdeﬁnitionwhichgivesthevolume
µ
X
(
J
m
(
X
))

M
k
of
X
virtualdimension
n
L

n
togive
µ
X
(
J
m
(
X
))virtualdimension0.Itseemstobeessentiallyamatteroftastewhichdeﬁnitiononeuses.
Justkeepthisinmindwhilebrowsingthroughdiﬀerentsourcesintheliteraturetoavoidunnecessary
confusion.
19
OfcourseDenefandLoeseralsosetupmotivicintegrationoversingularspaces[12]buttheir
approachdiﬀersfromtheonediscussedhereinthesensethattheyassignavolumetothe
formula
deﬁningaconstructiblesetratherthantothesetof(
k
-rational)pointsitself.Thustheyelegantly
avoidanyproblemswhichariseif
k
issmall.
20
Atthispointwebetterset
J

1
(
X
)=
X
and
π

1
=
π
0
=
π
toavoiddealingwiththecase
e
=0
separately.