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

Informations

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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|||

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

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
VigregraduatecourseinSaltLakeCity(2005).SpecialthanksgoestoKarenSmith
forencouragementtostartthisprojectandtoJuliaGordonfornumerouscomments,
suggestionsandcarefulreading.
2.
Geometricmotivicintegration
Wenowassumethat
k
isalgebraicallyclosedandofcharacteristiczero.Infact,
thereisonepoint(seesection4.1)wherewewillassumethat
k
=
C
inordertoavoid
sometechnicalitieswhichariseifthefieldisnotuncountable.Thusthereadermay
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
[2].AlsoCraw[7]wasveryhelpfulasafirstreading.Wealsorecommendthearticles
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
-adicintegrationindetail,
whichwedonotdiscussinthesenotesatall.Theabovementionedreferencesarealso
agreatsourcetolearnaboutthevariousdifferentapplicationsthetheoryhadtodate.
Wewilldiscussnoneofthemexceptforcertainapplicationstobirationalgeometry.
Wewillnowintroducethebuildingblocksofthetheory.Theseare:
(1)Thevalueringofthemeasure:
M
ˆ
k
,alocalizedandcompletedGrothendieck
.gnir(2)Adomainofintegration:
J

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

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

2.1.
Thevalueringofthemotivicmeasure.
Herewealreadygravelydepartfrom
anyprevious(classical?)theoryofintegrationsincethevaluesofourmeasuredonot
liein
R
.Insteadtheylieinahugering,constructedfromtheGrothendieckringof
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
leadstothesameobjectbecause
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
arcspacescontaininformationaboutsingularities,forexamplerationalsingularitiesof
X
canbedetectedbytheirreducibilityofthejetschemesforcompleteintersections.
Insubsequentinvestigationsheandhiscollaboratorsshowthatcertaininvariantsof
birationalgeometry,suchasthelogcanonicalthresholdofapair,forexample,canbe
readofffromthedimensionsofcertaincomponentsofthejetschemes,see[36,15,17]
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
,
orspaceoftruncatedarcs.It’scharacterizingpropertyisthatitisrightadjointtothe
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
)isnotadomainincharacteristiczero.Itisexpected
thoughthatthelocalizationmapisnotinjectiveandthat
M
k
isadomainandthatthecompletion
map
M
k
−→M
ˆ
k
isinjective.ButrecentlyNaumann[38]foundinhisdissertationzero-divisorsin
K
0
(Var
k
)for
k
afinitefieldandthesearenon-zeroevenafterlocalizingat
L
–thusforafinitefield
M
k
isnotadomain.Forinfinitefields(
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
ascanbeseenalreadybylookingatthe
tangentbundle
TX
=
J
1
(
X
)whichiswell-knowntobeabundleifandonlyif
X
is
smooth.Infact,overasingular
X
thejetschemesneednotevenbeirreduciblenor
reducedandcanalsobebadlysingular.
2.3.
Analgebraofmeasurablesets.
Theprototypeofameasurablesubsetof
J

(
X
)isa
stableset
.Theyaredefinedjustrightsothattheyreceiveanatural
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
andmakessomeadjustmentsfollowing
[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
.Denef,Loeser[13]andCraw[7]useanadditionalfactor
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.