Geometric motivic integration - IMJ-PRGpoint/geometric-integration-handout.pdf · Denef and Loeser developed a theory of geometric motivic integration on arbitrary algebraic varieties

Motivation: p-adic integrationFrom Zp to k[[t]]

Geometric motivic integration

Johannes Nicaise

Universite Lille 1

Modnet Workshop 2008

Johannes Nicaise Geometric motivic integration


Introduction

Kontsevich invented motivic integration to strengthen thefollowing result by Batyrev.

Theorem (Batyrev)

If two complex Calabi-Yau varieties are birationally equivalent, thenthey have the same Betti numbers.

Batyrev proved this result using p-adic integration and the WeilConjectures. Kontsevich observed that Batyrev’s proof could be“geometrized”, avoiding the passage to finite fields and yielding astronger result: equality of Hodge numbers. The key was toreplace the p-adic integers by C[[t]], and p-adic integration bymotivic integration.



Kontsevich presented these ideas at a famous “Lecture at Orsay”in 1995, but never published them. The theory was developed andgeneralized in the following directions:

Denef and Loeser developed a theory of geometric motivicintegration on arbitrary algebraic varieties over a field ofcharacteristic zero.They also created a theory of arithmetic motivic integration,with good specialization properties to p-adic integrals in ageneral setting, using the model theory of pseudo-finite fields.The motivic integral appears here as a universal integral,specializing to the p-adic ones for almost all p.



Loeser and Sebag constructed a theory of motivic integrationon formal schemes and rigid varieties, working over anarbitrary complete discretely valued field with perfect residuefield.

Cluckers and Loeser built a very general framework for motivicintegration theories, based on model theory.

We will only discuss the so-called naıve geometric motivicintegration on smooth algebraic varieties.



Outline

1 Motivation: p-adic integration

2 From Zp to k[[t]]The Grothendieck ring of varietiesArc spacesMotivic integrals and change of variables



Basic references:

J. Denef & F. Loeser, “Geometry on arc spaces of algebraicvarieties”, European Congress of Mathematics, Vol. 1(Barcelona, 2000), Progr. Math. 201, 2001

W. Veys, “Arc spaces, motivic integration and stringyinvariants”, Advanced Studies in Pure Mathematics 43 , Proc.of “Singularity Theory and its applications, Sapporo, 16-25september 2003” (2006)



p-adic integration

We fix a prime number p. For any integer n ≥ 0, we consider thecompact group Zn

p = (Zp)n with its unique Haar measure µ such

that µ(Znp) = 1.



Definition

We consider, for each m, n ≥ 0, the natural projection

πm : Znp → (Zp/p

m+1)n

A cylinder C in Znp is a subset of the form (πm)−1(Cm), for some

m ≥ 0 and some subset Cm of (Zp/pm+1)n.



Lemma

For any cylinder C, the series (p−n(m+1)|πm(C )|)m≥0 is constantfor m � 0, and its limit is equal to the Haar measure of C .More precisely, if we choose m0 ≥ 0 such thatC = (πm0)

−1(πm0(C )), then for m ≥ m0

p−n(m+1)|πm(C )| = p−n(m0+1)|πm0(C )| = µ(C )



Proof.

For m ≥ m0, C can be written as a disjoint union

C =⊔

a∈πm(C)

(a + (pm+1Zp)n)

so by translation invariance of the Haar measure and the fact that(pm+1Zp)

n has measure p−(m+1)n, we see that the measure of Cequals

p−(m+1)n|πm(C )|



The Grothendieck ring of varietiesArc spacesMotivic integrals and change of variables

From Zp to k[[t]]

If we identify Zp with the ring of Witt vectors W (Fp), then themap πm simply corresponds to the truncation map

W (Fp) → Wm+1(Fp) : (a0, a1, . . .) 7→ (a0, a1, . . . , am)




The idea behind the theory of motivic integration is to make asimilar construction, replacing W (Fp) by the ring of formal powerseries k[[t]] over some field k, and the map πm by the truncationmap

k[[t]] → k[t]/(tm+1) :∑i≥0

ai ti 7→

m∑i=0

ai ti

The problem is to give meaning to the expression |πm(C )| if C is a“cylinder” in k[[t]]n for infinite fields k, and to find a candidate toreplace p in the formula

µ(C ) = p−n(m+1)|πm(C )|)




But interpreting the coefficients of a power series as affinecoordinates, the set

(k[[t]]/(tm+1))n

gets the structure of the set of k-points on an affine space

A(m+1)nk , and if we restrict to cylinders C such that πm(C ) is

constructible in A(m+1)nk , we can use the Grothendieck ring of

varieties as a universal way to “count” points on constructiblesubsets of an algebraic variety.

The cardinality p of Fp is replaced by the “number” of points onthe affine line A1

k ; this is the “Lefschetz motive” L.




The price to pay is that we leave classical integration theory sinceour value ring will be an abstract gadget (the Grothendieck ring ofvarieties) instead of R.




The Grothendieck ring of varieties

k any field; k-variety= separated reduced k-scheme of finite type.

Definition (Grothendieck ring of k-varieties)

K0(Vark) abelian group

generators: isomorphism classes [X ] of k-varieties X

relations: [X ] = [X \ Y ] + [Y ] for Y closed in X (“scissorrelations”)

Ring multiplication: [X1] · [X2] = [(X1 ×k X2)red ]

L = [A1k ] Mk = K0(Vark)[L−1]

for X not reduced: [X ] := [Xred ]




If X is a k-variety and C a constructible subset of X , then C canbe written as a disjoint union of locally closed subsets (subvarieties)of X and this yields a well-defined class [C ] in K0(Vark).

If a morphism of k-varieties Y → X is a Zariski-locally trivialfibration with fiber F , then [Y ] = [X ] · [F ] in K0(Vark). Indeed,using the scissor relations and Noetherian induction we can reduceto the case where the fibration is trivial.




By its very definition, the Grothendieck ring is a universal additiveand multiplicative invariant of k-varieties.

specialization morphisms of rings

χtop : K0(Vark) → Z (Euler characteristic) k = C

] : K0(Vark) → Z (number of rational points) k finite

So in a certain sense, taking the class [X ] of a k-variety X is themost general way to “count points” on X .




Arc spaces

Let X be a variety over k. For each integer n ≥ 0, we consider thefunctor

Fn : (k − alg) → (Sets) : A 7→ X (A[t]/(tn+1))

Proposition

The functor Fn is representable by a separated k-scheme of finitetype Ln(X ), called the n-th jet scheme of X . If X is affine, then sois Ln(X ).




Note: Saying that Ln(X ) represents the functor Fn simply meansthe following: for any k-algebra A, there exists a bijection

φAn : Ln(X )(A) → Fn(A) = X (A[t]/tn+1)

such that for any morphism of k-algebras h : A → B, the square

Ln(X )(A)φA

n−−−−→ X (A[t]/tn+1)y yLn(X )(B)

φBn−−−−→ X (B[t]/tn+1)

commutes. This property uniquely determines the k-schemeLn(X ).




Idea of proof: We only consider the affine case

X = Spec k[x1, . . . , xr ]/(f1, . . . , f`)

i.e. X is the closed subvariety of Ark defined by the equations

f1 = . . . = f` = 0.Consider a tuple of variables (a1,0, . . . , a1,n, a2,0, . . . , ar ,n) and thesystem of congruences

fj(n∑

i=0

a1,i ti , . . . ,

n∑i=0

ar ,i ti ) ≡ 0 mod tn+1

for j = 1, . . . , `.




If we replace the variables ai ,j by elements of a k-algebra A, thesecongruences express exactly that the tuple

(a1,0 + . . .+ a1,ntn, . . . , ar ,0 + . . .+ ar ,nt

n) ∈ Ark(A[t]/tn+1)

lies in X (A[t]/(tn+1)).




Developing, for each j ,

fj(n∑

i=0

a1,i ti , . . . ,

n∑i=0

ar ,i ti )

into a polynomial in t and putting the coefficient of t i equal to 0for i = 0, . . . , n yields a system of `(n + 1) polynomial equationsover k in the variables ai ,j , and these define the scheme Ln(X ) as

a closed subscheme of Ar(n+1)k .




Example: Let X be the closed subvariety of A2k = Spec k[x , y ]

defined by the equation x2 − y3 = 0. Then a point of L2(X ) withcoordinates in some k-algebra A is a couple

(x0 + x1t + x2t2, y0 + y1t + y2t

2)

with x0, . . . , y2 ∈ A such that

(x0 + x1t + x2t2)2 − (y0 + y1t + y2t

2)3 ≡ 0 mod t3




(x0 + x1t + x2t2)2 − (y0 + y1t + y2t

2)3 ≡ 0 mod t3

Working this out, we get the equations(x0)

2 − (y0)3 = 0

2x0x1 − 3(y0)2y1 = 0

(x1)2 + 2x0x2 − 3y0(y1)

2 − 3(y0)2y2 = 0

and if we view x0, . . . , y2 as affine coordinates, these equationsdefine L2(X ) as a closed subscheme of A6

k .




For any m ≥ n and any k-algebra A, the truncation map

A[t]/tm+1 → A[t]/tn+1

defines a natural transformation of functors Fm → Fn, so byYoneda’s Lemma we get a natural truncation morphism ofk-schemes

πmn : Lm(X ) → Ln(X )

This is the unique morphism such that for any k-algebra A, thesquare

X (A[t]/tm+1) −−−−→ X (A[t]/tn+1)

φAm

y yφAn

Lm(X )(A)πm

n−−−−→ Ln(X )(A)

commutes.




Since the schemes Ln(X ) are affine for affine X and Ln(.) takesopen covers to open covers, the morphisms πm

n are affine for anyk-variety X , and we can take the projective limit

L(X ) = lim←−n

Ln(X )

in the category of k-schemes.




The scheme L(X ) is called the arc scheme of X . It is notNoetherian, in general. It comes with natural projection morphisms

πn : L(X ) → Ln(X )

For any field F over k, we have a natural bijection

L(X )(F ) = X (F [[t]])

and the points of these sets are called F -valued arcs on X . Themorphism πn maps an arc to its truncation modulo tn+1.




So by definition, a F -valued arc is a morphism

ψ : Spec F [[t]] → X

i.e. a point of X with coordinates in F [[t]]. The image of theclosed point of Spec F [[t]] is called the origin of the arc anddenoted by ψ(0); it is obtained by putting t = 0. An arc should beseen as an infinitesimal disc on X with origin at ψ(0).




We have L0(X ) = X and L1(X ) is the tangent scheme of X . Thetruncation morphism π0 : L(X ) → X maps an arc ψ to its originψ(0). A morphism of k-varieties h : Y → X induces morphisms

L(h) : L(Y ) → L(X )

Ln(h) : Ln(Y ) → Ln(X )

commuting with the truncation maps.




If X is smooth over k, of pure dimension d , then the morphisms

πmn are Zariski-locally trivial fibrations with fiber Ad(m−n)

k (useetale charts and the fact that A[t]/(tm+1) → A[t]/(tn+1) is anilpotent immersion for any k-algebra A).Intuitively, arcs are local objects on X and any smooth variety ofpure dimension d looks locally like an open subvariety of Ad

k ; butan element of Ln(Ad

k )(A) is simply a d-tuple of elements inA[t]/tn+1.




If X is singular, the spaces Ln(X ) and L(X ) are still quitemysterious. They contain a lot of information on the singularitiesof X .




Example: Let us go back to our previous example, where X ⊂ A2k

was given by the equation x2 − y3 = 0. For any k-algebra A, apoint on L(X ) with coordinates in A is given by a couple

(x(t) = x0 + x1t + x2t2 + . . . , y(t) = y0 + y1t + y2t

2 + . . .)

with xi , yi ∈ A, such that x(t)2 − y(t)3 = 0. Working this outyields an infinite number of polynomial equations in the variablesxi , yi and these realize L(X ) as a closed subscheme of theinfinite-dimensional affine space A∞k = Spec k[x0, y0, x1, y1, . . .].




The truncation map

πn : L(X ) → Ln(X )

sends (x(t), y(t)) to

(x0 + . . .+ xntn, y0 + . . .+ ynt

n)

and (if A is a field) the origin of (x(t), y(t)) is simply the point(x0, y0) on X .




Note: If k has characteristic zero, one can give an elegantconstruction of the schemes Ln(X ) and L(X ) using differentialalgebra. Assume that X is affine, say given by polynomialequations

f1(x1, . . . , xr ) = . . . = f`(x1, . . . , xr )

in affine r -space Ark = Spec k[x1, . . . , xr ]. Consider the k-algebra

B = k[y1,0, . . . , yr ,0, y1,1, . . .]

and the unique k-derivation δ : B → B mapping yi ,j to yi ,j+1 foreach i , j .




Then L(X ) is isomorphic to the closed subscheme of Spec Bdefined by the equations

δ(i)(fq(y1,0, . . . , yr ,0)) = 0

for q = 1, . . . , ` and i ∈ N. The point with coordinates yi ,j

corresponds to the arc

(∑j≥0

y1,j

j!t j , . . . ,

∑j≥0

yr ,j

j!t j)




Motivic integrals and change of variables

Copying the notion of cylinder and the description of its Haarmeasure, we can define a motivic measure on a class of subsets ofthe arc space L(X ). From now on, we assume that X is smoothover k, of pure dimension d .




Definition

A cylinder in L(X ) is a subset C of the form (πm)−1(Cm), withm ≥ 0 and Cm a constructible subset of Lm(X ).

Note that the set of cylinders in L(X ) is a Boolean algebra, i.e. itis closed under complements, finite unions and finite intersections.




Definition-Lemma

Let C be a cylinder in L(X ), and choose m ≥ 0 such thatC = (πm)−1(Cm) with Cm constructible in Lm(X ). The value

µ(C ) := [πm(C )]L−d(m+1) ∈Mk

does not depend on m, and is called the motivic measure µ(C ) ofC .

This follows immediately from the fact that the truncationmorphisms πn

m are Zariski-locally trivial fibrations with fiber

Ad(n−m)k .




Example

If C = L(X ), then µ(C ) = L−d [X ].

The normalization factor L−d is added in accordance with thep-adic case, where the ring of integers gets measure one (ratherthen the cardinality of the residue field).Note that the motivic measure µ is additive w.r.t. finite disjointunions.




In the general theory of motivic integration, one constructs a muchlarger class of measurable sets and one defines the motivic measurevia approximation by cylinders. This necessitates replacing Mk bya certain “dimensional completion”. We will not consider thisgeneralization here.




Definition

We say that a function

α : L(X ) → N ∪ {∞}

is integrable if its image is finite, and if α−1(i) is a cylinder foreach i ∈ N.We define the motivic integral of α by∫

L(X )L−α =

∑i∈N

µ(α−1(i))L−i ∈Mk




The central and most powerful tool in the theory of motivicintegration is the change of variables formula. For its precisestatement, we need some auxiliary notation. For any k-variety Y ,any ideal sheaf J on Y and any arc

ψ : Spec F [[t]] → Y

on Y , we define the order of J at ψ by

ordtJ (ψ) = min{ordtψ∗f | f ∈ Jψ(0)}

where ordt is the t-adic valuation. In this way, we obtain a function

ordtJ : L(Y ) → N ∪ {∞}

whose fibers over N are cylinders.




Theorem (Change of variables formula)

Let h : Y → X be a proper birational morphism, with Y smoothover k, and denote by Jach the Jacobian ideal of h. Let α be anintegrable function on L(X ), and assume that ordtJach takes onlyfinitely many values on each fiber of α ◦ L(h) over N. Then∫

L(X )L−α =

∫L(Y )

L−((α◦L(h))+ordtJach)

in Mk .




The very basic idea behind the change of variables formula is thefollowing: if we denote by V the closed subscheme of Y defined bythe Jacobian ideal Jach, and by U its image under h, then themorphism h : Y − V → X − U is an isomorphism. Combined withthe properness of h, this implies that

L(h) : L(Y )− L(V ) → L(X )− L(U)

is a bijection; but L(V ) and L(Y ) have measure zero in L(Y ),resp. L(X ) (w.r.t. a certain more refined motivic measure) so it isreasonable to expect that there exists a change of variablesformula.




The jet spaces Ln(Y ), however, are “contracted” under themorphism

Ln(h) : Ln(Y ) → Ln(X )

and this affects the motivic measure of cylinders. The “contractionfactor” is measured by the Jacobian.


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Geometric motivic integration - IMJ-PRGpoint/geometric-integration-handout.pdf · Denef and Loeser developed a theory of geometric motivic integration on arbitrary algebraic varieties