Seattle lectures on motivic integration

Jan Denef

Description

Niveau: Supérieur, Doctorat, Bac+8
Seattle lectures on motivic integration Franc¸ois Loeser Preliminary notes (January 17, 2006) These notes are the written-up version of a series of 4 lectures given at the Summer Institute. Though I tried as much as possible to keep the basic structure of the lectures as well as their rather informal style, some flesh has also been added to the bones. Motivic integration being born exactly ten years ago, nothing could be more timely than the proposition by the organizers of the Institute to review the achievements of the past decade in a series of lectures. I would like to thank them for providing me such a unique opportunity. Lecture 1: Before Motivic Integration 1.1. Modifications. One may start the whole story of motivic integration with a somewhat intriguing result obtained by Jan Denef and myself in 1987 and only published in 1992 [23]. At the time we certainly would never have guessed the fantastic developments that would arise later. Let us consider a smooth complex algebraic variety X and a closed nowhere dense subscheme F . By a log-resolution h : Y ? X of (X,F ) we mean a proper morphism h : Y ? X with Y smooth such that the restriction of h: Y \ h?1(Fred)? X \ Fred is an isomorphism, and h?1(Fred) is a divisor with simple normal crossings.

• almost all


• prime ideals


• theorems between -adic


• complex algebraic


• rational function


• local ring


• varieties

Sujets

• Almost all
• Prime ideal
• Rational function
• Local ring

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 19
• Langue: English

Extrait

Seattle lectures on motivic integration

¸ Francois Loeser

Preliminary notes (January 17, 2006)

These notes are the written-up version of a series of 4 lectures given at the Summer Institute. Though I tried as much as possible to keep the basic structure of the lectures as well as their rather informal style, some flesh has also been added to the bones. Motivic integration being born exactly ten years ago, nothing could be more timely than the proposition by the organizers of the Institute to review the achievements of the past decade in a series of lectures. I would like to thank them for providing me such a unique opportunity.

Lecture 1: Before Motivic Integration

1.1. Modifications.start the whole story of motivic integrationOne may with a somewhat intriguing result obtained by Jan Denef and myself in 1987 and only published in 1992 [23 the time we certainly would never have guessed the]. At fantastic developments that would arise later. Let us consider a smooth complex algebraic varietyXand a closed nowhere dense subschemeF a log-resolution. By

h:Y→X

of (X F) we mean a proper morphismh:Y→XwithYsmooth such that the restriction ofh: Y\h −1 (F red )→X\F red is an isomorphism, andh −1 (F red irsmaolrcwriotshs)iimspaldeinvoisgn.seW denote byE i ,iinA, the set of irreducible components of the divisorh −1 (F red ). Hence, by definition theE i  If’s are smooth and intersect transversally.h:Y→X is log-resolution of (X F) for someF, we callha DNC-modification. ForI⊂A, we set E I := \ E i i∈I and [

E I ◦ :=E I \E j . j /∈I 1

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FRAN¸COISLOESER

IfIan ideal sheaf defining the closed subschemeis FofXandh −1 (I)O Y is locally principal, we defineN i (I), the multiplicity ofIalongE i , by h ∗ (I)O Y ' O Y  − X N i (I)E i  . i∈A

IfIis principal, generated by a functiong, we writeN i (g) forN i (I). Similarly, one defines integersν i called log discrepancies, by the equality of, divisors K Y =h ∗ K X + X (ν i −1)E i . i∈A LetX Ifa complex algebraic variety (not necessarily smooth).be Xis proper, X(C) is compact and we may define its Euler Characteristic as Eu(X) := X (−1) i rkH i (X(C)C).

i There is a unique way to extend Eu additively to the category of all complex algebraic varieties, by requiring that

Eu(X) = Eu(X 0 ) + Eu(X\X 0 ) forX 0 closed inX just set. Indeed, Eu(X) := X (−1) i rkH i c (X(C)C) i

whereH i c (C) stands for cohomology with compact supports. The following result was obtained in 1987 and published in 1992:

1.1.1. Theorem(Denef and Loeser [23]).(1)Leth:Y→Xbe a DNC modification between smooth complex algebraic varieties. We have († () EuX) = X Eu (E I ◦ ) I⊂A Q i∈I ν i .

(2)LetFbe a nowhere dense subscheme ofXdefined by an idealIand let h:Y→Xbe a log-resolution of(X F) the rational function. Then Eu (E I ◦ ) (‡)Z top,F (s) := I⊂ X A Q i∈I N i (I)s+  ν i does not depend on the log-resolutionh.

1.1.2. Remarks.The result also holds in the complex analytic setting. Initially 2) was stated only whenIis principal, but the proof is the same in general.

The original proof of Theorem 1.1.1 was quite surprizing at the time, since it used integration overp-adic numbers to prove a purely complex statement. That proof, we shall explain now, used also the change of variables formula forp-adic integrals, expression ofpterm of number of points on varieties over-adic integrals in finite fields, and computing Euler characteristics as limits of number of points on varieties over finite fields.

LECTURES ON MOTIVIC INTEGRATION

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1.2. Quick review ofp-adic integration.Most of the material in this sub- section is detailled in the book [56]. Letp We endowbe a prime number.Qwith thep-adic valuation ord p :Q × →Z and thep-adic norm|x| p :=p −ord p (x) ,|0| p = 0, and consider its completionQ p with ring of integers Z p :={x∈Q p ;|x| p ≤1}. More generally one can consider a fieldKwith a valuation ord :K × →Z, extended toKby ord(0) =∞. We denote byO K the valuation ringO K ={x∈ K|ord(x)≥0}and we fix an uniformizing parameter$ element of valuation an, i.e. 1 inO K  ring. TheO K is a local ring with maximal idealM K ofO K generated by$. We shall assume the residue fieldk:=O K /M K is finite withq=p e elements. One endowsKwith a norm by setting|x|:=q −ord(x) forxinK shall furthermore. We assumeKis complete for| |. It follows in particular that the abelian groups (K n +) are locally compact, hence they have a canonical Haar measureµ n , unique up to multiplication by a non zero constant, so we may assumeµ n (O n K ) = 1. The measureµ n is the uniqueR-valued Borel measure onK n which is invariant by translation and such thatµ n (O K n ) = 1. instance For µ n (a+$ m O n K ) =q −mn and for any measurable subsetAofK n and anyλinK, µ n (λA) =|λ| n µ n (A). More generally, for everygin GL n (K), µ n (gA) =|detg|µ n (A). Iffis, say, aK-analytic function onA, we set Z A |f|µ n := Z A |f||dx|:= m X ∈Z µ n (ord(f) =m)q −m  assuming the series P m∈Z µ n (ord(f) =m)q −m is convergent inR. More generally, one can define similarly R A |f| s |dx|by X µ n (ord(f) =m)q −ms m∈Z

whenever it makes sense. For instance, whenn= 1, we have, fors >0 inR, Z x∈O K ,ord(x)≥m |x| s |dx|= j≥ X m q −sj Z ord(x)=j |dx| = X q −sj (q −j −q −j−1 ) j≥m = (1−q −1 )q −(s+1)m /(1−q −(s+1) ).

The formula for change of volumes under GL n (K)-action is a very special form of the following fundamental change of variables formula

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FRANC¸ OIS LOESER

1.2.1. Proposition(Thep-adic change of variables formula).Letf= (f 1  . . .  f n ) be aK-analytic isomorphism between open subsetsUandVofK n . Then

µ n|V =|Jacf|f ∗ (µ n|U ) whereJacfis the determinant of the jacobian matrix(∂f i /∂x j )off other. In terms: Z V ϕµ n = Z U (ϕ◦f)|Jacf|µ n  for every integrable functionϕonV.

LetXbe and-dimensional smoothK One assigns to any-analytic manifold. K-analyticd-differential formωonXa measureµ ω :=|ω|as follows. Take an atlas{(U φ U )}ofX. Write (φ U −1 ) ∗ ω |U =f U dx 1 ∧ ∙ ∙ ∙ ∧dx d . IfAis small enough to be contained in someU, we set µ ω (A) := Z φ U (A) dx|. |f U || It follows from the change of variables formula that the measure may be extended uniquely by additivity to anyAin a way which is independent of the choice of the atlas. Assume nowXis a (smooth) closedd-dimensional submanifold ofO N K  is. There a canonical measureµ X onX Fordefined as follows. any subsetI={i 1 <∙ ∙ ∙< i d } of cardinalitydof{1 . . .  N}, we consider the measureµ X,I onXinduced by dx i 1 ∧ ∙ ∙ ∙ ∧dx i d onXand we setµ X := sup I µ X,I . The canonical volume ofXis vol(X) :=µ X (X relation between the). The volume vol(X by) and counting points is the following. DenoteX n the image ofX in the finite set (O K /$ n O K ) N . IfXis smooth overO K , then vol(X) =|X 1 |q −d . In general, ifXis smooth overK, vol(X) =|X n |q −nd forn0. For singularX, one may define vol(Xlimit of the volume of the com-) as the plement inXof a tubular neighborhood of small radius around the singular locus, andbyaresultofOesterle´[79]: vol(X lim) =|X n |q −nd . n7→∞ 1.3. Sketch of proof of Theorem 1.1.1.For simplicity, we shall assume thatX=A d andF=f −1 (0), withfa polynomial inC[x 1  . . .  x d ] but the proof in general works just the same. Let us first prove 2). We shall writeZ top,f (s) forZ top,F (s). We shall make the assumption that the coefficients offall lie in the same number fieldK, i.e.fis in K[x 1  . . .  x d ] (in general, we can only assume they lie in a field of finite type over Qof the proof still remains the same)., but the basic idea  for every prime Now idealPin the ring of integersO K , we denote byK P the corresponding local field, with ring of integersO P and residue fieldk P .