1227067

Nombres de Betti virtuels des ensembles symétriques
par arcs et équivalence de Nash après éclatements
Goulwen Fichou
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Goulwen Fichou. Nombres de Betti virtuels des ensembles symétriques par arcs et équivalence de
Nash après éclatements. Mathématiques [math]. Université d’Angers, 2003. Français. �tel-00004279�
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D'ANGERS
UNIVERSITE
Annee : 2003
NÆ d'ordre : 580
NOMBRES DE BETTI VIRTUELS
DES ENSEMBLES SYMETRIQUES PAR ARCS
ET EQUIVALENCE DE NASH APRES ECLATEMENTS
THESE
DE DOCTORAT
Spe ialite : Mathematiques
COLE DOCTORALE D'ANGERS
E
Presentee et soutenue publiquement
le 28 Novembre 2003
a
l'universite d'Angers
par Goulwen FICHOU
Devant le jury i-dessous :
Mi hel MERLE,
Rapporteur, Professeur, Universite de Ni e Sophia-Antipolis
Wim VEYS,
Rapporteur, Professeur, University of Leuven
Mi hel GRANGER,
Examinateur, Professeur, Universite d'Angers
Krzysztof KURDYKA,
Examinateur, Professeur, Universite de Savoie
Franois LOESER,
Examinateur, Professeur, E ole Normale Superieure
Dire teur de these : Adam Parusinski, Professeur, Universite d'Angers
Nom et oordonnees du laboratoire : U.M.R NÆ 6093 asso iee au CNRS
2 Bd Lavoisier, 49045 Angers edex 01, Fran e
Table des matieres
Introdu tion
5
English introdu tion
9
partie 1. Cadre et resultats prin ipaux
13
Chapitre 1. Outils prin ipaux
1. Pre isions sur les varietes algebriques reelles
2. E latements
3. Resolution des singularites
4. Theoreme de fa torisation faible
15
15
17
19
22
Chapitre 2. Integration motivique en geometrie reelle
1. Espa es des ar s
2. Mesure sur l'espa e des ar s
3. Formule de hangement de variables
25
25
26
27
Chapitre 3. Resultats prin ipaux
1. Nombres de Betti virtuels
2. Equivalen e de Nash apres e latements
3. Fon tions z^eta
31
31
35
38
partie 2. Generalized Euler hara teristi of ar -symmetri sets
43
Chapitre 4. Ar -symmetri sets
1. Ar -symmetri sets and losure
2. Nonsingular ar -symmetri sets
3. Ar -symmetri sets and resolution of singularities
45
45
48
49
Chapitre 5. Virtual Betti numbers
1. Generalized Euler hara teristi s of ar -symmetri sets
2. Virtual Betti numbers and Nash isomorphisms
53
53
61
partie 3. Zeta fun tions and blow-Nash equivalen e
67
Chapitre 6. Zeta fun tions
1. Zeta fun tions and Denef & Loeser Formula
2. Motivi integration and the proof of Denef & Loeser formula
69
69
74
Chapitre 7. An invariant of the blow-Nash equivalen e
1. Blow-Nash equivalen e
2. Appli ation to Brieskorn polynomials
79
79
84
3
TABLE DES MATIE RES
4
3. Proof of theorem 7.6
Bibliographie
87
95
Introdu tion
L'integration motivique est une theorie re ente, initiee en 1995 par M. Kontsevi h
[21℄, puis developpee depuis par J. Denef et F. Loeser prin ipalement [5, 6℄. Elle
onsiste en une integration sur les espa es d'ar s formels des varietes algebriques
sur un orps de ara teristique zero, ave notamment une formule de hangement de
variables fondamentale, qui s'applique typiquement a une resolution des singularites.
J. Denef et F. Loeser introduisent ainsi des fon tions z^eta motiviques d'une appli ation algebrique. Elles ontiennent des informations sur les singularites, et permettent par exemple de de nir une bre de Milnor motivique dans le as omplexe,
in arnation de la bre de Milnor habituelle.
Cependant, en geometrie algebrique reelle, si la theorie de l'integration motivique
existe toujours, elle est moins aisement appli able faute de ara teristiques d'Euler
generalisees pertinentes. En e et, pour obtenir des mesures al ulables sur les espa es
d'ar s, il est ne essaire de disposer de tels objets, et le seul onnu jusqu'a maintenant
etait la ara teristique d'Euler a supports ompa ts. Elle est toutefois peu utilisable
du fait de l'identi ation des dimensions qu'elle realise.
Or, re emment, C. M Crory et A. Parusinski [27℄ ont donne un autre exemple,
suivant une idee de F. Bittner [3℄, d'une ara teristique d'Euler generalisee pour les
varietes algebriques reelles : le polyn^ome de Poin are virtuel. De ni a partir d'invariants additifs appeles nombres de Betti virtuels, eux-m^emes deduits des nombres
de Betti usuels a oeÆ ients dans 2ZZ, e polyn^ome de Poin are virtuel possede lui
de bonnes proprietes par rapport a l'integration motivique.
La premiere partie de ette these onsiste en une generalisation de es nombres
de Betti virtuels, et don de e polyn^ome de Poin are virtuel, a une ategorie
d'ensembles plus large que les varietes algebriques reelles, a savoir les ensembles
symetriques par ar s. Introduits par K. Kurdyka en 1988 [23℄ dans le but d'etudier les
\ omposantes rigides" des varietes algebriques reelles, les ensembles symetriques par
ar s lui ont permis de demontrer notamment un theoreme de Borel, qui a longtemps
resiste aux mathemati iens : un endomorphisme inje tif d'une variete algebrique
reelle est surje tif ( f. [24℄ ; voir aussi [28℄).
On montre de plus que es nombres de Betti virtuels sont des invariants des
ensembles symetriques par ar s en un sens plus analytique, et plus pre isement
5
6
INTRODUCTION
de Nash. Cette propriete supplementaire est primordiale pour l'appli ation faite a
l'etude des germes de fon tions analytiques reelles au hapitre 7.
L'etude des germes de fon tions analytiques reelles est un sujet diÆ ile, en partiulier dans le hoix d'une bonne relation d'equivalen e entre germes. Si l'equivalen e
topologique est trop faible, l'equivalen e C 1 est trop forte, alors que l'equivalen e
bi-lips hitz possede du module. En 1985, T.-C. Kuo a introduit une nouvelle relation
d'equivalen e, apres avoir onstate que la elebre famille de Whitney, topologiquement triviale, mais possedant une in nite de lasses au sens C 1 , etait triviale analytiquement apres un e latement. Appelee equivalen e analytique apres e latements,
ette relation d'equivalen e est de nie par la ondition que deux germes de fon tions analytiques reelles sont equivalents s'ils sont topologiquement equivalents par
un homeomorphisme qui devient analytique apres e latements des espa es de depart
et d'arrivee. Cette relation, qui possede de bonnes proprietes de trivialite obtenues
via l'integration le long de hamps de ve teurs bien hoisis, est ependant diÆ ile a
etudier, notamment du fait qu'on en onnaisse peu d'invariants.
Dans e travail, on etudie un as parti ulier de ette relation pour les germes
de fon tions de Nash, pour lequel on sait de nir un invariant gr^a e a l'integration
motivique et le polyn^ome de Poin are virtuel. Pour ela, on ajoute des hypotheses
d'algebri ite et une ondition de type Nash (pour plus de details, voir la de nition
7.1), mais on ne sait alors plus si ette nouvelle relation, appelee malgre tout
equivalen e de Nash apres e latements, reste une relation d'equivalen e. On donne
prin ipalement deux resultats on ernant l'equivalen e de Nash apres e latements.
D'une part ette relation onserve des proprietes de trivialite, bien qu'on ne
puisse plus utiliser les m^emes arguments d'integration le long de hamps de ve teurs
dans le adre Nash. Toutefois, a l'aide de proje tions dans des voisinages tubulaires
et d'un lemme de trivialisation pour les ensembles de Nash d^u a T. Fukui, S. Koike,
et M. Shiota [12℄, on obtient tout de m^eme la nitude du nombre de lasses, au sens
de l'equivalen e de Nash apres e latements, d'une famille de fon tions de Nash de
parametrage Nash. On montre de plus que es lasses on ident ave les lasses au
sens de l'equivalen e analytique apres e latements.
D'autre part, de faon analogue a J. Denef et F. Loeser, on de nit une fon tion
z^eta nave et des fon tions z^eta ave signe pour un germe de fon tions analytiques
reelles. Ces fon tions sont de nies a l'aide du polyn^ome de Poin are virtuel et onstituent des invariants pour l'equivalen e de Nash apres e latements. Elles permettent
par exemple de determiner les lasses, pour l'equivalen e de Nash apres e latements,
mais aussi pour l'equivalen e analytique apres e latements, des polyn^omes de Brieskorn a deux variables.
Notons que S. Koike et A. Parusinski avaient deja de ni, de maniere similaire, des
fon tions z^eta a partir de la ara teristique d'Euler a supports ompa ts, qui sont des
invariants pour l'equivalen e analytique apres e latements. Leurs fon tions z^eta ne
peuvent ependant determiner, a elles seules, les lasses d'equivalen e des polyn^omes
de Brieskorn a deux variables pour l'equivalen e analytique apres e latements, du
fait des moins bonnes proprietes de la ara teristique d'Euler a supports ompa ts.
Ce travail est redige en trois parties prin ipales. La premiere regroupe quelques
pre isions sur les varietes algebriques reelles, des rappels sur les e latements, ainsi
INTRODUCTION
7
que les enon es de deux fameux theoremes pre ieux par la suite, la resolution des
singularites de H. Hironaka et le theoreme de fa torisation faible des appli ations
birationnelles. Cette partie se poursuit ave une mise au point sur l'integration
motivique telle qu'elle est utilisee dans e travail, et se termine ave un resume des
resultats prin ipaux de ette these, deja evoques i-dessus.
Les deux parties suivantes, redigees en anglais par sou i d'^etre omprehensible
pour le plus grand nombre de le teurs possible, forment un expose plus detaille de es
resultats. Ainsi, la deuxieme partie est onsa ree aux ensembles symetriques par ar s,
a la de nition des nombres de Betti virtuels, et a leur invarian e par isomorphisme
de Nash.
En n, 'est dans la troisieme et derniere partie que l'on de nit, et etudie, la fon tion z^eta nave et les fon tions z^eta ave signe d'un germe de fon tions analytiques
reelles, ainsi que l'equivalen e de Nash apres e latements. Le resultat prin ipal est
i i le theoreme d'invarian e des fon tions z^eta qui lie es deux objets. Le as des
polyn^omes de Brieskorn a deux variables est nalement traite a titre d'exemple.
English introdu tion
In the study of real analyti fun tion germs, the hoi e of a good equivalen e
relation between germs is a ru ial topi . Whereas the topologi al equivalen e is too
oarse and the C 1 -equivalen e too ne, blow-analyti equivalen e, a notion introdued by T.-C. Kuo in 1985 (see [22℄, and [10℄ for a survey) seems to behave better,
espe ially with respe t to niteness properties. In this Ph. D. thesis, we will fous on a parti ular ase of blow-analyti equivalen e, alled blow-Nash equivalen e,
for whi h we add algebrai data ; note that we do not know whether blow-Nash
equivalen e is an equivalen e relation or not.
Let f; g : (Rd ; 0) ! (R; 0) be Nash fun tion germs. Then f and g are said to
be blow-Nash equivalent if there exist two algebrai modi ations
f : Mf ; f 1 (0)
! ( d ; 0) and g : Mg ; g 1(0) ! ( d ; 0);
R
R
and a Nash-isomorphism,
that is an analyti isomorphism with semi-algebrai graph,
: Mf ; f 1 (0) ! Mg ; g 1 (0) whi h respe ts the multipli ity of the ja obian
determinants of f and g and whi h indu es a homeomorphism h between neighbourhoods of 0 in Rd su h that f = g Æ h. Here, by a modi ation of f , we mean
a proper birational map whi h is an isomorphism over the omplement of the zero
lo us of f and su h that f Æ is in normal rossing. One an de ne su h a relation
on Nash sets, and S. Koike ([18, 19℄) proved that niteness properties hold in this
ase. However in the ase of germs of fun tions, the question of moduli is still open
in general. In hapter 7, we prove that there is no moduli for a Nash family with
isolated singularities under some algebrai assumptions on modi ations. In partiular, an algebrai family of isolated singularities does not admit moduli for the
blow-Nash equivalen e.
A ommon issue for blow-analyti equivalen e and blow-Nash equivalen e is to
prove that, when it is the ase, two given germs of real analyti fun tions are not
equivalent. The diÆ ulty rises in the la k of invariants known for these relations. Up
to now, just one kind of invariants have been known : the Fukui invariants. With an
analyti fun tion germ f , the Fukui invariants asso iate the set of possible orders
n of series f Æ (t) = an tn + ; an 6= 0, for : (R; 0) ! (Rd ; 0) an analyti ar
([9, 17℄). There exists also a version of the Fukui invariants related to the sign of f .
9
10
ENGLISH INTRODUCTION
Using motivi integration ombined with the onstru tion of a omputable motivi invariant for ar -symmetri s sets, the virtual Betti numbers, we introdu e zeta
fun tions Z (T ); Z (T ) of a real analyti fun tion germ that belong to Z[u; u 1℄[[T ℄℄,
and take into a ount not only the orders of the series f Æ (t) but also the geometry
of the sets n (f ) of ar s that realize a given order n (for pre ise de nitions, see
se tion 1 of hapter 6). These zeta fun tions are similar to the motivi zeta fun tions
of Denef & Loeser [5℄.
We prove that our zeta fun tions are invariants of the blow-Nash equivalen e.
The proof is dire tly inspired by the work of Denef & Loeser via their formulae for
the zeta fun tions in terms of a modi ation of the zero lo us of the analyti fun tion
germ (propositions 6.3, 6.6). It uses the powerfull ma hinery of motivi integration,
a theory introdu ed by M. Kontsevit h in 1995 [21℄ and developped further by J.
Denef and F. Loeser [5, 6, 7, 8℄, in parti ular the fundamental hange of variables
formula (6.11).
In order to dispose of omputable invariants, motivi integration requires omputable measures, or in other words generalized Euler hara teristi s. A generalized
Euler hara teristi is an additive and multipli ative invariant de ned on the level of
the Grothendie k group of varieties. In our setting of the blow-Nash equivalen e, we
need invariants of the Zariski onstru tible sets over real algebrai varieties n (f )
(real algebrai variety is the sense of [4℄), and we ask it to be respe ted by Nash isomorphisms. It leads naturally to the ategory of Nash varieties, and more generally
of ar -symmetri sets.
Ar -symmetri sets have been introdu ed in 1988 by K. Kurdyka [23℄ in order
to study \rigid omponents" of real algebrai sets. Ar -symmetri sets enabled him
to prove the Borel theorem [24℄ whi h states that inje tive endomorphisms of real
algebrai sets are surje tive. With a slightly di erent de nition of ar -symmetri
sets, A. Parusinski [28℄ has proven the same result by using the fa t that these sets
form a onstru tible ategory.
In hapter 5, we give onditions, inspired by a result of F. Bittner [3℄, on an
invariant de ned on onne ted omponents of ompa t nonsingular real algebrai
varieties su h that it extends to an additive invariant on the onstru tible ategory
of ar -symmetri sets. Additive means that (A) = (B ) + (A n B ) for a losed
in lusion B A of ar -symmetri sets. Let us stress the fa t that the unique su h
additive invariant known up to now in the real ase is the lassi al Euler hara teristi with ompa t supports, and as a matter of fa t it is the unique generalized
Euler hara teristi for semi-algebrai sets up to homeomorphism [29℄.
As a fundamental example, we prove that the Betti numbers with 2ZZ- oeÆ ient
de ned on onne ted omponents A of ompa t nonsingular real algebrai sets by
bk (A) = dim Hk (A; 2ZZ), give su h an additive invariant k on ar -symmetri sets for
ea h k 2 N ( orollary 5.5),P alled k-Virtual Betti number. We make them multipliA
k
vative by putting (A) = dim
e polynomial
k (A)u 2 Z[u℄, alled virtual Poin ar
k=0
of A.
This invariant is di erent from the lassi al Euler hara teristi with ompa t
supports, and in parti ular it is not a topologi al invariant.
Moreover it respe ts
dimension as put in light by the formula deg (A) = dim(A) (see remark 5.12)
whereas the Euler hara teristi with ompa t supports may identify the dimension.
These numbers have been proven to be additive invariant of real algebrai varieties
ENGLISH INTRODUCTION
11
re ently by C. M Crory and A. Parusinski in [27℄ ; in this thesis, we extend the
virtual Betti numbers to the more general ontext of ar -symmetri sets, and we
prove the invarian e not only under algebrai isomorphisms but also under Nash
isomorphisms (see 5.16).
Note that the virtual Betti numbers over real algebrai sets have been introdued independentyly by C. M Crory and A. Parusinski [27℄, and by B. Totaro [31℄.
Moreover S. Koike and A. Parusinski ([20℄) have de ned in the same way zeta fun tions, by using the lassi al Euler hara teristi with ompa t supports, whi h are
invariant for blow-analyti equivalen e. The advantage of our zeta fun tions, whose
invarian e is proven only for blow-Nash equivalen e, is that the Virtual Betti numbers have a better behaviour with respe t to algebrai ity and analyti ity that the
lassi al Euler hara teristi with ompa t supports whi h is merely topologi al.
Premiere partie
Cadre et resultats prin ipaux
CHAPITRE 1
Outils prin ipaux
Avant de rentrer dans les details de e travail, ommenons par rappeler quelques
de nitions et pre iser ertaines notions. Tout d'abord, la notion de variete algebrique
reelle est abordee, ar l'utilisation dans e travail de resultats enon es dans le adre
des s hemas et appliques dans elui des points merite quelques pre isions. C'est
l'objet de la premiere se tion.
Ensuite, on rapelle dans la deuxieme se tion e qu'est l'e latement d'une variete
par rapport a une sous-variete, ette notion etant omnipresente dans e travail. En
parti ulier, elle est fondamentale pour la resolution des singularites (troisieme se tion) et le theoreme de fa torisation faible des appli ations birationnelles (quatrieme
se tion de e hapitre).
Ces deux derniers resultats, de demonstration tres diÆ ile, mais d'utilisation tres
pratique, meritent eux-aussi d'^etre mis en valeur ar leur utilisation est primordiale
dans ette these.
1. Pre isions sur les varietes algebriques reelles
Le nom de varietes algebriques reelles est utilise, dans la litterature, pour deux
situations di erentes. D'un ote la notion de varietes de nies a partir des s hemas,
pour lesquelles on emploie les termes de varietes algebriques sur R, de l'autre elle
de varietes vues omme ensembles des solutions reelles d'un systeme d'equations
polynomiales a oeÆ ients reels. Pour e se ond point de vue, la referen e lassique
est l'ouvrage de J. Bo hnak, M. Coste et M. F. Roy [4℄.
Les travaux de ette these se situent dans le adre des varietes algebriques reelles
au sens de [4℄ (par exemple les nombres de Betti virtuels, f. orollaire 5.5, sont de nis
a partir des nombres de Betti lassiques, 'est-a-dire de la dimension de l'homologie a
oeÆ ients dans 2ZZ de l'ensemble des points d'une variete reelle ; de m^eme la notion
d'ensemble symetrique par ar s s'exprime dans e adre), mais utilisent de maniere
essentielle des resultats de geometrie algebrique sur un orps de ara teristique zero
(resolution des singularites, theoreme de fa torisation faible) qui s'expriment plus
naturellement dans le adre des s hemas.
Cette partie a pour but de justi er les utilisations de es resultats dans le adre
des varietes algebriques reelles au sens de [4℄. Il ne s'agit pas, bien entendu, de
donner i i un ours omplet sur le sujet, mais juste de xer les notions et enon er les
resultats utilises par la suite. Pour ommen er, rappelons les de nitions de varietes
dans les deux sens.
Definition 1.1. Une variete algebrique sur R est un s hema integre et separe
de type ni sur R.
15
16
1. OUTILS PRINCIPAUX
Exemple 1.2. La variete algebrique sur R asso iee a l'ideal J R[X1 ; : : : ; Xn ℄
est le s hema aÆne de ni par l'ensemble
R [ X1 ; : : : ; X n ℄
Spe
J
R[X1 ;:::;X ℄
des ideaux premiers de
, muni de la topologie de Zariski et du fais eau des
J
R[X1 ;:::;X ℄
.
fon tions regulieres de ni par
J
Remarque 1.3.
(1) Soit x un point ferme de X . L'anneau lo al de X en x a pour orps residuel
R ou C . Si le orps r
esiduel est R, on dit que x est un point reel de X , sinon
x est un point omplexe. On note par X (R) l'ensemble des points reels. Par
ailleurs, notant X (C ) l'ensemble des points fermes de la variete omplexe
XC = X Spe R Spe C , on a une in lusion naturelle X (R ) X (C ).
(2) On peut voir un s hema sur R omme un s hema sur C muni d'une involution. Les points reels orrespondent alors aux points xes de ette involution.
On ne donne la de nition de variete algebrique reelle au sens de [4℄ que dans le
as aÆne ar les grasmaniennes, et don en parti ulier les espa es proje tifs, sont
des varietes aÆnes dans e adre ( f. Theoreme 3.4.4, [4℄). Tout d'abord rappelons
la notion d'ensemble algebrique reel.
Definition 1.4. Un ensemble algebrique reel aÆne de Rn est le lieu des zeros
d'un nombre ni d'equations polynomiales.
Un tel ensemble V devient un espa e topologique ave la topologie de Zariski, et
est dote naturellement d'un fais eau de fon tions regulieres dont les se tions globales
sont donnees par
R [X1 ; : : : ; Xn ℄
J (V ) ;
ou J est l'ideal asso ie a V .
Definition 1.5. Une variete algebrique reelle est un espa e topologique X , muni
d'un fais eau de fon tions a valeurs reelles, isomorphe a un ensemble algebrique
V Rn muni de la topologie de Zariski, et de son fais eau des fon tions regulieres.
Remarque 1.6. Une variete algebrique aÆne X sur R possede un ensemble de
points reels, notes X (R ), qui est naturellement un ensemble algebrique reel. Dans le
as de l'exemple 1.2, 'est exa tement l'ensemble des zeros ommuns aux elements
de J . En parti ulier, on asso ie de faon naturelle une variete aÆne X sur R a un
ensemble algebrique reel aÆne V , ave X (R) = V .
On peut de nir des notions de points reguliers, points singuliers, de dimension,
de de omposition en irredu tible, et ... A haque fois, il onvient d'^etre prudent dans
les relations entre les deux adres. A titre d'illustration, notons que la dimension
d'un ensemble algebrique reel est la m^eme que la dimension de la variete algebrique
aÆne sur R asso iee. Cependant, la dimension d'un ensemble algebrique irredu tible
reel n'est pas for ement la m^eme en haque point ( ontrairement a e qui se passe en
geometrie omplexe par exemple) omme l'illustre l'exemple du parapluie de Whitney ( f. gure 1). Par ontre, par de nition, la dimension d'une variete algebrique
reelle non singuliere est la m^eme en tous ses points.
n
n
2. ECLATEMENTS
17
Pour e qui est des points lisses, nous avons les relations suivantes :
Proposition 1.7.
(1) Si X est une variete algebrique aÆne sur R lisse, alors X (R) est une
variete algebrique reelle lisse.
(2) Si V est une variete algebrique reelle lisse aÆne, alors la variete algebrique
aÆne X sur R asso iee veri e
X (R ) Reg(X );
ou Reg(X ) designe l'ensemble des points lisses de X .
Remarque 1.8. Il est faux en general que la variete algebrique X sur R soit lisse,
des singularites omplexes pouvant appara^tre (par exemple dans (x2 + 1)2 = 0).
La notion de ompa ite s'exprime en terme de proprete dans le adre des s hemas.
De la m^eme maniere que pre edemment :
Proposition 1.9. Si X est une variete algebrique aÆne sur R propre, alors
X (R) est une variete algebrique reelle ompa te.
Remarque 1.10.
(1) Il est faux que la variete algebrique sur R asso iee a un ensemble algebrique
reel aÆne et ompa t reste propre des que la dimension des varietes est plus
grande que 0.
(2) La ompa ti ation d'une variete algebrique X sur R peut avoir plus de
points reels que X m^eme si X (R) est ompa te.
Les morphismes algebriques ne sont pas non plus onserves lorsque l'on passe
des ensembles algebriques aux s hemas, ar des fra tions rationnelles sans p^ole reel
peuvent avoir des p^oles omplexes. Par ontre :
Proposition 1.11. Si h : V1 ! V2 est une appli ation birationnelle entre deux
ensembles algebriques reels, il existe une appli ation birationnelle eh : X1 ! X2
entre les varietes algebriques aÆnes sur R asso iees telle que la restri tion de eh a
X1 (R) oin de ave h.
Demonstration. eh est donnee par les m^emes fra tions rationnelles que h.
2. E latements
Les e latements sont tres utilises en geometrie, aussi bien pour etudier les varietes
singulieres que pour les varietes lisses. Ils onstituent un ingredient essentiel de la
demonstration du fameux theoreme d^u a H. Hironaka [16℄ de resolution des singularites pour les varietes algebriques sur un orps de ara teristique zero ( f. se tion
3), mais aussi de l'etude de la geometrie birationnelle des varietes, ave notamment
le theoreme de fa torisation faible des appli ations birationnelles ( f. se tion 4).
En plus de es appli ations, on utilise egalement les e latements dans e travail
pour :
{ enon er une hypothese du theoreme 5.3 que doit veri er une ara teristique
d'Euler generalisee,
18
1. OUTILS PRINCIPAUX
{ etudier, de maniere generale, les equivalen es analytiques apres e latements,
et demontrer en parti ulier le theoreme 7.6 sur les lasses pour l'equivalen e
de Nash apres e latements d'une famille de germes de fon tions de Nash parametree au sens Nash.
Dans ette partie sur les e latements, variete signi e variete analytique aussi
bien que algebrique, reelle ou bien omplexe. On de nit i i un e latement par une
propriete universelle, et on enon e ensuite l'existen e et l'uni ite a isomorphisme
pres de ette notion. Pour une de nition pre ise du fais eau image inverse, on peut
onsulter [15℄ par exemple.
Definition 1.12. Soient X et X 0 deux varietes, : X 0 ! X un morphisme et
J un fais eau d'ideaux oherent sur X . On dit que : X 0 ! X est un e latement
de X par rapport a J si
(1) le fais eau d'ideaux image inverse 1 J OX de J par est un fais eau
d'ideaux inversible sur X ,
(2) pour tout morphisme f : Y ! X tel que f 1J OX est un fais eau d'ideaux
inversible sur Y , il existe un unique morphisme f 0 : Y ! X 0 fa torisant
f , 'est-a-dire tel que f = Æ f 0.
Remarque 1.13.
(1) Lorsque le fais eau d'ideaux oherent J provient d'une sous-variete fermee
Y , on parle d'e latement de X par rapport a Y .
(2) L'hypersurfa e de X 0 de nie par le fais eau d'ideaux inversible 1 J:OX
est appelee diviseur ex eptionnel de l'e latement.
Theoreme 1.14. L'e latement d'une variete par rapport a un fais eau d'ideaux
oherent sur ette variete existe et est unique a isomorphisme pres.
Remarque 1.15. L'uni ite est dire te en vertu de la propriete universelle de
l'e latement. L'existen e ne essite par ontre une onstru tion. A titre d'exemple,
traitons le as de l'e latement de R2 au point (0; 0).
Exemple 1.16. Posons
M := f(x; d) 2 R2 P1 ; x 2 dg;
ou P1 designe la droite proje tive reelle. La proje tion : M ! R2 par rapport
a la premiere oordonnee est l'e latement de R2 au point (0; 0). Re ouvrant M par
deux artes isomorphes a R2 , provenant des artes habituelles sur P1 , on obtient
des formules pour al uler de maniere expli ite, a savoir (X; Y ) = (X; XY ) et
(X; Y ) = (XY; Y ).
Notons que, topologiquement, M est homeomorphe a une bande de Mobius et
n'est don pas orientable.
Le resultat suivant est une onsequen e immediate de la propriete universelle de
l'e latement.
Corollaire 1.17. Soient X et Y deux varietes, f : Y ! X un morphisme, et
J un fais eau d'ideaux oherent sur X . Notons par X : X 0 ! X et Y : Y 0 ! Y
les e latements respe tifs de X et Y par rapport au fais eau d'ideaux oherent J
3. RE SOLUTION DES SINGULARITE S
19
et a l'image inverse f 1J:OX de J par f . Alors il existe un unique morphisme
f 0 : Y 0 ! X 0 ompletant et rendant le diagramme
Y0 _
f0
X0
_ _/
Y
X
Y
f
/
X
ommutatif.
De plus, si f est une immersion fermee, f 0 l'est aussi.
Remarque 1.18. Dans le as ou Y est une sous-variete fermee de X , la sousvariete Y 0 de X 0 est appelee transformee stri te de Y par l'e latement X .
Le as parti ulier d'un e latement d'une variete lisse par rapport a un entre
lisse est remarquable, que e soit pour la resolution plongee des singularites ou pour
l'etude de la geometrie birationnelle des varietes algebriques. De plus, dans le adre
de la geometrie algebrique reelle, le morphisme d'e latement devient surje tif ave
es hypotheses, e qui n'est pas le as en general (voir l'exemple 1.21). Notons en
parti ulier :
Proposition 1.19. Soit X une variete reelle lisse et soit C X une sousvariete stri te lisse. Si X : X 0 ! X designe l'e latement de X par rapport a C ,
alors X 0 est lisse et est surje tif.
3. Resolution des singularites
La resolution des singularites est un sujet que l'on peut faire remonter a Newton pour le as des ourbes planes. Il a ete formule ensuite pour les dimensions
superieures vers la n du 19e sie le, pour ^etre nalement demontre par H. Hironaka
dans le as des varietes algebriques sur un orps de ara teristique zero, et pour les
varietes analytiques reelles, en 1964. H. Hironaka a ensuite traite le as des varietes
analytiques omplexes. Le resultat n'est pas onnu pour les varietes algebriques sur
un orps de ara teristique quel onque.
La demonstration de H. Hironaka est tres diÆ ile, et non onstru tive. Elle utilise
notamment les e latements a entres lisses. E. Bierstone et P. Milman [2℄ ont donne,
en 1997, une preuve plus elementaire reposant sur un invariant lo al qui determine
les entres d'e latements a hoisir.
On enon e i i le resultat, sous sa forme plongee qui nous sera utile par la suite,
dans le adre des varietes algebriques.
Theoreme 1.20. Soit X une sous-variete algebrique d'une variete algebrique
ambiante M non singuliere sur un orps de ara teristique zero.
f ainsi qu'un morphisme propre
Il existe une variete algebrique non singuliere M
f
: M ! M tels que :
(1) est un isomorphisme en dehors du lieu singulier Sing X de X ,
(2) la transformee stri te Xe de X est non singuliere,
(3) la transformee stri te Xe et le diviseur ex eptionnel E = 1 (Sing X ) sont
simultanement a roisements normaux (i.e. lo alement il existe un systeme
de oordonnees tel que Xe est un sous-espa e de oordonnees et E une
reunion d'hyperplans de oordonnees).
20
1. OUTILS PRINCIPAUX
Fig. 1. Resolution du parapluie de Whitney W .
~
W
W
π
E
De plus, le morphisme birationnel peut ^etre hoisi omme etant la omposition
d'un nombre ni d'e latements de entres lisses.
Exemple 1.21.
(1) Considerons le parapluie de Whitney W d'equation zx2 = y2 dans R3 .
L'axe des z est le lieu singulier, et on peut i i resoudre les singularites a
l'aide d'un seul e latement , elui de R3 par rapport a et axe (voir gure
1). On peut de rire et e latement ave deux artes isomorphes a R3 par
les formules :
(X; Y; Z ) = (X; XY; Z )
et
(X; Y; Z ) = (XY; Y; Z ):
Dans la premiere arte, l'equation de nissant le parapluie de Whitney devient X 2 (Z Y 2 ) = 0 ; alors X = 0 est l'equation du diviseur ex eptionnel
f . Dans la se onde arte, on
E , et Z Y 2 = 0 elle de la transformee stri te W
obtient de faon similaire Y = 0 pour le diviseur ex eptionnel et ZX 2 = 1
pour la transformee stri te.
(2) Considerons le usp C d'equation y2 = x3 dans R2 . Il suÆt en ore une fois
d'un seul e latement 1 pour resoudre les singularites de la transformee
strite C1 de C . Cependant C1 et le diviseur ex eptionnel E1 ne sont pas a
roisements normaux. Pour resoudre les singularites au sens du theoreme
1.20, il est ne essaire d'e e tuer deux nouveaux e latements 2 et 3 (voir
la gure 2).
En termes d'equations, il suÆt a haque fois d'etudier une seule arte,
elle ou se situe la singularite. Ainsi, par le premier e latement on obtient
X12 (Y12 X1 ) = 0, la transformee stri te etant dans ette arte une parabole tangente en (0; 0) au diviseur ex eptionnel. Puis, apres le deuxieme
e latement, il vient X22 Y23 (Y2 X2 ) = 0 ; notons qu'i i les deux diviseurs
3. RE SOLUTION DES SINGULARITE S
21
Fig. 2. Resolution du usp.
E1
E1
C
π3
C1
E1
C3
E3
C2
π1
π2
E3
E2
C3
π3
E2
Fig. 3. Diagramme de resolution du usp.
(E3 ,6)
(E1 ,2)
C3
(E2 ,3)
ex eptionnels apparaissent dans la m^eme arte, e qui ne sera pas le as
apres le dernier e latement pour lequel on obtient X32 Y36 (X3 1) = 0 dans
une arte et X36 Y33 (1 Y3 ) = 0 dans l'autre. On peut representer ette
su ession d'e latements par le diagramme de la gure 3 qui permet de
s hematiser le pro essus de resolution. On garde de plus en memoire, pour
haque diviseur ex eptionnel, la multipli ite ave laquelle il appara^t.
Remarque 1.22. Le theoreme de desingularisation est omnipresent dans e travail. Il permet par exemple de montrer que :
22
1. OUTILS PRINCIPAUX
{ les ensembles symetriques par ar s irredu tibles sont moralement des omposantes onnexes de varietes algebriques reelles ( f. proposition 4.14),
{ toute variete lisse admet une ompa ti ation lisse, e qui est utilise dans la
demonstration du theoreme 5.3,
{ deux ensembles symetriques par ar s isomorphes au sens de Nash ont les
m^emes nombres de Betti virtuels ( f. theoreme 5.17), en jouant sur le fait
qu'un e latement algebrique est aussi analytique.
C'est, de plus, un des ingredients essentiels pour de nir et etudier l'equivalen e
analytique apres e latements ( f. se tion 1 du hapitre 7).
4. Theoreme de fa torisation faible
Le theoreme de fa torisation faible des appli ations birationnelles peut s'enon er
omme suit : un isomorphisme birationnel entre deux varietes non singulieres ompa tes sur un orps de ara teristique zero peut se de omposer omme une suite
d'e latements et de ontra tions de entres non singuliers.
Ce resultat, dont l'origine remonte sans doute a l'e ole italienne de geometrie
algebrique, est enon e sous sa forme forte (voir la remarque 1.24) par H. Hironaka
des 1964 dans son elebre arti le sur la resolution des singularites [16℄. De nombreux
mathemati iens se sont pen hes depuis sur le probleme, et on doit sa demonstration
a D. Abramovi h, K. Karu, K. Matsuki et J. Wlodar zyk en 2002 [1℄, a la suite de
travaux de J. Wlodar zyk [32℄. Cette demonstration, diÆ ile, utilise de nombreuses
notions de geometrie algebrique, notamment la theorie des varietes torodales et le
obordisme birationnel.
Theoreme 1.23. Soit : X1 99 K X2 un isomorphisme birationnel entre deux
varietes non singulieres ompa tes X1 et X2 sur un orps de ara teristique zero,
et soit U X1 un ouvert sur lequel est un isomorphisme. Alors il existe une
su ession de morphismes birationnels entre des varietes algebriques non singulieres
et ompa tes
X1 = V0 _1 _ _/ V1 _
veri
(1)
(2)
(3)
2
_ _/
_ _/ _ _ 1_/ Vl
_ _ _/ Vi _ _+1_/ Vi+1 _ +2
i
i
i
l
l
1 _ _ _/ Vl
= X2
ant
= l Æ l 1 Æ Æ 2 Æ 1 ,
i est un isomorphisme en restri tion a U ,
i : Vi 99 K Vi+1 , ou bien i 1 : Vi+1 99 K Vi , est un e latement de entre lisse
et irredu tible disjoint de U .
Remarque 1.24. La forme forte du theoreme de fa torisation des appli ations
birationnelles dit qu'on peut hoisir les e latements i de telle sorte qu'on e e tue
dans un premier temps uniquement des e latements, puis dans un se ond temps
uniquement des ontra tions. Ce n'est pour l'instant qu'une onje ture.
Ce resultat est utilise dans ette these dans le adre des ensembles algebriques
reels au sens de [4℄. La proposition suivante justi e ette utilisation.
Proposition 1.25. Le theoreme 1.23 reste vrai si X1 et X2 sont des ensembles
algebriques reels au sens de [4℄.
4. THE ORE ME DE FACTORISATION FAIBLE
23
Demonstration. L'ensemble algebrique reel Xi , pour i 2 f1; 2g, peut ^etre plonge
dans un espa e aÆne RN pour N assez grand. Il orrespond alors de faon naturelle
aux points reels d'un s hema aÆne Yi sur R integre de type ni. L'isomorphisme
birationnel entre X1 et X2 s'etend lui aussi aux s hemas aÆnes ( f. se tion 1).
Notons par Yei une resolution des singularites d'une ompa ti ation de Yi , pour
i 2 f1; 2g. Ces deux varietes algebriques Ye1 et Ye2 sur R restent birationnelles, et Yei
ontient un sous-ensemble Xi0 isomorphe a Xi puisque et ensemble est ompose de
points lisses de Yi , pour i 2 f1; 2g.
On peut desormais appliquer le theoreme de fa torisation faible a l'isomorphisme
birationnel entre Ye1 et Ye2 , e qui a heve la demonstration de la proposition, par
restri tion a X10 et X20 .
Remarque 1.26. Dans e travail, on utilise le theoreme de fa torisation faible
pour la demonstration du theoreme 5.3. On doit y omparer deux ompa ti ations lisses d'une m^eme variete lisse. Ces deux ompa ti ations sont automatiquement birationnelles, et le theoreme de fa torisation faible permet de se ontenter
de les omparer dans le as parti ulier ou l'isomorphisme birationnel est juste un
e latement.
CHAPITRE 2
Integration motivique en geometrie reelle
L'integration motivique a ete introduite par M. Kontsevi h [21℄ en 1995, et
developpee ensuite par J. Denef et F. Loeser [5, 6℄. Elle fournit des moyens puissants
pour etudier les singularites algebriques via une integration sur les espa es d'ar s
des varietes algebriques. Citons notamment la formule de hangement de variables
de Kontsevi h ( f. theoreme 2.2), dont on rappelle la demonstration i i, qui est
un des resultats essentiels de ette theorie. Elle est utilisee dans e travail pour
demontrer l'invarian e de fon tions z^eta, onstruites a la maniere de elles de J.
Denef et F. Loeser, par rapport a l'equivalen e de Nash apres e latements des germes
de fon tions analytiques reelles (voir le hapitre 7). Plus pre isement, on enon e au
hapitre 6 des formules, dites de Denef & Loeser, qui sont une onsequen e dire te
de ette formule de hangement de variables de Kontsevi h.
L'objet de e hapitre est de presenter la theorie de l'integration motivique telle
qu'elle est utilisee dans ette these, 'est-a-dire dans un adre plus elementaire que
dans [6℄. Notons en parti ulier que l'on travaille i i ave des varietes algebriques
reelles, les espa es d'ar s sont introduits pour des varietes lisses, et le theoreme
fondamental de hangement de variables est etabli ave l'espa e d'arrivee qui est un
espa e aÆne.
On rappelle ainsi la de nition des espa es d'ar s d'une variete algebrique reelle,
puis la mesure, les ensembles mesurables et l'integrale. On enon e puis demontre,
pour nir, le theoreme de hangement de variables de Kontsevi h.
1. Espa es des ar s
Soient X une variete algebrique reelle lisse et Y X une sous-variete fermee,
non ne essairement lisse. On onsidere l'espa e des ar s
L(X; Y ) = f : (R; 0) ! (X; Y ); ar formel, (0) 2 Y g
et l'espa e des ar s tronques a l'ordre n + 1, quotient de L(X; Y ) par la relation
d'equivalen e 1 2 si et seulement si 1 (t) 2 (t) mod tn+1 dans un systeme de
oordonnees lo ales. On note Ln (X; Y ) l'espa e des ar s tronques a l'ordre n + 1,
n : L ! Ln la tron ation des ar s a l'ordre n +1 et nm : Lm ! Ln les tron ations
pour m n.
Exemple 2.1.
(1) Dans le as d'un espa e aÆne,
L(Rd ; Y ) = f 2 R[[t℄℄d ; (0) 2 Y g
et on peut identi er Ln (Rd ; Y ) ave
fa0 + a1t + + antn; a0 2 Y; a1; : : : ; an 2 Rd g:
25
26
2. INTE GRATION MOTIVIQUE EN GE OME TRIE RE ELLE
En parti ulier, pour Y = 0 2 Rd , l'espa e des ar s tronques Ln (Rd ; 0) est
tout simplement isomorphe a Rnd .
(2) Dans la as ou Y = X , l'espa e L0 (X ) = L0 (X; X ) est en fait X , alors que
L1 (X ) est l'espa e tangent a X . Par exemple, dans le as du er le unite
S 1 de R2 ,
L1(S 1 ) = f(a0 + a1t; b0 + b1t); (a0 + a1 t)2 + (b0 + b1 t)2 1 mod t2 g
= f(a0 + a1 t; b0 + b1 t); a20 + b20 = 1; a0 a1 + b0 b1 = 0g:
L'espa e des ar s est une variete algebrique de dimension in nie, plus orre tement de nie en passant par les s hemas : a une variete algebrique reelle (aÆne)
est naturellement asso iee une variete algebrique X sur R ( f. remarque 1.6) dont
les points reels sont isomorphes a la variete algebrique reelle de depart. On de nit
alors l'espa e des ar s tronques Ln(X ) sur X omme etant le s hema representant
le fon teur
R[t℄
R ! HomR (Spe n+1 ; X )
t R[t℄
de ni sur la ategorie des R-algebres. C'est une variete algebrique sur R de dimension nie. L'espa e des ar s est alors de ni omme la limite proje tive des espa es
tronques.
Les espa es d'ar s onsideres plus haut orrespondent en fait, dans le adre des
s hemas sur R, aux points reels de es varietes sur R.
Le hoix de travailler sur les points, plut^ot que sur les s hemas, vient du fait que
la mesure sur l'espa e des ar s que l'on etudie dans e travail est un invariant des
points. En e et ette mesure provient des nombres de Betti virtuels, introduits dans
le hapitre 5, qui sont eux-m^emes des invariants des points.
2. Mesure sur l'espa e des ar s
On introduit maintenant une mesure sur l'espa e des ar s d'une variete lisse. Elle
prend ses valeurs dans un lo alise M(R) de l'anneau de Grothendie k des varietes
algebriques reelles.
L'anneau de Grothendie k des varietes algebriques reelles est le quotient de l'anneau libre engendre par les symboles [X ℄, pour X une variete algebrique reelle, par
l'ideal engendre par les relations :
{ [X ℄ [Y ℄ si X et Y sont isomorphes,
{ [X ℄ [Y ℄ [X n Y ℄ si Y X est une in lusion fermee,
{ [X Y ℄ [X ℄ [Y ℄:
L'anneau M(R) est simplement le lo alise de l'anneau de Grothendie k des varietes
algebriques reelles en le symbole de l'espa e aÆne L = [R℄.
Cet anneau a une propriete universelle par rapport aux ara teristiques d'Euler
generalisees ( f. de nition 5.1), et permet de travailler dans un adre general pour
l'integration motivique. Dans le hapitre 6, on utilise ette theorie ave le polyn^ome
de Poin are virtuel qui est un invariant plus on ret, et qui donne lieu par exemple
a des al uls expli ites pour les fon tions z^eta.
On introduit maintenant les ensembles mesurables de l'espa e des ar s. Un sousensemble A L est dit stable s'il existe un sous-ensemble onstru tible reel C Ln
tel que A = (n ) 1 (C ) pour un ertain n 2 N .
3. FORMULE DE CHANGEMENT DE VARIABLES
27
On ne onsidere i i que des varietes lisses et, dans e as, la tron ation n est
surje tive et les tron ations nm : Lm ! Ln, pour m n, sont des brations
lo alement triviales de bre R(m n)d ave d = dim X . La quantite
(A) = [n (A)℄ L (n+1)d
est alors onstante pour n suÆsamment grand si A est un sous-ensemble stable de
L, et 'est par de nition la mesure de A, a valeurs dans M(R).
Les fon tions integrables sont des appli ations : A ! M(R), pour A stable,
dont les bres sont des sous-ensembles stables de L et dont l'image Im() est nie.
L'integrale de sur A est par de nition
Z
A
d =
X
2Im()
1( ) :
3. Formule de hangement de variables
Soit h : (M; E ) ! (Rd ; 0) un morphisme birationnel propre, ave E un diviseur
a roisements normaux, entre des voisinages analytiques de E dans M et de 0 dans
d
R . Notons
h : L(M; E ) ! L(Rd ; 0) et hn : Ln (M; E ) ! Ln (Rd ; 0)
les appli ations induites au niveau des espa es d'ar s.
Le determinant ja h de la matri e ja obienne Ja h de h peut ^etre al ule en
hoisissant des oordonnees lo ales sur M . Il ne depend pas du hoix de es oordonnees, et don l'ordre ordt ja (t) h en t du determinant de la matri e ja obienne
evalue sur un ar (t) de L(M; E ) est bien de ni.
La formule de hangement de variables de Kontsevi h s'enon e alors omme suit
dans notre adre.
Theoreme 2.2. ([21, 6℄) Soit A L(R d ; 0) un sous-ensemble stable, et supposons que ordt ja h soit borne sur h 1 (A). Alors
(A) =
Z
h 1 (A)
L
ordt ja h d:
Remarque 2.3. Ce resultat est utilise par la suite ave h qui est une resolution
des singularites pour une fon tion. Il permet de traduire des informations, peut-^etre
ompliquees, sur les singularites de la fon tion en termes de donnees, plus simples
voire ombinatoires, sur la resolution
Nous allons demontrer e resultat en plusieurs etapes. Le lemme suivant onstitue
un point essentiel de la demonstration.
Posons
e = f 2 L(M; E ); ordt ja h (t) = eg;
ou e est un entier stri tement positif, et e;n = n (e ).
Lemme 2.4. Soient e 1 et n 2e des entiers. Alors hn de nit une bration
triviale par mor eaux au dessus de hn (e;n ) de bre Re .
On aura besoin du resultat intermediaire suivant, appli ation de la formule de
Taylor.
Lemme 2.5. Soient e 1 et n 2e des entiers.
28
2. INTE GRATION MOTIVIQUE EN GE OME TRIE RE ELLE
(1) Soient 1 ; 2 2 L(M; E ). Si 1 2 e et h( 1 )
n e+1 .
2 2 e et 1 2 mod t
(2) Pour 2 e;n , on a hn1 hn ( ) ' Re .
h( 2 ) mod tn+1, alors
Demonstration. Il suÆt de demontrer le resultat dans le as parti ulier ou M est un
espa e aÆne isomorphe a Rd , et on montre alors que
8 2 e; 8v 2 R[[t℄℄; 9u 2 R[[t℄℄d : h( + tn e+1u) = h( ) + tn+1v;
en exhibant u de maniere re ursive.
En e et, prenons 2 e, et notons M (t) la matri e M (t) = te (Ja h) 1 . C'est
une matri e a oeÆ ients dans R[[t℄℄ ar le determinant de Ja h est d'ordre e, et
la formule de Taylor appliquee a h( + tn e+1u) implique
u = M (t)v + tM (t)O(u2 ):
En e et
h( + tn e+1 u) = h( ) + tn e+1 Ja h(u) + t2(n e+1) O(u2 );
et don en remplaant h( + tn e+1 u) par h( ) + tn+1 v et en utilisant l'inegalite
n 2e 0, il vient
t e Ja h(u) = v + tO(u2 );
et il ne reste plus qu'a multiplier a gau he par la matri e M (t).
Or la formule u = M (t)v + tM (t)O(u2 ) permet de onstruire u de faon re ursive
a partir de v.
Maintenant, pour al uler la bre hn1 hn ( ) pour 2 e;n, on remarque que
les al uls pre edents permettent d'aÆrmer que
hn1 hn ( ) = f (t) + tn e+1u mod tn+1 ; u 2 R[[t℄℄d ; Ja h( )(u) 0 mod teg:
Or la matri e Ja h est equivalente dans l'ensemble des matri es a oeÆ ients dans
R [[t℄℄ a une matri e diagonale de diagonale (te1 ;: : : ; te ) par elimination de Gauss,
ave de plus e1 + + ed = e. Ainsi hn1 hn ( ) appara^t omme un espa e aÆne
de dimension e.
d
Remarque 2.6.
(1) Le lemme 2.5 implique en parti ulier que l'ensemble h (e ) est stable ( hoisir n 2e pour le voir).
(2) Il ressort de la demonstration que, de plus, l'ensemble e est stable (en
hoisissant i i n e). En e et, si 2 e, la formule de Taylor implique
que
h ( (t) + ute+1 ) h( (t)) mod t2e+1 :
Le lemme 2.5 aÆrme alors que (t) + ute+1 2 e.
Demonstration du lemme 2.4. On onstruit une se tion S de hn de la faon
suivante. Soit s : Ln (Rd ; 0) ! L(Rd ; 0) la se tion de n qui pousse un ar tronque a
l'ordre n sur le polyn^ome orrespondant. Prenons 2 hn (e;n ). L'image re iproque
de s( ) par h existe ar la ourbe s( ) n'est pas in luse totalement dans le lieu
singulier de l'appli ation birationnelle h d'apres le lemme 2.5 (rappelons que e 1).
3. FORMULE DE CHANGEMENT DE VARIABLES
29
On de nit alors S ( ) omme la proje tion, par n, de ette image re iproque, par
h , de dans Ln(M; E ) :
S : hn (e;n )
L( d ; 0)
s
/
R
h 1
/
L(M; E )
n
/
Ln(M; E ) :
Notons que h 1 n'est evidemment pas de nie de maniere globale sur L(Rd ; 0),
neanmoins l'appli ation S est un morphisme par mor eaux, i.e. il existe une partition nie de hn (e;n) en sous-varietes algebriques fermees telle que sur haque
strate la se tion S est un morphisme. Ainsi hn est une bration triviale en restri tion a ha une de es strates, de bre Re d'apres le lemme 2.5.
Demonstration du theoreme 2.2. Notons que A est une reunion nie d'ensembles
de la forme A \ h (e ) ar l'image de ordt ja h est nie par hypothese. Il suÆt
don , par additivite, de demontrer le theoreme pour un tel ensemble. Soit e 1,
et hoisissons n 2e. On se retrouve dans la situation du diagramme ommutatif
suivant :
h
/ h (e ) \ A
e \ h 1 (A)
n
n
n(e \ h 1 (A)) h / n(h (e ) \ A)
ave h bije tive et hn surje tive.
Or
[hn1 n(h (e ) \ A) ℄ = Le [n (h (e ) \ A)℄
d'apres le lemme 2.4, et don (A \ h (e )), qui vaut L (n+1)d [n (A \ h (e ))℄
par de nition, est egal a
(n+1)d L e [h 1 (h ( ) \ A)℄:
L
n n e
Toujours par de nition de la mesure, on obtient :
(A \ h (e )) = L e (h 1 (A) \ e);
qui est nalement egale a
n
Z
par de nition de l'integrale.
h 1 (A)\e
L
ordt ja h d
CHAPITRE 3
Resultats prin ipaux
L'objet initial de la these etait d'arriver a utiliser l'integration motivique en
geometrie algebrique reelle, dans le but de onstruire des invariants pour des singularites. Cette theorie de l'integration motivique ne essite la onnaissan e de ara teristiques d'Euler generalisees pour les varietes algebriques reelles, 'est-a-dire
d'invariants additifs et multipli atifs qui permettent de onstruire des mesures alulables sur les espa es des ar s. Or, si on dispose en geometrie algebrique omplexe
de ara teristiques d'Euler generalisees utilisables en integration motivique, e n'est
pas le as en geometrie algebrique reelle ou seule la ara teristique d'Euler a supports
ompa ts est onnue. Elle ne laisse ependant pas beau oup d'espoir pour l'utilisation en integration motivique ar elle identi e des espa es de dimension di erente ;
par exemple (P1 ) = 0.
Rappelons qu'en geometrie algebrique omplexe les stru tures de Hodge mixtes
fournissent des ara teristiques d'Euler generalisees pertinentes ; il n'appara^t pas
lairement omment l'on pourrait en deduire des ara teristiques d'Euler generalisees
reelles, m^eme en passant par la omplexi ation du fait qu'une variete algebrique
reelle admet plusieurs omplexi ations qui sont seulement birationnelles.
Cependant C. M .Crory et A. Parusinski [27℄ ont de ni, utilisant un resultat de
F. Bittner [3℄, le polyn^ome de Poin are virtuel qui est un exemple de ara teristiques
d'Euler generalisees utilisable en integration motivique. Ce polyn^ome , onstruit
a partir des nombres de Betti virtuels des varietes algebriques reelles, est a oeÆients entiers relatifs et il veri e notamment deg (X ) = dim X . Il onserve don la
dimension.
Dans ette these, on generalise la de nition des nombres de Betti virtuels, et du
polyn^ome de Poin are virtuel, aux ensembles symetriques par ar s (qui ontiennent
les varietes algebriques reelles), et on montre de plus que es nombres de Betti
virtuels sont invariants au sens de Nash. On applique alors l'integration motivique,
ave la mesure provenant du polyn^ome de Poin are virtuel, pour etudier les germes
de fon tions analytiques reelles. On onstruit en parti ulier des fon tions z^eta que
l'on prouve ^etre des invariants pour un as parti ulier de la relation d'equivalen e
analytique apres e latements, appelee l'equivalen e de Nash apres e latements.
1. Nombres de Betti virtuels
1.1. Ensembles symetriques par ar s. Les ensembles symetriques par ar s
ont ete introduits en 1988 par K. Kurdyka [23℄ dans le but d'etudier les " omposantes rigides" des varietes algebriques reelles. La de nition que l'on donne i i est
legerement di erente de l'originale, elle permet en parti ulier de onsiderer des ensembles symetriques par ar s non fermes. Elle est due a A. Parusinski [28℄, et elle
orrespond en quelque sorte aux ensembles symetriques par ar s onstru tibles et
proje tifs de K. Kurdyka.
31
3. RE SULTATS PRINCIPAUX
32
Definition 3.1. Un ensemble semi-algebrique A Pn est symetrique par ar s
si la ondition suivante est veri ee : pour tout ar analytique :℄
que (℄ 1; 0[) A, il existe > 0 tel que (℄0; [) A.
1; 1[
!
P
n
tel
Remarque 3.2. Les ensembles symetriques par ar s se omportent bien par
rapport aux operations booleennes et forment une ategorie onstru tible [28℄. En
parti ulier, ils sont stables par unions et interse tions nies, ainsi que par passage
au omplementaire.
Exemple 3.3.
(1) Les varietes algebriques reelles, et aussi les omposantes onnexes de varietes
algebriques reelles ompa tes sont des exemples elementaires d'ensembles
symetriques par ar s (par le prin ipe des zeros isoles).
(2) La nappe de dimension 2 du parapluie de Whitney est un exemple d'ensemble symetrique par ar s qui n'est pas une variete algebrique ( f. exemple
4.2.2).
On donne maintenant quelques de nitions qui sont des extensions naturelles des
de nitions pour les varietes algebriques reelles.
On de nit ainsi les ensembles symetriques par ar s irredu tibles omme etant
les ensembles qui ne peuvent s'e rire omme la reunion de deux sous-ensembles
symetriques par ar s fermes et propres. Notons qu'il existe une de omposition d'un
ensemble symetrique par ar s en reunion nie d'ensembles symetriques par ar s
irredu tibles.
On de nit de plus la partie lisse d'un ensemble symetrique par ar s omme
etant l'interse tion de et ensemble ave la partie lisse, au sens algebrique, de son
adheren e de Zariski. Notons egalement que la dimension d'un ensemble symetrique
par ar s est sa dimension en tant qu'ensemble semi-algebrique, et elle- i on ide
ave la dimension algebrique de son adheren e de Zariski. En n, on dit que deux
ensembles symetriques par ar s sont isomorphes si leur adheren e de Zariski sont
birationnelles par un isomorphisme birationnel ontenant les ensembles symetriques
par ar s de depart dans son support.
Les resultats suivants ont pour inter^et d'illustrer les bonnes proprietes des ensembles symetriques par ar s.
Proposition 3.4.
(1) (A. Parusinski, [28℄) Tout ensemble symetrique par ar s A admet un plus
petit ensemble symetrique par ar s ferme le ontenant. On appelle adheren e
symetrique par ar s et ensemble, note AAS .
(2) Si A est symetrique par ar s, alors dim AAS n A < dim A.
(3) Les ensembles symetriques par ar s lisses et ompa tes sont isomorphes a
des reunions de omposantes onnexes de varietes algebriques reelles lisses
et ompa tes.
A l'aide de la resolution des singularites pour les varietes algebriques reelles, on
peut aussi mimer une resolution des singularites pour les ensembles symetriques par
ar s.
1. NOMBRES DE BETTI VIRTUELS
33
Proposition 3.5. (K. Kurdyka, [23℄) Pour un ensemble symetrique par ar s
irredu tible A, notons X une variete algebrique reelle de m^eme dimension le ontenant. Soit : Xe ! X une resolution des singularites de X . Alors il existe une
unique omposante onnexe Ae de Xe telle que (Ae) = Reg(A).
De maniere plus elementaire, on peut aussi parler d'e latement d'un ensemble
symetrique par ar s lisse et ompa t le long d'un sous-ensemble symetrique par ar s
lisse et ompa t en termes de omposantes onnexes de varietes algebriques reelles
lisses et ompa tes. Ce resultat merite d'^etre mis en eviden e en vue du theoreme
3.8 i-dessous.
On note par BlY (X ) l'e latement de X le long de Y , Bl venant de l'anglais
\blowing-up".
Proposition 3.6. Soient Y X des varietes algebriques reelles lisses et ompa tes, ave dim Y < dim X , et soit A une omposante onnexe de X .
Alors l'e latement : BlY (X ) ! X de X par rapport a Y est surje tif, et
1 (A) est une omposante onnexe de BlY (X ).
1.2. Polyn^ome de Poin are virtuel. Pour utiliser l'integration motivique,
on a besoin de onna^tre des invariants additifs et multipli atifs des varietes.
Definition 3.7. Soit une appli ation de nie sur les ensembles symetriques
par ar s et a valeurs dans un anneau ommutatif. Si veri e
{ (A) = (B ) ou A et B sont deux ensembles symetriques par ar s isomorphes,
{ (A) = (B ) + (A n B ) ou B est un sous-ensemble symetrique par ar s de A
ferme dans A,
alors est un invariant additif des ensembles symetriques par ar s. Si de plus est
multipli ative, 'est-a-dire (A B ) = (A) (B ) pour des ensembles symetriques
par ar s A et B , alors est une ara teristique d'Euler generalisee des ensembles
symetriques par ar s.
Suivant le resultat de F. Bittner [3℄, qui traite le as des varietes algebriques sur
un orps de ara teristique nulle, on peut donner des onditions sur un invariant,
a isomorphisme algebrique pres, des omposantes onnexes de varietes algebriques
reelles lisses et ompa tes pour qu'il puisse s'etendre en un invariant des ensembles
symetriques par ar s. Les arguments les de la demonstration sont la resolution
des singularites ( f. hapitre 1, se tion 3) et le theoreme de fa torisation faible des
appli ations birationnelles ( f. hapitre 1, se tion 4).
Notons que e resultat, et le orollaire qu'on en deduit, sont des generalisations
aux ensembles symetriques par ar s de resultats de C. M Crory et A. Parusinski
[27℄ on ernant les varietes algebriques reelles.
Theoreme 3.8. Soit une appli ation de nie sur les omposantes onnexes des
varietes algebriques reelles lisses et ompa tes et a valeurs dans un anneau ommutatif. Considerons les onditions suivantes :
{ (;) = 0,
{ (A) = (B ) ou A et B sont des omposantes onnexes de varietes algebriques
reelles lisses et ompa tes isomorphes en tant qu'ensembles symetriques par
ar s,
{ sous les hypotheses et ave les notations de la proposition 3.6,
1 (A) 1 (A) \ 1 (A \ Y ) = (A) (A \ Y );
34
3. RE SULTATS PRINCIPAUX
{ (A B ) = (A) (B ) si A et B sont des omposantes onnexes de varietes
algebriques reelles lisses et ompa tes.
Si satisfait aux trois premieres onditions, alors s'etend en un invariant additif
des ensembles symetriques par ar s. Si de plus veri e la derniere ondition, est
une ara teristique d'Euler generalisee des ensembles symetriques par ar s.
Comme orollaire de e theoreme, on obtient l'existen e des nombres de Betti
virtuels pour les ensembles symetriques par ar s, qui sont des invariants additifs,
et l'existen e du polyn^ome de Poin are virtuel, qui est une ara teristique d'Euler
generalisee des ensembles symetriques par ar s.
Corollaire 3.9. Il existe un invariant additif i des ensembles symetriques
par ar s, appele i-eme nombre de Betti virtuel, qui on ide ave le i-eme nombre
de Betti bi a oeÆ ients dans 2ZZ sur les ensembles symetriques par ar s lisses et
ompa ts. En d'autres termes, si A est une reunion de omposantes onnexes de
varietes algebriques reelles lisses
P et ompa tes, alors i (A) = bi (A).
De plus l'appli ation = i0 i ui , de nie sur les ensembles symetriques par
ar s et a valeurs dans les polyn^omes a oeÆ ients dans Z, est une ara teristique
d'Euler generalisee des ensembles symetriques par ar s, appele polyn^ome de Poin are
virtuel.
Exemple 3.10.
(1) Par onstru tion, (P1 ) = 1 + u, et (point) = 1, don par additivite
(R ) = (P1 ) (point) = u.
(2) Une parabole est isomorphe a R don son polyn^ome de Poin are virtuel est
u.
Remarque 3.11.
(1) La demonstration du orollaire 3.9 est de nature topologique. On utilise en
e et la dualite de Poin are (d'ou les oeÆ ients 2ZZ pour avoir une orientation), la formule de proje tion liant le up et le ap produit, des suites
exa tes longues de paires, ainsi que la formule de Kunneth pour la multipli ativite du polyn^ome de Poin are virtuel.
(2) Les nombres de Betti virtuel ne sont pas des invariants topologiques ( f.
exemple 5.11.3). On montre i-dessous (theoreme 3.14) qu'ils sont par ontre
des invariants au sens de Nash.
On donne maintenant un exemple de al ul du polyn^ome de Poin are virtuel. Le
al ul permet de omprendre la onstru tion, dans la demonstration du theoreme
3.8, d'une ara teristique d'Euler generalisee a partir d'un invariant des ensembles
lisses et ompa ts. On utilise pour ela la resolution des singularites. Le theoreme
de fa torisation faible permet lui de montrer que ette ontru tion ne depend pas
des hoix.
Exemple 3.12. On al ule le polyn^ome de Poin are virtuel du parapluie de
Whitney d'equation zx2 = y2 . Tout d'abord, on ommen e par resoudre les singularites du parapluie de Whitney. Il suÆt pour ela de l'e later le long du man he
(l'axe des z), et on trouve alors un ylindre de base une parabole, ave pour diviseur
ex eptionnel une parabole. Notons W le parapluie, D le man he (qui est une droite),
et P la parabole.
2. EQUIVALENCE DE NASH APRE S E CLATEMENTS
35
Alors, par additivite,
(W ) = (W n D) + (D)
et par invarian e par rapport a l'isomorphisme deduit de l'e latement, on obtient
(W n D) =
Mais
P (R n point) :
P (R n point) = (P ) (R n point) par multipli ativite, d'ou nalement
(W ) = u(u 1) + u = u2 :
Bien que l'on de nisse les nombres de Betti virtuels de maniere algebrique, on
montre aussi qu'ils onstituent un invariant des ensembles symetriques par ar s en
un sens plus analytique, et plus pre isement Nash. On de nit i-dessous une notion
d'isomorphisme de Nash entre ensembles symetriques par ar s, et on enon e ensuite
le theoreme d'invarian e orrespondant.
Definition 3.13. Soient A1 et A2 des ensembles symetriques par ar s. S'il existe
deux varietes analytiques ompa tes V1 et V2 ontenant A1 et A2 respe tivement et
: V1 ! V2 un isomorphisme analytique tels que
{ V1 ; V2 et sont de plus semi-algebriques,
{ (A1 ) = A2 ,
alors A1 et A2 sont dits isomorphes au sens de Nash.
Theoreme 3.14. Les nombres de Betti virtuels sont invariants par isomorphisme
de Nash entre ensembles symetriques par ar s.
Remarque 3.15. Ce resultat est important pour l'appli ation a l'equivalen e de
Nash apres e latements.
2. Equivalen e de Nash apres e latements
L'etude des germes de fon tions analytiques reelles est un sujet diÆ ile, en parti ulier dans le hoix d'une bonne relation d'equivalen e entre germes :
\deux germes de fon tions analytiques reelles f; g : (Rd ; 0) ! (R; 0) sont
equivalents au sens () si et seulement s'il existe h : (Rd ; 0) ! (Rd ; 0) veri ant
() tel que f Æ h = g"
ave () qui peut ^etre rempla e par C 0; C 1 , bi-lips hitz, et ...
Si l'equivalen e topologique (i.e. h est un homeomorphisme) est trop grossiere,
l'equivalen e au sens C 1 est quant a elle trop ne omme le montre le fameux exemple
de la famille de Whitney qui possede une in nite de lasses d'equivalen e.
T.-C. Kuo [22℄ a remarque qu'apres e latement, ette famille est analytiquement
triviale. Cette onstatation l'a amene a de nir, en 1985, l'equivalen e analytique
apres e latements :
-deux germes de fon tions analytiques reelles f; g : (Rd ; 0) ! (R; 0) sont
equivalents analytiquement apres e latements s'il existe un homeomorphisme
lo
al : (Rd ; 0) ! (Rd ; 0), deux modi ations reelles f : Mf ; f 1 (0) ! (Rd ; 0)
et g : Mg ; g 1 (0) ! (Rd ; 0) ainsi qu'un isomorphisme analytique au niveau des
36
espa es modi es : Mf ; f
3. RE SULTATS PRINCIPAUX
1 (0)
M ; 1 (0) tels
!
Mf ; f 1 (0)
g
g
que le diagramme
Mg ; g 1 (0)
/
f
g
(Rd ; 0) M
MMM
MMM
f MMM&
(R; 0)
/ (R d ; 0)
qq
qqq
q
q
g
qx qq
soit ommutatif.
Pour une de nition pre ise de modi ation reelle, nous renvoyons a la se tion 1
du hapitre 7.
En fait, ela n'est pas la de nition originale de l'equivalen e analytique apres
e latements, mais une ondition ne essaire et suÆsante enon ee par T.-C. Kuo
dans [22℄. C'est sous ette forme ependant qu'elle nous sera utile. En e et, nous
n'etudions pas dire tement ette relation d'equivalen e, mais un as parti ulier pour
lequel nous ajoutons des hypotheses d'algebri ite. Notons qu'on ne sait pas si ette
nouvelle relation, appelee equivalen e de Nash apres e latements, est une relation
d'equivalen e. Ce n'est ependant pas trop g^enant dans le sens ou d'une part, 'est
la relation d'equivalen e analytique apres e latements qui est la relation que l'on
souhaite veritablement omprendre, et d'autre part, pour les exemples onnus et
etudies, les hypotheses supplementaires de l'equivalen e de Nash apres e latements
sont generalement satisfaites.
Definition 3.16. Deux germes f; g : (R d ; 0) ! (R; 0) de fon tions de Nash sont
equivalents au sens de Nash apres e latements s'il existe un homeomorphismelo al
: (Rd ; 0) ! (Rd ; 0), des morphismes propres birationnels f : Mf ; f 1 (0) !
(Rd ; 0) et g : Mg ; g 1 (0) ! (Rd ; 0) tels que f Æ f et ja f (respe tivement g Æ g
et ja g ) sont a roisements normaux
et un isomorphisme
de Nash (i.e. analytique
et semi-algebrique) : Mf ; f 1 (0) ! Mg ; g 1 (0) tels que
{ le diagramme
Mf ; f 1 (0)
/
Mg ; g 1 (0)
f
g
(Rd ; 0) M
MMM
MMM
f MMM&
(R; 0)
/ (R d ; 0)
q
q
qqq
q
q
g
qx qq
soit ommutatif,
{ respe te les multipli ites des determinants ja obiens de f et g .
Remarque 3.17.
(1) Les di eren es ave l'equivalen e analytique apres e latements, en plus du
fait qu'on se restreigne aux germes de fon tions de Nash, sont don que
2. EQUIVALENCE DE NASH APRE S E CLATEMENTS
37
les modi ations reelles sont i i algebriques, leur determinant ja obien est
a roisements normaux ave le lieu des zeros des fon tions, l'isomorphisme
analytique devient un isomorphisme de Nash, et de plus et isomorphisme
de Nash doit respe ter les determinants ja obiens des modi ations.
(2) L'inter^et d'introduire une telle relation est qu'ave es hypotheses, les analogues en reel des fon tions z^eta motiviques de Denef & Loeser, al ulees
ave le polyn^ome de Poin are virtuel , deviennent des invariants pour
ette relation (voir la se tion suivante sur les fon tions z^eta ; e resultat est
naturel en vertu de la formule de Denef & Loeser, f. proposition 3.22). On
obtient don par e biais un invariant pour un type de relations peu aise
a etudier (pour l'equivalen e analytique apres e latements, seuls les invariants de Fukui [17℄, et les fon tions z^eta de C. M Crory et A. Parusinski
[27℄ etaient onnus).
On ne sait pas ependant si es fon tions z^eta sont aussi des invariants
pour l'equivalen e analytique apres e latements.
(3) Notons que l'hypothese re lamant que doit respe ter les determinants
ja obiens des modi ations provient de l'utilisation de la formule de Denef
& Loeser pour la demonstration de l'invarian e des fon tions z^eta.
Cependant ette hypothese n'est pas seulement ne essaire pour ette
demonstration, elle a aussi un sens propre dans le adre de l'equivalen e
analytique apres e latements. Notamment, T. Fukui, T.-C. Kuo et L. Paunes u ( f. [11℄) ont propose re emment de modi er quelque peu la de nition
de ette relation d'equivalen e en demandant que l'isomorphisme analytique
soit egalement un isomorphisme entre les lieux ritiques des modi ations
reelles.
Pour l'equivalen e analytique apres e latements, une te hnique eÆ a e permet
de donner des resultats de trivialite, et par e biais de prouver que deux germes
donnes sont equivalents en les plaant dans une famille : l'integration le long d'un
hamp de ve teurs bien hoisi ( f. [22℄). Cependant on ne peut re ourir a une telle
pratique dans le adre de l'equivalen e de Nash, ar on quitte e monde de Nash (typiquement, l'integration d'une equation di erentielle lineaire a oeÆ ients onstants
fait appara^tre des exponentiels). Neanmoins, gr^a e a un resultat de trivialisation
pour les ensembles de Nash d^u a T. Fukui, S. Koike et M. Shiota [12℄, ouple a
des approximations des niveaux des germes de fon tions par des proje tions dans
des voisinages tubulaires, on peut ontourner ette diÆ ulte et produire le theoreme
suivant.
Theoreme 3.18. Soit F : (R d ; 0) P
! (R; 0) une appli ation de type Nash,
ave P un ensemble de Nash di eomorphe a un simplexe ouvert d'un espa e eu lidien. On suppose que les appli ations F (:; p) ont une singularite isolee a l'origine
pour tout p 2 P . On suppose de plus que F admet une resolution algebrique des
singularites, 'est-a-dire qu'apres un nombre ni d'e latements algebriques, F est a
roisements normaux.
Alors la famille de germes de fon tions analytiques reelles indexee par P est
onstituee d'un nombre ni de lasses pour l'equivalen e de Nash apres e latements.
De plus, ha une de es lasses est triviale pour l'equivalen e analytique apres
e latements.
38
3. RE SULTATS PRINCIPAUX
La demonstration de e resultat ( f. hapitre 7, se tion 3) permet egalement
d'enon er le orollaire suivant, qui, dans le adre de l'equivalen e analytique apres
e latements, est un resultat lassique.
Corollaire 3.19. Soit ft : (R d ; 0)
! ( ; 0), t 2 I , ave I
un intervalle de
R , une famille de polyn^
omes homogenes par poids (de m^eme poids) parametrisee
de maniere Nash. Alors ette famille fft gt2I est onstituee d'une seule lasse pour
l'equivalen e de Nash apres e latements.
R
Exemple 3.20. Cet exemple est utile dans la se tion 2 du hapitre 7 pour
determiner les lasses des polyn^omes de Brieskorn a deux variables. On montre
i i que x3 + y6 et x3 y6 sont equivalents au sens de Nash apres e latements. Pour
ela onsiderons la famille de polyn^omes homogenes par poids de poids (2; 1) donnee
par
2t 6
1 t2 4
xy +
y ; t 2 [ 1; 1℄:
ft (x; y) = x3 + 3
1 + t2
1 + t2
Pour tout t 2 [ 1; 1℄ le polyn^ome ft a une singularite isolee en (0; 0), don le orollaire pre edent implique que f 1 et f1 , 'est-a-dire x3 y6 et x3 + y6 , sont equivalents
au sens de l'equivalen e de Nash apres e latements.
3. Fon tions z^eta
Dans [7℄, J. Denef et F. Loeser de nissent des fon tions z^eta motiviques pour
un polyn^ome. Ces fon tions z^eta ontiennent des informations sur les singularites,
notamment en lien ave la bre de Milnor. Dans e travail, on de nit de maniere
analogue des fon tions z^eta asso iees a un germe de fon tions analytiques reelles. On
montre, omme resultat prin ipal, que es fon tions z^eta sont des invariants pour
l'equivalen e de Nash apres e latements.
Soit f : (Rd ; 0) ! (R; 0) un germe de fon tions analytiques reelles. Pour les
notations de l'espa e des ar s, on se refere au hapitre 2. Soit n le sous-ensemble
de Ln (Rd ; 0) de ni par
n = f 2 Ln (Rd ; 0); ord f Æ = ng:
Cet ensemble est un onstru tible des varietes algebriques reelles, et don a e titre
possede une valeur par le polyn^ome de Poin are virtuel .
On de nit alors la fon tion z^eta nave de f omme etant la serie formelle
Zf (T ) =
X
n1
(n )u
T
nd n
a oeÆ ients dans Z[u; u 1℄.
De faon analogue, on de nit des fon tions z^eta ave signe. Posons
+n = f 2 Ln (Rd ; 0); f Æ (t) = tn + g
et
n = f 2 Ln (Rd ; 0); f Æ (t) = tn + g:
Les fon tions z^eta de f ave signe sont alors de nies par
Zf+(T ) =
X
n1
(+n )u
T
nd n
3. FONCTIONS ZE^ TA
et
Zf (T ) =
X
n1
(n )u
39
T :
nd n
Il s'agit egalement de series formelles a oeÆ ients dans Z[u; u 1℄.
Remarque 3.21. Le terme u
qui appara^t dans la de nition des fon tions
z^eta est juste un terme orre tif, qui permet de simpli er les al uls par la suite.
Il peut ^etre omplique de al uler de telles series sur des exemples. Les formules
suivantes, appelees formules de Denef & Loeser, permettent d'e rire les fon tions
z^eta d'un germe de fon tions de Nash f de maniere elementaire en fon tion d'une
resolution des singularites de f . Elles onstituent de plus l'ingredient essentiel de la
demonstration de l'invarian e des fon tions z^eta pour l'equivalen e de Nash apres
e latements.
Notons que es formules sont une onsequen e de la formule de hangement de
variables en integration motivique (voir le hapitre 2).
On enon e d'abord la formule de Denef & Loeser pour la fon tion z^eta nave.
Proposition 3.22
. Soit f : (R d ; 0) ! (R; 0) un germe de fon tions de Nash.
Soit : M; 1 (0) ! (Rd ; 0) une appli ation propre birationnelle qui est un
isomorphisme en dehors du lieu des zeros de f . Supposons que f Æ et le determinant
ja obien ja de sont a roisements normaux simultanement.
Notons par [j 2J Ej la de omposition en omposantes irredu tibles de (f Æ) 1 (0),
et soit K J tel que 1 (0) = [k2K Ek . Notons en ore par EI0 , pour I J ,
l'ensemble (\i2I Ei ) n ([j 2J nI Ej ).
Posons en n Ni = multE f Æ et i = 1 + multE ja . Alors
X
Y u TN
:
Zf (T ) = (u 1)jI j EI0 \ 1 (0)
1 u TN
i2I
I 6=;
nd
i
i
i
i
i
Exemple 3.23. Soit f : (R 2 ; 0)
i
! ( ; 0) de nie par f (x; y) = x2 + y2 . On
R
obtient une modi ation de f satisfaisant aux hypotheses de la formule de Denef &
Loeser en e latant l'origine uniquement. Dans e as (f Æ ) 1 (0) est simplement le
diviseur ex eptionnel P1 de l'e latement, et N = 2; = 1 + 1. Ainsi
u 2T 2
Zf (T ) = (u 1)(u + 1)
:
1 u 2T 2
Notons que la fon tion z^eta de S. Koike et A. Parusinski est nulle sur et exemple
du fait que (P1 ) = 0.
Pour les fon tions z^eta ave signe, on introduit au prealable un re ouvrement
℄
0
;
EI , ou designe soit +, soit , des strates anoniques EI0 de Q
la faon suivante :
soit U un ouvert aÆne de M sur lequel f Æ est de la forme u i2I yiN , ou u est
une fon tion qui ne s'annule pas sur U . On note par RU l'ensemble
1
g
RU = f(y; t) 2 (EI0 \ U ) R; tm = u(y)
ou mI = pg d(Ni )i2I . Ces ensembles RU se re ollent le long des EI0 \ U pour donner
℄
0;
le re ouvrement E
I .
i
I
3. RE SULTATS PRINCIPAUX
40
Proposition 3.24. Sous les hypotheses et notations de la proposition pre edente,
les fon tions z^eta ave signe s'expriment en termes d'une modi ation ainsi :
Y u TN
X
℄
0;
1 (0)
\
Zf (T ) = (u 1)jI j 1 E
I
TN :
1
u
i2I
I=
6 ;
Exemple 3.25. La fon tion z^eta negative de f (x; y) = x2 + y 2 est evidemment
nulle, et la fon tion z^eta positive se al ule omme la fon tion z^eta nave i-dessus.
On obtient
u 2T 2
:
Zf+(T ) = (u + 1)
1 u 2T 2
Notons que sur et exemple Zf = (u 1)Zf+ .
i
i
i
Remarque 3.26.
i
(1) Contrairement a l'exemple de x2 + y2 , et ontrairement aussi au as des
fon tions z^eta de S. Koike et A. Parusinski, on n'a pas, en general, de
relation dire te entre Zf ; Zf+ et Zf .
Rappelons que la fon tion z^eta nave de S. Koike et A. Parusinski est
la somme de leurs fon tions z^eta ave signe. On ne peut ependant utiliser
un argument similaire au leur du fait que ℄ 1; 0[ et ℄0; +1[ ne sont pas
des ensembles symetriques par ar s !
En e et, leurs fon tions zeta ave signe ne sont pas de nies omme i i
a partir des ar s tels que f Æ (t) = tn + pour un ertain n, mais
par les ar s tels que f Æ (t) = btn + ave b > 0 ou b < 0. Or, si la
ara teristique d'Euler a supports ompa ts satisfait
(℄ 1; 0[) + (℄0; +1[) = (R );
une telle relation n'est m^eme pas de nie ave le polyn^ome de Poin are
virtuel.
(2) Il ne semble pas evident non plus d'obtenir pour nos fon tions z^eta une
formule de Thom-Sebastiani, omme ela existe pour les fon tions z^eta de
J. Denef, F. Loeser et de S. Koike, A. Parusinski. On obtient ependant
une telle formule dans le as parti ulier ou les fon tions en jeu sont de
signe identique (voir la proposition 6.9).
Revenons maintenant a l'equivalen e de Nash apres e latements. Pour deux
germes de fon tions de Nash equivalents en e sens, il existe par de nition un diagramme ommutatif qui relie des modi ations asso iees aux germes. On peut alors
al uler les fon tions z^eta de es germes par la formule de Denef & Loeser. Or, es
modi ations sont liees par un isomorphisme de Nash, et on sait que le polyn^ome
de Poin are virtuel est invariant par de tels isomorphismes. On obtient don le
resultat suivant qui fait le lien entre l'equivalen e de Nash apres e latements et
les fon tions z^eta, et qui onstitue une appli ation de l'integration motivique en
geometrie reelle :
Theoreme 3.27. La fon tion z^eta nave et les fon tions z^eta ave signe sont des
invariants de l'equivalen e de Nash apres e latements.
Exemple 3.28. Les polyn^omes de Brieskorn a deux variables sont les polyn^omes
de la forme xp yq , ave p; q 2 N . La lassi ation de es polyn^omes par rapport
3. FONCTIONS ZE^ TA
41
a la relation d'equivalen e analytique apres e latements est un resultat re ent [20℄
qui ne essite l'utilisation des invariants de Fukui ainsi que des fon tions z^eta de
S. Koike et A. Parusinski. Ces fon tions analytiques n'etant que des polyn^omes,
il est naturel de s'interesser a leur lasse au sens de l'equivalen e de Nash apres
e latements. On montre dans la se tion 2 du hapitre 7 qu'il suÆt d'utiliser nos
fon tions z^eta pour distinguer es lasses, et que, de plus, es lasses on ident ave
elles de l'equivalen e analytique apres e latements.
Deuxi
eme partie
Generalized Euler hara teristi of
ar -symmetri sets
CHAPITRE 4
Ar -symmetri sets
Ar -symmetri sets have been introdu ed by K. Kurdyka [23℄ in 1988 in order
to study \rigid omponents" of real algebrai varieties. These sets enabled him to
prove the so-salled Borel theorem, whi h states that an inje tive endomorphism of a
real algebrai variety is surje tive [24℄(A. Borel proved this theorem in the smooth
ase).
The ategory of ar -symmetri sets ontains the real algebrai varieties and, in
some sense, this ategory has a better behaviour that the one of real algebrai varieties, similar to omplex algebrai varieties. As an example, a losed and irredu ible
ar -symmetri set is onne ted whereas a losed and irredu ible real algebrai variety
may have as onne ted omponents as one wants !
In this hapter, we introdu e ar -symmetri sets, with a slightly di erent de nition that the one of K. Kurdyka, and we state several basi properties of them in
relation with losure, dimension and irredu ibility. Then, we study in more details
the nonsingular ar -symmetri sets and espe ially the resolution of singularities for
ar -symmetri sets. In parti ular, we emphasize the relations between ompa t nonsingular ar -symmetri sets and onne ted omponents of ompa t nonsingular real
algebrai varieties. These relations will be usefull in the next hapter when studying
generalized Euler hara teristi s of ar -symmetri sets.
1. Ar -symmetri sets and losure
We x a ompa ti ation of Rn , for instan e Rn Pn .
Definition 4.1. Let A Pn be a semi-algebrai set. We say that A is ar symmetri if, for every real analyti ar :℄ 1; 1[ ! Pn su h that (℄ 1; 0[) A,
there exists > 0 su h that (℄0; [) A.
Remark that an ar -symmetri set needs not to be an analyti variety ( f. [23℄,
example 1.2).
This de nition is the one of A. Parusinski [28℄. Note that a losed ar -symmetri
set is ne essarily ompa t. This de nition di ers from the one of K. Kurdyka [23℄
who only onsiders losed ar -symmetri sets in Rn . One an think about our ar symmetri sets as the proje tive onstru tible ar -symmetri sets of K. Kurdyka.
Example 4.2.
(1) A real algebrai variety is an ar -symmetri set. A onne ted omponent
of a ompa t real algebrai variety is also an ar -symmetri set.
(2) The 2-dimensional sheet N of the Whitney umbrella W of equation zx2 =
y2 (see gure 1) is ar -symmetri .
Indeed, if :℄ 1; 1[ ! P3 is an analyti ar whi h satis es :
{ (℄ 1; 0[) N ,
45
46
4. ARC-SYMMETRIC SETS
Fig. 1. The Whitney umbrella.
{ there exists 0 su h that (℄0; [) L, where L is the line W n N ,
then an not be di erent from 0. A tually if it is not the ase, de ne
an analyti ar e :℄ 1; 1[ ! P3 by e(t) = ( t). Then e(℄ 1; 0[) is
in luded in L whi h is ar -symmetri , therefore there exists 0 > 0 su h
that e(℄0; 0 [) = (℄ 0 ; 0[) is in luded in L. But this in lusion ontradi ts
the rst assumption on .
Remark that the ar -symmetri sets form a onstru tible ategory of semialgebrai sets in the sense of [28℄, denoted AS , that is :
{ AS ontains the algebrai sets,
{ AS is stable under set-theoreti operations [; \; n,
{ AS is stable by inverse images of AS -map (i.e. whose graph is in AS ) and by
images of inje tive AS -map,
{ ea h A 2 AS has a well-de ned fundamental lass with oeÆ ients in Z2.
In parti ular there is a notion of losure in AS (we refer to [28℄ for a proof).
Proposition 4.3. Every A 2 AS admits a smallest ar -symmetri set, denoted
by AAS , ontaining A and losed in Pn .
Remark 4.4. Ar -symmetri sets are not stable under the eu lidean losure.
Consider for example the regular part A of the Whitney umbrella zx2 = y2 (see
gure 1). The losure of A in AS is the entire Whitney umbrella.
We an de ne irredu ible ar -symmetri sets in the usual way : A 2 AS is
irredu ible if the existen e of a de omposition A = B [ C , with B and C losed
in A and ar -symmetri , implies that either B C or C B . Remark that an
irredu ible ar -symmetri set is not ne essarily onne ted, with our de nition of
ar -symmetri sets (as an example onsider a hyperbola in the plane). Nevertheless,
as proved in [23℄, an ar -symmetri set A admits a unique de omposition as a nite
union of irredu ible ar -symmetri sets losed in A. Note that AS - losure has a good
behaviour with respe t to irredu ibility :
AS
Proposition 4.5. If A 2 AS is irredu ible, then so is A .
Proof. Assume that AAS an be de omposed into AAS = B [ C with B and C
ar -symmetri and losed in AAS . Then B and C are losed and A splits in
A = A \ AAS = (A \ B ) [ (A \ C );
with A \ B and A \ C ar -symmetri and losed in A. But A is irredu ible so either
A \ B A \ C or the reverse in lusion holds. By symmetry, one an assume that
1. ARC-SYMMETRIC SETS AND CLOSURE
47
A \ B A \ C . Then A equals A \ C , so A is in luded in C and nally AAS is equal
to C be ause C is ar -symmetri and losed.
Remark 4.6. The assumption \B and C are losed in A", in the de nition of
an irredu ible ar -symmetri set, is essential. Indeed, let A be the regular part of the
Whitney umbrella. Then the AS - losure of A is the entire Whitney umbrella whi h
is the disjoint union of two ar -symmetri sets : A and the verti al line.
De ne the dimension of an ar -symmetri set to be its dimension as a semialgebrai set. Then, this dimension is equal also to the one of its Zariski losure in
the proje tive spa e [23℄ (re all that by real algebrai variety, we mean in the sense
of [4℄). Therefore, if A 2 AS , one has the equalities
dim A = dim AAS = dim AZ :
The following result relies the dimension of an ar -symmetri set and the dimension
of its ar -symmetri losure. It will be useful in the sequel when dealing with proofs
by indu tion, e.g. for theorem 5.3 and 5.17.
AS = A [ A n AAS . In parti ular
Proposition 4.7. Let A 2 AS . Then A
dim AAS n A < dim A.
AS
Proof. Note that, as a union of ar -symmetri sets, F = A [ A n A is ar symmetri . Moreover F an be de omposed into
AS
AS
F = A [ (A n A) [ (A n A ) = A [ A n A ;
thus F is losed. So the in lusion AAS F holds.
AS
Moreover A is in luded in AAS be ause AAS is losed, thus A n A AAS ,
and so F AAS . Consequently F = AAS .
We an adapt proposition 1.5 of [24℄ to our de nition of ar -symmetri sets. It
is another example of the good behaviour of irredu ible ar -symmetri sets.
Proposition 4.8. Let A 2 AS be irredu ible, and B A be a losed ar symmetri subset of A of the same dimension. Then B = A.
Proof. A an be de omposed into the union of two ar -symmetri sets losed in A as
AS
AS
follows : A = B [ (A n B \ A). Then, by irredu ibility of A, either B A n B \ A
AS
or A n B \ A B .
AS
In the se ond ase B is equal to A, and in the rst one B is in luded in (A n B )n
(A n B ). But this an not happen for the dimension of this ar -symmetri set is stri ly
less than dim B by proposition 4.7.
48
4. ARC-SYMMETRIC SETS
2. Nonsingular ar -symmetri sets
Let us de ne a nonsingular ar -symmetri set with relation to its Zariski losure
in the proje tive spa e.
Definition 4.9. The singular part of an ar -symmetri set A is Sing(A) =
A \ Sing(AZ ). In parti ular A is nonsingular if Sing(A) = ;.
Lemma 4.10. A nonsingular and onne ted ar -symmetri set is irredu ible.
Proof. Let C 2 AS be nonsingular and onne ted, and A C be a onne ted
ar -symmetri subset losed in C and of the same dimension as C . We are going to
prove that A = C .
Denote by A0 the semi-algebrai set onsisting of the part of maximal dimension
of A, and let A1 be the eu lidean losure of A0 in C . Then A1 is a losed semialgebrai subset of C , ontained in A.
But A1 is also open in C . A tually, take a 2 A1 ; there exists an open ball D,
with enter a, in luded in C . Then the dimension of D \ A0 and A oin ide, therefore
D is in luded in A be ause A is ar -symmetri (one an ll D with analyti ar s
whose interiors interse t A0 in non empty sets). Moreover D is in luded in A0 , by
de nition of A0 , and then A1 is an open neighbourhood of a.
Finally A1 is a onne ted omponent of C , so A1 = C . But A1 A, thus A = C .
Let us state a de nition of an isomorphism between ar -symmetri sets.
Definition 4.11. Let A; B 2 AS . Then A is isomorphi to B if and only if there
exist Zariski open subsets U and V in AZ and B Z ontaining A and B respe tively,
and an algebrai isomorphism : U ! V su h that (A) = B .
Remark 4.12. At this point, we are only interested in the algebrai point of
view, be ause we have in mind to study algebrai singularities of ar -symmetri
sets, and to use Hironaka's Desingularisation theorem. In se tion 2 of hapter 5,
where we prove the invarian e of the Betti numbers under Nash isomorphism, we
give another de nition of isomorphism between ar -symmetri sets (see de nition
5.16).
The following proposition laims that losed and nonsingular ar -symmetri sets
are very similar to ompa t nonsingular real algebrai varieties.
Proposition 4.13. Let A 2 AS be ompa t and nonsingular. Then A is isomorphi to a union of onne ted omponents of some ompa t nonsingular real algebrai
variety.
Proof. Let X = AZ be the Zariski losure of A in the proje tive spa e, and let
: Xe ! X be a resolution of the singularities of X . Remark that the three
spa es A; X and Xe have the same dimension and that A is isomorphi to the subset
1 (A) = Ae of Xe be ause A
Reg(X ) and Reg(X ) is a Zariski open subset of X
isomorphi to 1 Reg(X ) Xe .
S
Now, denote by Xe = i2I Ci the de omposition of Xe into onne ted omponents.
Ea h Ci ; i 2 I; is a losed and nonsingular ar -symmetri set, hen e irredu ible
by proposition 4.10. Therefore Ae \ Ci is either equal to Ci or empty be ause of
proposition 4.8, and so Ae is a union of onne ted omponents of Xe as laimed.
3. ARC-SYMMETRIC SETS AND RESOLUTION OF SINGULARITIES
49
3. Ar -symmetri sets and resolution of singularities
The following proposition is an adaptation of Theorem 2.6 of [23℄ to our de nition of ar -symmetri sets. It asserts that, up to desingularization, we an think
about an irredu ible ar -symmetri set as a onne ted omponent of a real algebrai
variety.
Proposition 4.14. Let A 2 AS be irredu ible. Let X be a ompa t real algebrai variety ontaining A with dim X = dim A, and : Xe ! X a resolution of
singularities for X ( f. [16℄). Then, there exists a unique onne ted omponent Ae of
Xe su h that (Ae) = Reg(A).
f0 be an irredu ible ar -symmetri omponent of dimension dim A of
Proof. Let A
1
f0 exists be ause the dimension of 1 (A) and A oin ides. Then
(A). Su h an A
AS
f0
f0 is ontained in some onne ted omponent A
e of X
e . A tually A
is irredu ible
A
f0 , f. proposition 4.5) and losed, therefore is onne ted by propo(be ause so is A
AS
sition 4.13. Now Ae0 is in luded in some onne ted omponent of Xe , and is equal
AS
f0 .
to this omponent by proposition 4.8. We an put Ae = A
Let us prove that the equality (Ae) = Reg(A) announ ed holds. In fa t, it suÆ es
to prove that
AS
dim (Ae) n (Ae) < dim A;
what will be done in the next lemma.
AS
This is suÆ ient for the following reasons. On one hand (Ae) is equal to AAS
by proposition 4.8, so dim AAS n(Ae) < dim A. Now Reg(A)\ AAS n(Ae) is an open
subset of AAS of dimension stri ly less than dim A, so Reg(A) \ AAS n (Ae) = ;.
This implies the in lusion Reg(A) (Ae).
On the other hand, if E denotes the ex eptional divisor of the resolution, the
in lusion (Ae n E ) Reg(AAS ) holds. However Reg(AAS ) is in luded in Reg(A)
be ause dim AAS n A < dim A by proposition 4.7.
Thus we have the following in lusions (Ae n E ) Reg(A) (Ae) whi h gives
the on lusion by taking the losure.
Lemma 4.15. Let A and Ae be as in the proof of proposition 4.14. Then
AS
dim (Ae)
n (Ae) < dim A:
AS
AS
Proof. Let us prove that the in lusion (Ae) (Ae) [ (E ) holds. Denote by
AS
F the set (Ae) [ (E ) . Remark that if F is losed and ar -symmetri , the lemma
is proved.
As is proper, (Ae) is losed and so is F . Now, let :℄ ; [ ! Pn be a real
analyti ar su h that int 1 (F ) 6= ;. Then either
AS
AS
int 1 ((E ) ) 6= ; and (℄ ; [) (E )
50
4. ARC-SYMMETRIC SETS
or
AS
int 1 ((Ae) n (E ) ) 6= ;:
In the latter ase, there exists a unique analyti ar e :℄ ; [ ! Pm su h that
Æ e = . One has int e 1 (Ae) 6= ;, therefore e(℄ ; [) Ae be ause Ae is ar symmetri . Finally (℄ ; [) (Ae) F , and thus F is ar -symmetri .
Remark 4.16. Denote by D the singular lo us of X and by E the ex eptional
divisor of the resolution of proposition 4.14. Then : Ae n E ! AAS n D is an
isomorphism of ar -symmetri sets (restri tion of an algebrai ismorphism).
If we add the assumption that A is nonsingular, then the on lusion of proposition 4.14 be omes simply (Ae) = A. Moreover : Ae n Ae n 1 (A) ! A
is an isomorphism of ar -symmetri sets, and Ae is lose to A is the sense that
dim Ae n 1 (A) < dim A.
One an des ribe with pre ision the di eren es between A and (Ae) in the
general ase. A tually the symmetri di eren e of A and (Ae) onsists of a semialgebrai set of dimension stri ly less than dim A, and more pre isely one has the
following proposition.
Proposition 4.17. Let A and Ae be as in proposition 4.14. Then we an des ribe
the di eren es between A and (Ae) as follows.
On one hand
A n ((Ae) \ A) = fx 2 Sing(A); dimx A < dim Ag
and on the other hand
(Ae) n A \ (Ae) = Reg(AAS n A) [ fx 2 Sing(AAS n A); dimx AAS = dim Ag
= fx 2 AAS n A; dim AAS = dim Ag:
x
Proof. For the rst equality, remark that A n ((Ae) \ A) is in luded in Sing(A) and
that Sing(A) splits in
fx 2 Sing(A); dimx A < dim Ag [ fx 2 Sing(A); dimx A = dim Ag:
But re all that (Ae) = Reg(A), and so
Sing(A) \ (Ae) = Sing(A) \ Reg(A) = fx 2 Sing(A); dimx A = dim Ag:
In the same way, the in lusion (Ae) n A \ (Ae) AAS n A holds and AAS n A
an be de omposed into
AAS n A = Reg(AAS n A) [ S [ S
1
where S1 and S2 are de ned by
2
S1 = fx 2 Sing AAS n A ; dimx AAS = dim Ag
and
S2 = fx 2 Sing AAS n A ; dimx AAS < dim Ag:
3. ARC-SYMMETRIC SETS AND RESOLUTION OF SINGULARITIES
51
Fig. 2. Resolution of the Whitney umbrella.
However S2 is disjoint from Reg(A) = (Ae), so (Ae) n A \ (Ae) is just in luded
in Reg(AAS n A) [ S1 . Moreover S1 is in luded in Reg(A), and the in lusions
Reg(AAS n A) AAS n D (Ae)
hold from remark 4.16, so the se ond equality is proved. The last one is just a
reformulation of the se ond one.
Example 4.18. Consider on e more the Whitney umbrella zx2 = y 2 in order to
illustrate the di erent possibilities in proposition 4.17. One an resolve the singularities of the Whitney umbrella by blowing-up along the z -axis.
Let (u; v; w) = (u; uv; w) denote the hart that ontains the hole stri t transform of the Whitney umbrella. Then, the equation of this stri t transform is w = v2 ,
while the equation of the ex eptional disisor is u = 0 ( gure 2).
On one hand, if A is just the regular part minus the ir le in dotted line, then A
is in luded in (Ae) and (Ae) n A \ (Ae) onsists of the ir le and the losed upper
part of the verti al line. On the other hand, if A is the entire Whitney umbrella,
then (Ae) is in luded in A and A n ((Ae) \ A) is the open bottom part of the verti al
line.
Let us nish this se tion by stating the parti ular ase of the \blowing-up" of a
real algebrai variety along a losed nonsingular ar -symmetri set. More pre isely,
in virtue of proposition 4.13, we just re all here the blowing-up of a nonsingular
real algebrai variety along a nonsingular enter, emphasizing the behaviour of a
onne ted omponent of a ompa t nonsingular real algebrai variety ( f. se tion 2
of hapter 1).
Proposition 4.19. Let Y X be ompa t nonsingular algebrai varieties su h
that dim Y < dim X , and let A X be a onne ted omponent of X . Denote by
: Xe ! X the blowing-up of X along Y . Then is surje tive and 1 (A) is a
onne ted omponent of Xe .
CHAPITRE 5
Virtual Betti numbers
In the theory of motivi integration, generalized Euler hara teristi s play the
role of a measure for ertain subsets of the ar spa e of a variety. In the rst se tion
of this hapter, we give a new example of su h a generalized Euler hara teristi
of ar -symmetri sets, onstru ted from the Betti numbers of ompa t nonsingular
ar -symmetri sets.
This example, alled the virtual Poin are polynomial, was already known for
real algebrai varieties ; it has been introdu ed independently by C. M Crory and A.
Parusinski [27℄ and by B. Totaro [31℄. Here we de ne the virtual Poin are polynomial
for the larger ategory of ar -symmetri sets. The way to perform this is, following
an idea of F. Bittner [3℄, to extend an invariant of the ompa t nonsingular ar symmetri sets to the whole ategory of ar -symmetri sets. The key ingredients
are the resolution of singularities, whi h enables to de ne the invariant for all ar symmetri sets, and the weak fa torisation theorem whi h simpli es the proof of the
independan e from the hoi es we have to make.
Moreover we prove, in the se ond se tion of this hapter, that the virtual Betti
numbers are invariants of the ar -symmetri sets under not only algebrai isomorphisms, but also Nash isomorphisms. This result will be useful when studying the
blow-Nash equivalen e of germs of real analyti fun tions in the next part.
1. Generalized Euler hara teristi s of ar -symmetri sets
Definition 5.1. An additive map on AS with values in an abelian group is a
map de ned on AS su h that
(1) for ar -symmetri sets A and B whi h are isomorphi , (A) = (B ),
(2) for a losed ar -symmetri subset B of A, (A) = (B ) + (A n B ).
If moreover takes values in a ommutative ring and satis es (A B ) = (A) (B ) for ar -symmetri sets A and B , then we say that is a generalized Euler
hara teristi on AS .
Remark 5.2.
(1) One an onstru t a universal generalized Euler hara teristi with values
in the Grothendie k ring of ar -symmetri sets. This Grothendie k ring is
the quotient of the free ring generated by the symbols [A℄, for A 2 AS , by
the ideal generated by the elements
{ [A℄ [B ℄ with A isomorphi to B ,
{ [A℄ [B ℄ [A n B ℄ if B is a losed ar -symmetri subset of A,
{ [A℄ [B ℄ [A B ℄.
But this ring is rather ompli ated, and we are interested in more omputable invariants.
53
54
5. VIRTUAL BETTI NUMBERS
(2) The Euler hara teristi with ompa t supports is a generalized Euler hara teristi on AS , and maybe the simplest one. It follows from the long
exa t sequen e for a pair of the ohomology with ompa t supports. A tually, if we just onsider semi-algebrai sets, with isomorphisms repla ed
by homeomorphisms, the Euler hara teristi with ompa t supports is the
unique generalized Euler hara teristi (see [29℄).
However, for omplex algebrai varieties, there exist a lot of su h generalized Euler hara teristi s, for example dedu ed from mixed Hodge
stru tures (see [5, 26℄).
The aim of the following theorem is to give suÆ iently good onditions, on
an invariant over the losed (i.e. ompa t) and nonsingular ar -symmetri sets,
su h that extends to an additive map on AS . We state the theorem in terms
of onne ted omponents of real algebrai varieties thanks to proposition 4.13. The
method is inspired by the one of F. Bittner [3℄, who proves the result for an algebrai
variety over a eld of hara teristi zero.
Theorem 5.3. Let be a map de ned on onne ted omponents of ompa t
nonsingular real algebrai varieties with values in an abelian group and su h that
P1: (;) = 0,
P2: if A and B are onne ted omponents of ompa t nonsingular real algebrai varieties whi h are isomorphi as ar -symmetri sets, then (A) =
(B ),
P3: with the notations and assumptions of proposition 4.19,
1 (A)
1 (A) \ 1 (A \ Y ) = (A) (A \ Y ):
Then extends uniquely to an additive map de ned on AS .
Remark 5.4. The property P3 of theorem 5.3 is a kind of additivity property
for nonsingular ar s-symmetri sets. A tually, if one a epts the additivity for , P3
is just the invarian e under the isomorphism indu ed by .
Before giving the proof of the theorem, let us state some onsequen es. First,
this result enables us to give another example of su h an additive map by onsidering
the homology with oeÆ ients in 2ZZ.
For i 2 N , denote by bi the i-th Betti number with oeÆ ients in 2ZZ, de ned by
bi () = dim Hi (; 2ZZ).
Corollary 5.5. There exist additive maps on AS with values in Z, noted i and
alled virtual Betti numbers, su h that i oin ides with the lassi al Betti numbers
bi on the onne ted omponent of ompa t nonsingular real algebrai varieties.
Remark 5.6. C. M Crory and A. Parusinski [27℄ have proven the same result
for real algebrai varieties. In parti ular, they have de ned the virtual Betti numbers
of real algebrai varieties. The proof below is inspired by their argument.
Proof of Corollary 5.5. Property P1 and P2 of theorem 5.3 are lear. Let us
prove property P3. In the situation of P3, put B = Y \ A and Ae = 1 (A). Then
e A
e \ 1 (B ) and (A; B )
the exa t sequen es with oeÆ ients in 2ZZ of the pairs A;
1. GENERALIZED EULER CHARACTERISTICS OF ARC-SYMMETRIC SETS
55
give the following ommutative diagram :
/
Hi
\ 1(B )
e A
e
1 A;
/
/
Hi Ae \ 1 (B )
Hi 1 (A; B )
/
Hi (B )
/
Hi(Ae)
/
Hi(A)
where the verti al arrows are indu ed by . Note that
e A
e \ 1 (B ) ! Hi 1 (A; B )
: Hi 1 A;
/
/
is an isomorphism be ause is a homeomorphism between Ae n Ae \ 1 (B ) and
A n B , and that : Hi (Ae) ! Hi(A) is surje tive by lemma 5.7. Now it is an easy
game to he k that the following sequen e :
0 ! Hi(Ae \ 1 (B )) ! Hi (B ) Hi (Ae) ! Hi(A) ! 0
is exa t, hen e
bi (Ae) bi Ae \ 1 (B ) = bi (A) bi (B ):
Then we an apply theorem 5.3.
! A be a degree one ontinuous map between smooth
e Z ) ! Hi (A; Z ) is surje tive.
topologi al manifolds. Then i : Hi (A;
2Z
2Z
Lemma 5.7. Let : Ae
Proof. This is an appli ation of Poin are duality and the proje tion formula. Denote
e Z ) ! Hi (A;
e Z ) the Poinby DA : H n i (A; 2ZZ) ! Hi(A; 2ZZ) and DAe : H n i (A;
2Z
2Z
are isomorphisms for the 2ZZ-oriented manifolds A and Ae. The proje tion formula
states
i Æ DAe Æ n i = DA ;
for is a degree one appli ation. Therefore the diagram :
Hn
n
DA
/
A; 2ZZ '
i
i
Hi A; 2ZZ
O
i
e Z)
e Z ) ' / Hi (A;
H n i (A;
2Z D e
2Z
A
e
is ommutative and thus i : Hi (A;
)
Z
2Z
! Hi(A; 2ZZ) is surje tive.
It turns out to be easy to adapt theorem 5.3 in order to obtain not only additive
maps but also generalized Euler hara teristi s.
Theorem 5.8. Let be as in theorem 5.3. Assume moreover that takes values
in a ommutative ring, and that for onne ted omponents of ompa t nonsingular
real algebrai varieties A and B , the relation (A B ) = (A)(B ) holds. Then the
extension of on AS of theorem 5.3 is a generalized Euler hara teristi .
The following orollary is an immediate onsequen e of the Kunneth formula.
56
5. VIRTUAL BETTI NUMBERS
Corollary 5.9. Let
be de ned on A 2 AS by
(A) =
Then
mial.
dim
XA
i=0
A)ui :
i(
is a generalized Euler hara teristi on AS , alled virtual Poin are polyno-
Remark 5.10. The name of \virtual Poin are polynomial" is inspired by [14℄,
where W. Fulton studies su h a virtual Poin are polynomial for omplex algebrai
varieties. It is related to the weighted hara teristi asso iated with mixed Hodge
stru tures.
Example 5.11.
(1) If Pk denotes the real proje tive spa e of dimension k, whi h is nonsingular
and ompa t, then (Pk ) = 1 + u + + uk . Now, ompa tify the aÆne
line A 1R in P1 by adding one point at the in nity. By additivity
(A 1R ) = (P1 ) (point) = u;
and so (A kR ) = uk .
(2) Let W be the Whitney umbrella, and L be the line in luded in W . Then
(W ) = (W n L) + (L)
by additivity. Moreover W n L is isomorphi , via the blowing-up of W along
L, to the stri t transform of W minus a parabola P , therefore
(W n L) = (A 1R P ) (P ) = (A 1R ) 1 (P ) = (u 1)u:
Finally (W ) = u2 .
(3) The real algebrai varieties
1, de ned by y2 = x2 (1 x2 ) for C1
of gure
2
2
2
and by (x + 1) + y 1 (x 1) + y2 1 = 0 for C2 , are not isomorphi
whereas they are learly homeomorphi . Indeed, one an ompute (C1 )
and (C2 ) by onsidering the resolutions given by blowing-up the singular
point of C1 and that of C2 . In this manner
(C1 ) = (C1 n P1 ) + (P1 )
by additivity, and
(C1 n P1 ) = (P1 n ftwo pointsg)
by isomorphism outside the singular lo us of C1 . Therefore
(C1 ) = (P1 ) (point) = u;
by additivity on e more.
In the same way :
(C2 ) = (C2 n P2 ) + (P2 ) = 2 (P1 n fpointg) + (P2 ) = 2u + 1:
Therefore is not a topologi al invariant ! Remark that the Euler hara teristi with ompa t supports (one an re over it by evaluating u at 1
in this example) does not distinguish these two urves, as it is a topologi al
invariant.
1. GENERALIZED EULER CHARACTERISTICS OF ARC-SYMMETRIC SETS
57
Fig. 1. resolution of C1 and C2
*
*
*
*
P
1
*
P
2*
C
1
C
2
Remark 5.12. The virtual Poin are polynomial has the following property,
as we will see in the next proof. Let A 2 AS . Then
dim(A) = deg (A) :
This property is interesting for motivi integration be ause it indu es a measure
whi h respe ts the dimension of mesurable sets, as opposed to the Euler hara teristi with ompa t supports. It will permit, in se tion 1 of hapter 7, to re over the
Fukui invariants of a real analyti germ of fun tions from the naive zeta fun tion
onstru ted via the virtual Betti numbers.
Let us give the proof of theorem 5.3 rst, and later of theorem 5.8.
Proof of theorem 5.3. We prove theorem 5.3 by indu tion on the dimension ;
the rank n indu tive hypothesis laims that is de ned on ar -symmetri sets of
dimension less than or equal to n, is invariant under isomorphisms of ar -symmetri
sets, and is additive.
For n = 0 the ar -symmetri sets are just nite unions of points and the result
is learly true. Assume that the indu tive hypothesis is true at rank n 1. We prove
the result at rank n in two steps :
(1) if is an additive map on the nonsingular elements of AS of dimension less
than or equal to n, then extends to an additive map on all ar -symmetri
sets of dimension less than or equal to n,
(2) if satis es property P1,P2 and P3, then extends to an additive map on
the nonsingular elements of AS of dimension less than or equal to n.
S
Step 1. Let A 2 AS of dimension n. There exists a strati ation AZ = S 2S S
of AZ with nonsingular algebrai strata, i.e. AZ is a disjoint union of lo ally losed
algebrai varieties (note in parti ular that we do not ask the strata to be onne ted).
Then S \ A, for ea h S 2 S , is a nonsingular ar -symmetri set, and thus (S \ A)
58
5. VIRTUAL BETTI NUMBERS
P
is de ned. Put (A) = S 2S (S \ A). One has to he k that (A) is well-de ned
and satis es the additivity property.
P
We show rstly that (A) is equal to S 2S (S \ A) in the ase where A is
nonsingular, by indu tion on the number of elements in S . Indeed, take N0 2 S ;
then
(A) = A n (A \ N0 ) + (A \ N0 );
and
X
A n (A \ N0 ) =
(S \ A)
S 2SnfN0 g
by indu tion, so the result follows.
Now, if S1 and S2 are two strati ations of AZ , one an nd a ommon re nement
S of S1 and S2 . The independen e in the nonsingular ase indu es that
X
X
X
(S \ A);
(S \ A) =
(S \ A) =
S 2S
S 2S2
S 2S1
thus does not depend on the hoi e of the strati ation.
Let us show nally that is additive. Take A and B in 2 AS , Swith B A,
of dimension less than or equal to n. One an hoose a strati ation S 2S S of AZ
Z
su h that B Z and A n B are unions of strata. Then
X
and
S 2S
(S \ B ) +
X
S 2S
S \ (A n B ) =
X
S 2S
(S \ B ) + S \ (A n B ) ;
S \ (A n B ) + (S \ B ) = (S \ A)
be ause the strata are nonsingular, so is additive.
Step 2. The se ond step onstitutes the heart of the work. De ne over the
nonsingular ar -symmetri sets of dimension n in the following way :
S
D1: if A = i2I Ai denotes
the de omposition of A into irredu ible ompoP
nents, put (A) = i2I (Ai ),
D2: if A 2 AS is nonsingular
and irredu ible, then de ne (A) by (A) =
1
e
e
(A) A n (A) , where Ae is the onne ted omponent of a resolution
of singularities of AZ given by remark 4.16.
We have to prove that is well-de ned, invariant under isomorphisms and additive
over the nonsingular elements of AS .
The following lemma will be useful in the sequel.
Lemma 5.13. Let A and B in AS be nonsingular, irredu ible and isomorphi .
Suppose that AZ and B Z are nonsingular, and denote by Ae AZ and Be B Z the
onne ted omponents ontaining A and B respe tively. Then
(Ae) (Ae n A) = (Be) (Be n B ):
Proof. By de nition of an isomorphism between ar -symmetri sets, we know that
AZ and B Z are birationally equivalent, and the weak fa torization theorem [1, 32℄
fa tors this birational isomorphism in a su ession of blowings-up and blowingsdown. In parti ular, we an assume that the birational isomorphism between AZ
1. GENERALIZED EULER CHARACTERISTICS OF ARC-SYMMETRIC SETS
59
and B Z is just a blowing-up : AZ ! B Z along a nonsingular variety C su h that
C \ B = ;. Note that 1 (Be) = Ae by proposition 4.19.
Now
(Be n B ) = (Be \ C ) + Be n (B [ C )
by the additivity indu tive hypothesis be ause dim Be n B < dim B by proposition
4.7. Moreover
(Be ) (Be \ C ) = (Ae) Ae \ 1 (C )
by property P3, and
Be n (B [ C ) = Ae n A [ 1 (C )
by the indu tive hypothesis on invarian e under isomorphisms. Therefore
(Be) (Be n B ) = (Ae) Ae \ 1 (C )
Ae n A [ 1 (C ) ;
whi h is equal to (Ae) (Ae n A) by the additivity indu tive hypothesis.
Let us he k that the de nition of , for the nonsingular and irredu ible ar symmetri sets of dimension n, does not depend on the hoi e of the resolution
of singularities of remark 4.16.
fi ! AZ , for i 2 f1; 2g,
Let A 2 AS be nonsingular and irredu ible, and let i : X
fi be the onne ted omponents of X
fi given
be resolutions of singularities of AZ . Let A
by proposition 4.14. One has to show that :
f1 ) A
f1 n 1 (A) = (A
f2 ) A
f2 n 1 (A) :
(A
1
2
But 1 1 (A) and 2 1 (A) are isomorphi irredu ible nonsingular ar -symmetri
fi ontaining 1 (A),
sets be ause i is an isomorphism on a Zariski open subsets of X
i
for i 2 f1; 2g. Therefore lemma 5.13 applies and is well-de ned.
Now let us show that is invariant under isomorphisms of ar -symmetri sets.
The proof is very similar to the last one. Let A and B in AS be nonsingular,
irredu ible and isomorphi . Then there exists Zariski open subsets U and V in AZ
and B Z respe tively, and an algebrai isomorphism : U ! V su h that (A) = B .
Choose resolutions of singularities
: Xe ! AZ and : Ye ! B Z
A
B
for AZ and B Z respe tively. Then A 1 (A) and B 1 (B ) are isomorphi as ar symmetri sets, and then by lemma 5.13 :
A 1 (A) = B 1 (B ) :
Moreover A 1 (A) equals (A) be ause both are equal to (Ae) Ae n A 1 (A) ,
where Ae is the onne ted omponent of Xe given by proposition 4.14. In the same
way the equality B 1 (B ) = (B ) holds, hen e
(A) = A 1 (A) = B 1 (B ) = (B ):
60
5. VIRTUAL BETTI NUMBERS
In the ase where A and B are not irredu ible, it suÆ es to de ompose A and B
in irredu ible omponents, and to apply the property D1 be ause an isomorphism
between ar -symmetri sets respe ts the irredu ible omponents.
Finally, let us he k that is additive. Let B A be an in lusion of nonsingular
ar -symmetri sets. Note that, by de nition of , we need to prove the result only
in the ase where A is irredu ible.
If A and B have the same dimension, then B AS = AAS by proposition 4.8, and
so B Z = AZ . Now hoose a resolution of singularities : Xe ! AZ for AZ . If Ae
denotes the onne ted omponent of Xe given by proposition 4.14 for A, then it is
also the omponent asso iated to B , and therefore
(B ) = (Ae) Ae n 1 (B ) :
Now
Ae n 1 (B ) = Ae n 1 (A) + 1 (A) n 1 (B )
by the indu tive hypothesis on additivity. As a onsequen e :
(B ) = (A) (A n B )
be ause 1 (A) n 1 (B ) is equal to (A n B ) by the invarian e under isomorphisms in dimensions smaller that n.
If dim B < dim A, hoose a resolution of singularities : Xe ! AZ for AZ .
Z
Then it is also a resolution of singularities of A n B = AZ . Then
(A n B ) = (Ae) Ae n 1 (A n B ) :
Now
Ae n 1 (A n B ) = Ae n 1 (A) [ 1 (B ) = Ae n 1 (A) + 1 (B )
by the indu
tive assumption, and on e more by the indu tive assumption one has
1
(B ) = (B ). Finally
(A n B ) = (A) (B ):
This a hieves the proof of step 2, and thus the proof of theorem 5.3.
As it was the ase for the previous proof, we are going to prove theorem 5.8 by
indu tion on the dimension. The following relations will be useful :
(1)
(t
k
i=1
Ai ) =
k
X
i=1
(Ai );
where the union of the ar -symmetri sets Ai ; i = 1; : : : ; k is disjoint, and
(2)
(A) = (Ae) (Ae n A);
where A is a nonsingular ar -symmetri set whose ar -symmetri losure Ae is nonsingular.
Proof of theorem 5.8. Put, as an indu tive hypothesis at rank n, that is multipli ative for all ar -symmetri sets of dimension stri ly less than or equal to n.
2. VIRTUAL BETTI NUMBERS AND NASH ISOMORPHISMS
61
Remark that we an restri t our attention to the nonsingular ase be ause, by
onsidering strati ations of ar -symmetri sets with nonsingular strata, we prove
the multipli ativity dire tly with formula (1).
Assume therefore that A and B are nonsingular ar -symmetri sets of dimension
less than or equal to n ; suppose that dim A = n for instan e.
In the ase where A is ompa t, the result follows from another indu tion, nite
this one, on the dimension of B : indeed, resolving the singularities of B Z , one an
assume that B Be , where Be is the nonsingular ar -symmetri losure of B . Then,
by (2),
(A B ) = (A Be) A (Be n B ) :
However (A Be) = (A)(Be) for they are ompa t and nonsingular, and
A (Be n B ) = (A) Be n B )
as we an see by stratifying Be n B with nonsingular strata and using the indu tive
assumption of the se ond indu tion, be ause dim Be n B < dim B by lemma 4.7.
Consequently
(A B ) = (A) (Be ) (Be n B ) = (A)(B ):
If A is no longer ompa t, then ompa tify A and B in Ae and Be respe tively, and
e B
e are nonsingular, even if it means resolving singularities, as before.
assume that A;
Then, by additivity,
(A B ) = (Ae B ) (Ae n A) Be + (Ae n A) (Be n B ) :
The multipli ativity of the rst two terms omes from the pre eding ase (in the
se ond one, stratify the possibly singular set Ae n A), and the multipli ativity of the
third is obtained by the indu tive assumption for max(dim Ae n A; dim Be n B ) < n by
lemma 4.7. Therefore
(A B ) = (Ae)(B ) (Ae n A)(Be) + (Ae n A)(Be n B ) = (A)(B );
and theorem 5.8 is proven.
2. Virtual Betti numbers and Nash isomorphisms
The de nition of an isomorphism between ar -symmetri sets, we gave in hapter
4, is algebrai , via birational morphisms. But ar -symmetri sets are also losely
related to analyti obje ts. As an example, the following proposition emphasizes the
good behaviour of the virtual Poin are polynomial with respe t to the ompa t
algebrai varieties whi h are nonsingular as analyti varieties.
Re all that by bi (X ) wePdenote the i-th Betti number of X with oeÆ ients in
dim X
Z
i
,
2Z and let us put b(X ) = i=0 bi (X )u .
Proposition 5.14. Let X be a ompa t algebrai variety whi h is nonsingular
as an analyti spa e. Then the virtual Poin are polynomial evaluated on X is equal
to b(X ).
62
5. VIRTUAL BETTI NUMBERS
Proof. One an desingularize the algebrai singularities of X by a sequen e of
blowings-up with smooth enters ([2, 16℄). At ea h step of the desingularization,
one has the following relation, where BlC X designs the blowing-up of X along the
nonsingular subvariety C , and E is the ex eptional divisor :
(BlC X ) (E ) = (X ) (C );
be ause the blowing-up is birational, and
b(BlC X ) b(E ) = b(X ) b(C );
be ause X and C are smooth and the blowing-up is a degree one morphism ( f.
orollary 5.5).
Remark that (E ) and (C ) are equal to b(E ) and b(C ) respe tively by de nition
of , be ause E and C are nonsingular and ompa t ar -symmetri sets. The same
is true for Xe , the desingularization of X . Then (X ) and b(X ) an be expressed by
the same formulae in terms of for the former, and b for the latter, where the spa es
involved are nonsingular and ompa t. Therefore, for ea h one of these spa es, and
b oin ide, and then (X ) is equal to b(X ).
Remark 5.15.
(1) The proposition makes sense be ause a real algebrai variety whi h is nonsingular as an analyti spa e is not ne essarily nonsingular as an algebrai
variety ( f. [4℄, example 3.3.12.b).
(2) We will see later, in the proof of theorem 5.17, that the assumption \X is
an algebrai variety" an be repla ed by the weaker \X is a semi-algebrai
set" or, what is the same as X is already an analyti variety, by \X is a
Nash manifold".
In order to relate the analyti aspe t of ar -symmetri sets to the behaviour of
the virtual Poin are polynomial , we propose the following de nition of a Nash
isomorphism between ar -symmetri sets.
Definition 5.16. Let A; B 2 AS . Assume that there exist ompa t analyti
varieties V1 ; V2 ontaining A; B respe tively, and also an analyti isomorphism from V1 to V2 su h that (A) = B . If moreover one an hoose V1 ; V2 to be semialgebrai sets and to be a semi-algebrai map, then we say that A and B are Nash
isomorphi .
We want the virtual Poin are polynomial to be still an invariant for this new
de nition of isomorphism and, a tually, this is the ase.
Theorem 5.17. Nash isomorphi ar -symmetri sets have the same value under
the virtual Poin are polynomial.
Remark 5.18.
(1) This result is one of the two key ingredients, with the hange of variables
formula in motivi integration, of the proof of the fa t that the zeta fun tions of a real analyti germ of fun tions, that we onstru t in se tion 1 of
hapter 6, are invariants of the blow-Nash equivalen e (theorem 7.12).
2. VIRTUAL BETTI NUMBERS AND NASH ISOMORPHISMS
63
(2) Note that, by de nition of the virtual Poin are polynomial as a polynomial whose oeÆ ients are the virtual Betti numbers, proposition 5.14 and
theorem 5.17 remain obviously true if we repla e by i (and b by bi in
proposition 5.14).
Proof of theorem 5.16. On e more, we are going to prove the result by an indu tion on the dimension. As a rst step, let us generalize the result of proposition
5.14.
Step 1. Let A be a ompa t ar -symmetri set whi h is also a nonsingular
analyti subspa e of the Zariski losure X of A. Then (A) is equal to b(A).
In order to prove this laim, one wants to apply the same method as in the proof
of proposition 5.14. But, if C is a smooth enter of blowing-up for X , it is not true in
general that C \ A is still nonsingular, so the equality (C \ A) = b(C \ A) does no
longer hold. In order to solve this problem, onsider the algebrai normalization Xe of
X . There exists Ae Xe the analyti normalization of A ([25℄), whi h is analyti ally
isomorphi to A be ause A is nonsingular as an analyti spa e. Then b(A) is equal
to b(Ae) be ause b is invariant under homeomorphisms.
Moreover (A) is equal to (Ae) ; a tually the algebrai normalization is a birational map, hen e it is an algebrai isomorphism outside ompa t subvarieties E
and D of Xe and X respe tively, of dimension stri ly less than dim X = dim A. Thus
(Ae n E ) is equal to (A n D) by orollary 5.5, and the algebrai normalization,
restri ted to Ae \ E , is an analyti isomorphism onto A \ D, so (Ae \ E ) is equal to
(A \ D) by the indu tive assumption.
Note that Xe is lo ally analyti ally irredu ible as a normal spa e, therefore Ae is a
union of onne ted omponents of Xe . Now it is true that C \ Ae is nonsingular when
C is nonsingular, and the method of the proof of proposition 5.14 applies, therefore
(Ae) is equal to b(Ae). It follows that
(A) = (Ae) = b(Ae) = b(A):
Step 1 is a heived.
Step 2. Let A1 and A2 be Nash isomorphi ar -symmetri sets. Let us prove the
theorem in the parti ular ase where A1 and A2 are nonsingular ar -symmetri sets
and moreover, with the assumptions of the de nition of a Nash isomorphism, the
ompa t analyti variety V1 and V2 are supposed to be smooth as analyti spa es.
{ First we show that (A2 AS ) = (A1 AS ).
Remark that A2 AS is a union of onne ted omponents of V2 by proposition
4.8. Thus A2 AS is also nonsingular as an analyti variety and (A2 AS ) is equal
to b(A2 AS ) by step 1.
Moreover A2 AS is isomorphi to A1 AS by . Indeed, 1 (A2 AS ) is a losed
ar -symmetri set be ause have an ar -symmetri graph and is ontinuous,
and it ontains A1 , so A1 AS 1 (A2 AS ). The reverse in lusion omes from
the fa t that the image by an inje tive map with ar -symmetri graph of
an ar -symmetri set is still an ar -symmetri set (re all that AS form a
onstru tible ategory, f. hapter 4). Consequently, A1 AS is nonsingular as
64
5. VIRTUAL BETTI NUMBERS
an analyti variety be ause so is A2 AS and is an analyti isomorphism,
hen e (A1 AS ) equals b(A1 AS ) by the rst step.
Remark also that b(A2 AS ) is equal to b(A1 AS ) be ause is a homeomorphism between these two smooth ompa t topologi al varieties.
These equalities imply that (A2 AS ) equals (A1 AS ).
{ Then, remark that (A1 AS n A1 ) is equal to (A2 AS n A2 ). Indeed this follows
from the indu tive hypothesis, for A1 AS nA1 and A2 AS nA2 are Nash isomorphi
ar -symmetri sets of dimension stri tly less than dim A2 .
{ Finally (A1 ) is equal to (A2 ). A tually
(A ) = (A AS ) (A AS n A )
1
1
1
1
and
(A2 ) = (A2 AS ) (A2 AS n A2 );
and we have proved that the se ond members are equal, so (A1 ) equals (A2 ) by
additivity of the virtual Poin are polynomial .
Step 3. Redu tion of the problem to Step 2.
Let A1 and A2 be Nash isomorphi ar -symmetri sets. By de nition of a Nash
isomorphism, there exist ompa t analyti varieties V1 and V2 ontaining A and B
respe tively, and an analyti isomorphism : V1 ! V2 su h that (A1 ) = A2 , and
moreover V1 and V2 are semi-algebrai sets and is a semi-algebrai map.
Denote by X1 and X2 the Zariski losures of V1 and V2 respe tively.
As a rst step, we are going to obtain a regular morphism rather than a semialgebrai map between V1 and V2 . Denote by the graph of . This graph is semialgebrai and analyti , thus ar -symmetri . Then the proje tion pi from Z = Z
onto Xi , for i 2 f1; 2g, is a regular morphism whose restri tion to is an analyti
isomorphism onto Vi . Moreover, the preimages by these restri tions of A1 and A2
oin ide, so one an put B = p 1 (A), where A = Ai Vi = V Xi = X for
i 2 f1; 2g and p : Z ! X denotes the natural proje tion. Therefore B is an ar symmetri set whi h is Nash isomorphi to A, and the issue is now to prove that
(B ) equals (A).
In order to do this, we want to ome down to nonsingular obje ts.
Desingularize X by a sequen e of blowings-up with respe t to oherent algebrai
sheaves of ideals (this is possible by [2, 16℄). By blowing-up Z with respe t to the
orresponding inverse image ideal sheaves with respe t to p, one has at ea h step
a regular morphism whi h lifts the proje tion p : Z ! X to the orresponding
blowing-up by the universal property of algebrai blowing-up. Let X : Xe ! X
denotes the resolution of singularities of X and Z : Ze ! Z the orresponding
omposition of blowings-up of Z . If pe denotes the morphism obtained between Ze
and Xe by the universal property, one has the following diagram :
Ze
e
p
/ e
X
X
Z
Z
p
/
X
2. VIRTUAL BETTI NUMBERS AND NASH ISOMORPHISMS
65
whi h is ommutative.
Moreover pe restri ted to the analyti stri t transform e of is an analyti
isomorphism onto the stri t transform Ve of V be ause so is p between and V
(here we onsider the blowing-up as an analyti one).
e
Z j
e
pj e
/ Ve
X jV
V
Now we redu e the problem to the ase where A and B are nonsingular by
the indu tive hypothesis. A tually the singular part of A and B are not ne essarily
ex hanged by pj , but
Sing(A) [ pj 1 Sing(B )
and
Sing(B ) [ pj Sing(A)
are Nash isomorphi by the restri tion of pj . Moreover the dimension of these ar symmetri sets is stri ly less than dim A = dim B , so they have the same image by
thanks to the indu tive hypothesis. Let us denote by A0 and B 0 the respe tive
omplements of these sets in A and B . Now A0 and B 0 are nonsingular.
As A0 is nonsingular, it is isomorphi to its preimage in the desingularization
e
X of X , in the sense of de nition 4.11. Consequently B 0 is also isomorphi to its
preimage in e by ommutativity
diagram.
As a onsequen e (A0 ) and
of the rst
0
1
0
1
0
(B ) are equal to (A ) and (B ) respe tively, and we have redu ed
the problem to step 2.
pj
/
Troisieme partie
Zeta fun tions and blow-Nash
equivalen e
CHAPITRE 6
Zeta fun tions
The zeta fun tions of a Nash fun tion germ (that is semi-algebrai and analyti )
we onsider in this hapter are dire tly inspired by the work of J. Denef & F.
Loeser [5℄ on their motivi zeta fun tions. In parti ular, our zeta fun tions are
de ned by onsidering the image, under the virtual Poin are polynomial, of ertain
onstru tible real algebrai subsets of the ar spa e of an aÆne spa e.
These zeta fun tions ontain some informations on the Nash fun tion germs, and
we onne t it with the blow-analyti equivalen e in hapter 7.
In this hapter, after the de nition and some examples of zeta fun tions, we
fo us on the Denef & Loeser formulae. These formulae enable to ompute the zeta
fun tions in terms of a modi ation of the Nash fun tion germ we onsider. It give
some possibilities to ompute more easily these zeta fun tions, and it is also a key
ingredient for the appli ation to blow-Nash equivalen e in hapter 7.
Note that we state also a Thom-Sebastiani formula for Nash germs of the same
sign, but we do not obtain su h a formula in the general ase.
1. Zeta fun tions and Denef & Loeser Formula
We rst de ne the zeta fun tions for a germ of real analyti fun tions. Then we
give a formula to ompute these zeta fun tions in terms of a modi ation in the ase
where the germ is Nash.
Denote by L the spa e of ar s at the origin 0 2 Rd , de ned by :
L = L(Rd ; 0) = f : (R; 0) ! (Rd ; 0) : formalg;
and by Ln the spa e of trun ated ar s at the order n + 1 :
Ln = Ln(Rd ; 0) = f 2 L : (t) = a1 t + a2t2 + antn; ai 2 Rd g;
for n 0 an integer . Let n : L ! Ln and n;i : Ln ! Li , with n i, be the
trun ation morphisms.
Consider f : (R d ; 0) ! (R; 0) a real analyti fun tion germ. We de ne the naive
zeta fun tion Zf (T ) of f as the following element of Z[u; u 1℄[[T ℄℄ :
Zf (T ) =
X
n1
(n )u
T ;
nd n
where
n = f 2 Ln : ord(f Æ ) = ng = f 2 Ln : f Æ (t) = btn + ; b 6= 0g:
Similarly, we de ne zeta fun tions with sign by
Zf+(T ) =
X
n1
(+n )u
T
nd n
and Zf (T ) =
69
X
n1
(n )u
T ;
nd n
70
6. ZETA FUNCTIONS
where
+n = f 2 Ln : f Æ (t) = +tn + g and n = f 2 Ln : f Æ (t) = tn + g:
Remark that n and n , for n 1, are Zariski onstru tible subsets of Rnd ,
hen e belong to AS .
Remark 6.1. The monodromi zeta fun tion of J. Denef and F. Loeser is de ned
is the same way (but in the monodromi Grothendie k group, f. [5℄) by onsidering
the set n of trun ated ar s , with ordt f Æ (t) = n, su h that the oeÆ ient of
f Æ in front of tn is a n-th root of unity.
However, the zeta fun tions with sign of S. Koike and A. Parusinski [20℄ are
de ned by onsidering the set n of trun ated ar s, with ordt f Æ (t) = n, su h that
the oeÆ ient in front of tn is positive or negative. This is possible in their setting
be ause the sets ℄ 1; 0[ and ℄0; 1[ have a well de ned Euler hara teristi with
ompa t supports. But these semi-algebrai sets are not ar -symmetri , therefore
this onstru tion is no longer valid with the virtual Poin are polynomial.
Example 6.2. Let f : (R; 0) ! (R; 0) be de ned by f (x) = xk ; k 1. Then
f = am tm + + antn; am 6= 0g '
;
f = tm + + antn; am 6= 0g ' f1g ;
Rn
if n=mk,
otherwise.
Therefore (n ) = (u 1)un m if n = mk and 0 otherwise, hen e
X
Tk
T mk
:
= (u 1)
Zf (T ) =
(u 1)umk m
u
u Tk
m1
To ompute the zeta fun tions with sign, we have to onsider the ase k = 2p and
k = 2p + 1. If k = 2p, then n = ; and
n =
+
n
=
so
Zf+ (T ) =
If k = 2p + 1, then
n =
and thus
X
m1
2umk
m
R
X
m1
n m
if n=mk,
otherwise,
T mk
Tk
:
=2
u
u Tk
f = tm + + antn; am 6= 0g ' f1g ;
Zf+ (T ) = Zf (T ) =
R
m
umk
m
R
n m
if n=mk,
otherwise,
T mk
Tk
:
=
u
u Tk
It may be onvenient to express the zeta fun tions of a germ f in terms of a
modi ation of f , that is a proper birational map whi h is an isomorphism over the
omplement of the zero lo us of f , and su h that omposed with the modi ation
f be omes a normal rossing. As we are dealing with analyti germs, note that we
are only interested in analyti neighbourhoods of 0 in Rd and of the ex eptional
divisor in the modi ed spa e. A tually there exists a formula, alled Denef & Loeser
1. ZETA FUNCTIONS AND DENEF & LOESER FORMULA
71
formula, whi h enables to do this in the ase the germ is Nash. In the naive ase,
the Denef & Loeser formula is given by the following proposition.
Proposition
6.3. Let f : (Rd ; 0) ! (R; 0) be a Nash fun tion germ. Let :
1
M; (0) ! (Rd ; 0) be a modi ation of Rd su h that f Æ and the ja obian
determinant ja are normal rossings simultaneously, and assume moreover that
is an isomorphism over the omplement of the zero lo us of f .
Let (f Æ ) 1 (0) = [j 2J Ej be the de omposition into irredu ible omponents of
(f Æ ) 1 (0), and assume that 1 (0) = [k2K Ek for some K J .
Put Ni = multE f Æ and i = 1 + multE ja , and for I J denote by EI0 the
set (\i2I Ei ) n ([j 2J nI Ej ). Then
i
i
Zf (T ) =
Remark 6.4.
X
I 6=;
(u 1)jI j EI0 \ 1 (0)
u
TN
:
TN
1
u
i2I
Y
i
i
i
i
(1) The sets EI0 , for I 6= ;, form the so- alled anoni al strati ation of the
divisor (f Æ ) 1 (0).
(2) We have to assume that f is Nash, and not only analyti , in order the sets
EI0 to be ar -symmetri .
Example 6.5. Let fk : (R 2 ; 0) ! (R; 0) be de ned by fk (x; y) = xk + y k , k 2.
The blowing-up at the origin gives a suitable modi ation for f . Here (f Æ ) 1 (0)
onsists of just the ex eptional divisor P1 in the ase k even, and furthermore, in the
ase k odd, of the stri t transform of f whi h is a smooth urve rossing transversally
the ex eptional divisor. Then
(
if k is even,
(u2 1)u 2 1 uT 2 T
Zf =
1T
2T
u
u
(u 1) 1 u 2 T (u + (u 1) 1 u 1 T ) if k is odd.
Note in parti ular that for k 6= k0 , the zeta fun tions Zf and Zf 0 are di erent.
℄
0;
When we are dealing with signs, one de nes overings E
of EI0 , where I
designs either + or , in the following way.
Q
Let U be an aÆne open subset of M su h that f Æ = u i2I yiN on U , where
u is a unit. Let us put
1
g;
RU = f(x; t) 2 (EI0 \ U ) R; tm = u(x)
where m = g d(Ni ). Then the RU glue together along the EI0 \ U ( f. lemma 6.13)
℄
0;
to give E
I .
Proposition 6.6. With the assumptions and notations of proposition 6.3, one
an express the zeta fun tions with sign in terms of a modi ation as :
k
k
k
k
k
k
k
i
Zf (T ) =
X
I 6=;
(u 1)jI j
1
℄
0;
\ 1(0)
E
I
TN
:
1 u TN
i2I
Y
These formulae will be proven in se tion 2 below.
Example 6.7.
u
i
i
i
i
72
6. ZETA FUNCTIONS
(1) The ase of a normal rossings fun tion is parti ularly
simple to handle
Q
with. Let f : Rd ! R be de ned by f (x) = u(x) ki=1 xNi , with Ni 2 N .
Then
k
Y
u 1T N
Zf (T ) = (u 1)k
1T N :
1
u
i=1
Now, if there exists at least one Ni odd, then
1
Zf+ (T ) = Zf (T ) =
Z (T ):
u 1 f
On the other hand, if all the Ni are even, then Zf (T ) = 0 and
2
Zf+(T ) =
Z (T )
u 1 f
if u is positive, the onverse otherwise.
(2) Let f : R2 ! R be de ned by f (x; y) = x2 + y2 . As f is a positive fun tion,
then Zf (T ) = 0.
i
i
i
℄
0;+
We obtain a modi ation in the same way as in example 6.5, and E
is
I
here the boundary of a Mobius band, hen e homeomorphi to P1 . Therefore
1
u 2T 2
Z (T ):
=
Zf+(T ) = (u + 1)
1 u 2T 2 u 1 f
(3) Let f : R2 ! R be de ned by f (x; y) = x2 + y4 . One an solve the
singularities of f by two su essive blowings-up, and then one obtains that
the ex eptional divisor E has two irredu ible omponents E1 and E2 with
N1 = 2; 1 = 2; N2 = 4; 2 = 3. Therefore
u 3T 4
u 2T 2
u 3T 4
u 2T 2
+
(
u
1)
u
+
(
u
1)
u
:
Zf (T ) = (u 1)2
1 u 2T 2 1 u 3T 4
1 u 2T 2
1 u 3T 4
℄
0;+
℄
0;+
Moreover in this ase E
f1g and Ef2g are homeomorphi to a ir le minus
two points, so
u 2T 2
u 2T 2
u 3T 4
u 3T 4
Zf+ (T ) = 2(u 1)
+
(
u
1)
+
(
u
1)
:
1 u 2T 2 1 u 3T 4
1 u 2T 2
1 u 3T 4
Note that in this parti ular ase one has neither Zf (T ) = (u 1)Zf+ (T ) nor
Zf (T ) = u 2 1 Zf+(T ), whereas it was the ase in the previous examples.
Remark 6.8. It would be onvenient to dispose of a Thom-Sebastiani formula
in order to ompute the zeta fun tions of the fun tion f g, whi h is de ned by the
formula
f g(x; y) = f (x) + g(y);
from the ones of f and g, as it is the ase in [20, 5, 26℄. But it seems to be impossible
to nd in general su h formulae with our zeta fun tions. However in the parti ular
ase of two positive (respe tively negative) fun tions, one has the following formulae.
Proposition 6.9. Let f : (R d1 ; 0)
! ( ; 0) and g : ( d2 ; 0) ! ( ; 0) be two
R
positive or two negative real analyti fun tion germs.
R
R
1. ZETA FUNCTIONS AND DENEF & LOESER FORMULA
Let us put
Zf (T ) =
and
An = 1
X
n1
an T n ; Zg (T ) =
n
X
j =1
aj ; B n = 1
X
n1
bn T n
n
X
j =1
73
bj :
P
Then the naive zeta fun tion of f g : Rd1 +d2 ! R is Zf g (T ) = n1 n T n ,
where
n = a n Bn + A n bn + a n bn :
Example 6.10.
(1) Let h : R2 ! R be de ned by h(x; y) = x2 + y2 . Re all that ( f. example
6.5) :
X T 2n
:
Zf (T ) = (u2 1)
u2n
n1
Putting f (x) = g(x) = x2 , then h = f g and by example 6.2 we get that
a2n = b2n = uu 1 and a2n+1 = b2n+1 = 0, hen e A2n = A2n+1 = u1 . Then,
2
by proposition 6.9, we rederive 2n = uu2 1 and 2n+1 = 0.
(2) Let f and g be de ned by f (x) = x2 and g(y) = y4 , and onsider
f g(x; y) = x2 + y4 :
The odd oeÆ ients of the naive zeta fun tion of f g are zero be ause f
and g are positive, and it is easy to verify that
1
u 1
a2n = n ; A2n = n ;
u
u
and
u 1
1
b4n = n ; b4n+2 = 0; B4n = n = B4n+2 :
u
u
Therefore
u2 1
u 1
; 4n+2 = 3n+1 ;
4n =
3
n
u
u
whi h was not so lear on the expression of the naive zeta fun tion of f g
omputed with the Denef & Loeser formula, in example 6.7.3.
Proof of proposition 6.9. Remark rst that
und1 An = (f 2 Ln; ord(f Æ ) > ng):
A tually, the spa e Ln an be de omposed into the disjoint union
Ln = n;11(1 ) t : : : t n;n1 (n ) t f 2 Ln; ord(f Æ ) > ng:
Hen e, by additivity of , one gets
n
n
n
und1 =
n
X
j =1
ai und1 + (f
2 Ln; ord(f Æ ) > ng);
and the remark is proved.
Now take ( 1 ; 2 ) in Ln (Rd1 ) Ln (Rd2 ) = Ln(Rd1 +d2 ). Then ord(f Æ 1 + g Æ 2 )
is greater than n if and only if ord(f Æ 1 ) and ord(g Æ 2 ) are greater than n, be ause
f and g are of the same sign. Therefore we have to distinguish the three ases :
74
6. ZETA FUNCTIONS
{ ord(f Æ 1 ) = n and ord(g Æ 2 ) > n,
{ ord(f Æ 1 ) > n and ord(g Æ 2 ) = n,
{ ord(f Æ 1 ) = n and ord(g Æ 2 ) = n.
The omputation gives :
n (f g) =
n (f ) und2 Bn + und1 An n (g) +
n (f )
n (g) :
2. Motivi integration and the proof of Denef & Loeser formula
The proof of Denef & Loeser formula, whi h is a simpli ation of the one of [8℄,
theorem 2.2.1 to our setting, uses the theory of motivi integration on ar spa es for
real algebrai varieties (for motivi integration, we refer to hapter 2). In parti ular,
we will use the hange of variables formula of Kontsevi h.
For the onvenien e of the reader, we re all brie y these notions before proving
theorems 6.3 and 6.6. Take : M; 1 (0) ! (Rd ; 0) a real modi ation, and de ne the ar spa e
asso iated to M; 1 (0) by :
L M; 1 (0) = f : ( ; 0) ! M; 1 (0) ; is formalg:
The trun ated ar spa e Ln onsists of the ar s of L, but trun ated at the order
n + 1, for an integer n 0.
Denote by n : L ! Ln the natural trun ation morphism, for n 2 , where L
denotes either L M; 1 (0) or L( d ; 0). A subset A L is alled stable if there
exist a onstru tible set C Ln and some n 0 su h that A = n 1 (C ). Then we
R
N
R
an de ne the measure in Z[u; u 1℄ of su h a stable set A, with respe t to the virtual
Poin are polynomial , by
(A) = u
(n+1)d
n (A) ;
for n large enough (note that n(A) is well de ned sin e Zariski onstru tible
real algebrai varieties are ar -symmetri sets). Indeed, (A) does not depend on n
be ause the natural proje tions Ln+1 ! Ln are lo ally trivial brations with ber
d
R .
Let us re all now the de nition of integrals. Let : A ! Z[u; u 1℄ be a map
with a nite image and whose bers are stable sets. Then the integral of over A
with respe t to is de ned by :
Z
d =
X
1( ) :
2Z[u;u 1℄
We an state the Kontsevi h hange of variables formula. Re all that ja denotes the ja obian determinant of Ja .
A
Proposition 6.11. ([21, 6℄) Let A
fun tion ordt ja is bounded on
L( d ; 0) be stable, and suppose that the
1 (A).
Z
(A) =
R
Then
1 (A)
u
ordt ja d
:
2. MOTIVIC INTEGRATION AND THE PROOF OF DENEF & LOESER FORMULA
75
Before giving the details of the proof of Denef & Loeser formula, we x some
notations. The
modi ation indu es appli ations (respe tively ;n ) between
L M; 1 (0) and L(Rd ; 0) respe tively Ln M; 1 (0) and Ln(Rd ; 0) . Put
Zn(f ) = n 1 (n) and Zn(f Æ ) = 1 Zn(f ) :
Moreover, for e 1, put
e = f 2 L M; 1 (0) ; ordt ja (t) = eg;
and Zn;e(f Æ ) = Zn (f Æ ) \ e .
Let us state a preliminary lemma.
Lemma 6.12. Let : M; 1 (0)
! ( d ; 0) be a modi
ation of Rd su h that
f Æ and the ja obian determinant ja are normal rossings simultaneously, and
assume moreover that is an isomorphism over the omplement of the zero lo us of
f.
Let (f Æ ) 1 (0) = [j 2J Ej be the de omposition into irredu ible omponents of
(f Æ ) 1 (0), and assume that 1 (0) = [k2K Ek for some K J .
Put Ni = multE f Æ and i = 1 + multE ja , and for I J denote by EI0 the
set (\i2I Ei ) n ([j 2J nI Ej ). Then there exists 2 N su h that the naive zeta fun tion
Zf (T ) of f equals
R
i
i
ud
X
n1
Tn
X
e n
u
e
X
I 6=;
(f
2 Ln(M; EI0 ) \ n(e); ord f Æ Æ = ng)
and the zeta fun tions with sign Zf(T ) equal
ud
X
n1
Tn
X
e n
u
e
X
I 6=;
(f
2 Ln(M; EI0 ) \ n(e); f Æ Æ (t) = tn + g):
Proof. Let us prove the lemma for Zf sin e the argument is the same for Zf.
For n 1, ZnP
(f ) is stable, so
Zn(f ) is de ned and equals u (n+1)d (n),
hen e Zf (T ) = ud n1 Zn (f ) T n .
Moreover Zn (f Æ ) equals the disjoint union [e1Zn;e (f Æ ), whi h is a nite
union. A tually, take 2 Zn (f Æ ) ; there exists I J su h that 0 ( ) 2 EI0 . Then
in a neighbourhood of (0), one an hoose oordinates su h that
f Æ = unit
and
ja = unit
Y
i2I
Y
i2I
yiN
yi
i
i
1;
where by unit we denote a non-vanishing Nash fun tion.
Let us write
= ( 1; : : P
: ; d ), and ki = ordt i , for i = 1; : : : ; d. Then the order
ordt f Æ (t) is equal to di=1 Ni ki = n and therefore :
ordt ja (t) =
d
X
i=1
(i
d
1
i 1 X
) Ni ki = max( i )n:
1)ki max(
i
i
Ni i=1
Ni
76
6. ZETA FUNCTIONS
Let = maxi ( N 1 ). Then we have shown that
[e1Zn;e(f Æ ) = [e nZn;e(f Æ );
where the union is nite.
Now Kontsevi h hange of variables formula indu es that
X
Zn(f ) = u e Zn;e(f Æ ) ;
e n
and then
X
X
u e Zn;e(f Æ ) :
Zf (T ) = ud T n
e n
n1
We are going to ompute Zn;e (f Æ ) using the fa t that Zn;e(f Æ ) equals the
disjoint union
G
Zn;e(f Æ ) \ 0 1 EI0 \ 1(0) :
I=
6 ;
Indeed, by additivity we nd :
X Zn;e(f Æ ) =
Zn;e(f Æ ) \ 0 1 EI0 \ 1(0) :
I=
6 ;
Choose I 6= ;. Then n Zn;e(f Æ ) \ 0 1 (EI0 \ 1 (0) is just the set
f (t) 2 Ln M; 1 (0) ; (0) 2 EI0 \ 1 (0); ordt f Æ = n; ordt ja = eg:
The results follows dire tly from the additivity of .
i
i
The proof of propositions 6.3 and 6.6 just onsists in omputing the value of the
virtual Poin are polynomial on the sets whi h appear in the formulae of lemma
6.12. Let us rst prove proposition 6.3.
Proof of proposition 6.3. Take 2 n Zn;e(f Æ ) \ 0 1 (EI0 \ 1 (0) . On a
neighbourhood of (0), one an hoose oordinates su h that
Y
f Æ = unit yiN
i2I
and moreover
Y
ja = unit yi 1 ;
i2I
1
0
1
hen e n (Zn;e (f Æ ) \ 0 EI \ (0) is isomorphi to
i
i
f 2 Ln M; 1 (0) ; (0) 2 EI0 \ 1 (0);
where ki = ordt
is isomorphi to
X
ki Ni = n;
X
i2I
i2I
for
i
2
I
.
As
a
onsequen
e
i
n Zn;e(f Æ ) \ 0 1 EI0 \ 1 (0)
G
k2A(n;e)
EI0 \ 1 (0)
( )jI j
R
Y
i2I
R
n ki
ki (i 1) = eg;
( n )d
R
where
A(n; e) is the subset of k 2 N d de ned by the equations
Pd
1)ki = e.
i=1 (i
jI j
Pd
i=1 Ni ki
= n and
2. MOTIVIC INTEGRATION AND THE PROOF OF DENEF & LOESER FORMULA
By taking the image by , we obtain the equality :
n
Zn;e(f Æ)\0 1 EI0 \ 1(0)
=
X
P =1 k
EI0 \ 1 (0) (u 1)jI j und
k2A(n;e)
77
d
i
i
;
hen e the naive zeta fun tion of f satis es :
P
X
X X X
u e =1 k T n :
Zf (T ) = (u 1)jI j EI0 \ 1 (0)
n1 e n k2A(n;e)
I=
6 ;
Remark that fk 2 A(n; e); n 1; e ng is in bije tion with N jI j , therefore
P
XX X
XY
Y u TN
u e =1 k T n =
(u T N )k =
:
1 u TN
n1 e n k2A(n;e)
i2I
k i2I
d
i
d
i
i
i
i
i
i
i
i
i
Finally
Zf (T ) =
X
I 6=;
(u 1)jI j EI0 \ 1 (0)
TN
;
1 u TN
i2I
Y
whi h is the Denef & Loeser formula.
u
i
i
i
i
i
The proof of proposition 6.6 is a little bit more ompli ated due to the fa t that
℄
0;
we have to introdu e a overing E
of EI0 in order to ompute Zf(T ). Re all that
I
Q
if U is an aÆne open subset of M su h that f Æ = u i2I yiN on U , where u denotes
a unit, then by RU we mean the set
1
RU = f(x; t) 2 (EI0 \ U ) R; tm = g;
u(x)
℄
0;
where mI = g di2I (Ni ). Then E
is the gluing of the R along the E 0 \ U .
i
I
I
Lemma 6.13. The RU glue together along EI0 \ U .
I
U
℄
0; does not depend on the hoi e
Proof. It suÆ es to prove that the de nition of E
I
of the lo al oordinates.
Let zi be another lo al system of oordinates on U su h
Q
N
that f Æ = v i2I zi . Then zi is proportional to yi for the indi es i in I , therefore
Q
zi = i yi for a non vanishing analyti fun tion i . So v(y) i2I Ni = u(y) and thus
f(x; t) 2 (EI0 \ U ) R; tm = u(1x) g ! f(x; t) 2 (EI0 \ U ) R; tm = v(1x) g
i
i
I
I
(x; t) 7
is an isomorphism.
u
! (x; t
Y
i2I
Ni
m
i
)
Proof of proposition 6.6. Let U be an aÆne open subset of M su h that f Æ =
N
on
i2I yi on U , where u is a unit. What we have to ompute is the value of
Y
W = f(x; y) 2 (EI0 \ U ) (R )jI j; u(x) yiN = 1g:
i2I
Q
i
i
78
6. ZETA FUNCTIONS
Denote
P by m the greatest ommon divisor of the Ni , i 2 I , and hoose ni , i 2 I su h
that i2I ni Ni = m. Assume that I = f1; : : : ; sg. Remark that W is isomorphi
to
1 ; Y yN =m = 1g;
0
W ; = f(x; y; t) 2 (EI0 \ U ) (R )jI j R ; tm =
u(x) i2I i
by
0
W ; ! W ; (x; y; t) 7 ! (x; tn1 y1 ; : : : ; tn ys):
The inverse is the morphism given by
Y
Y
Y
(x; y) 7 ! (x; ( yiN =m ) n1 y1 ; : : : ; ( yiN =m ) n ys ; yiN =m ):
i2I
i2I
i2I
0 ;
0 ;
Now it is easier to ompute (W ) be ause W ' RU (R )jI j 1 . A tually,
this last isomorphism omes from the fa t that at least one Nm is odd. Therefore
(W ) = (u 1)jI j 1 (RU ), and the same omputation as in the naive ase gives
the formula.
i
s
i
i
i
s
i
CHAPITRE 7
An invariant of the blow-Nash equivalen e
In this hapter, we de ne the blow-Nash equivalen e of Nash fun tion germs,
and we ompare it with the original blow-analyti equivalen e. The former is just a
parti ular ase of the latter for whi h we add algebrai data.
The blow-analyti equivalen e of real analyti fun tion germs is a notion due
to T.-C. Kuo [22℄. He introdu ed this equivalen e relation after noti ing that the
famous Whitney family, whi h is topologi ally trivial but has in nitely many C 1 equivalen e lasses, is analyti ally trivial after one blowing-up. For a survey on
blow-analyti equivalen e, we refer to [10℄. Note in parti ular that there exist triviality results, su h as the niteness of the number of lasses for an analyti ally
parametrized family of analyti fun tion germs with isolated singularities (proved
by T.-C. Kuo [22℄), or the existen e of resolutions via tori modi ations for an analyti family whose weighted homogeneous initial terms has an isolated singularity
(see [13℄).
We prove a triviality result for the blow-Nash equivalen e, under the assumption
of the existen e of algebrai resolutions. The proof, whi h is inspired by the ones
for the blow-analyti equivalen e ase, requires di erent te hniques be ause of the
parti ularities of the Nash setting. A tually, the integration along ve tor elds, whi h
enables to onstru t blow-analyti isomorphisms, needs to be repla ed by other
arguments (see se tion 3).
Moreover, we state and prove the main result of this hapter whi h is the invarian e of the zeta fun tions, onstru ted in hapter 6, with respe t to the blow-Nash
equivalen e. As an appli ation, we nally state in se tion 2 the blow-Nash equivalen e lasses of the two variables Brieskorn polynomials. In this example, already
studied by S. Koike and A. Parusinski [20℄ with the help of the Fukui invariants and
of their zeta fun tions, we only need the use of our zeta fun tions to draw up the
lassi ation.
1. Blow-Nash equivalen e
1.1. De nition. To begin with, let us re all that by an algebrai modi ation
of a real analyti fun tion germ f : (Rd ; 0) ! (R; 0), we mean a proper birational
algebrai morphism f : Mf ; f 1 (0) ! (Rd ; 0), between analyti neighbourhoods
of 0 in Rd and the ex eptional divisor f 1 (0) in Mf , whi h is an isomorphism over
the omplement of the zero lo us of f and for whi h f Æ is in normal rossing.
Definition 7.1.
(1) A map : (Rd ; 0) ! (Rd ; 0) is blow-Nash
if there exists a proper birational
algebrai morphism : M; 1 (0) ! (Rd ; 0) su h that Æ is Nash (i.e.
semi-algebrai and analyti ).
79
80
7. AN INVARIANT OF THE BLOW-NASH EQUIVALENCE
(2) Let f; g : (Rd ; 0) ! (R; 0) be two germs of Nash fun tions. They are said
to be blow-Nash equivalent if there exist two algebrai modi ations
! ( d ; 0) and g : Mg ; g 1 (0) ! ( d ; 0);
su h that f Æ f and ja f (respe tively g Æ g and ja g ) are in normal
f : Mf ; f 1 (0)
R
R
rossings and a Nash isomorphism (i.e. a semi-algebrai map whi h is an
analyti isomorphism)
between analyti neighbourhoods Mf ; f 1 (0)
and Mg ; g 1 (0) whi h preserves the multipli ities of the ja obian determinants of f and g along the omponents of the ex eptional divisors, and
whi h indu es a homeomorphism : (Rd ; 0) ! (Rd ; 0) su h that f = g Æ ,
as illustrated by the diagram :
Mf ; f 1 (0)
/
Mg ; g 1 (0)
f
g
(Rd ; 0) M
MMM
MMM
f MMM&
(R; 0)
/ (R d ; 0)
q
q
q
qqgq
q
q
qx q
whi h is ommutative.
Remark 7.2. We do not know whether the blow-Nash equivalen e is an equivalen e relation or not.
The blow-Nash equivalen e is a parti ular ase of the blow-analyti equivalen e,
for whi h we add algebrai data. Let us re all the de nition of blow-analyti equivalen e, a notion introdu ed by T.-C. Kuo in 1985 [22℄.
Definition 7.3.
(1) A proper surje tive real analyti map : Xe ! X between real analyti
spa es is a real analyti modi ation of X if its omplexi ation is
biholomorphi outside (N ), for N a thin subset of the omplexi ation
of X .
(2) A mapping h : X ! Y of real analyti spa es is blow-analyti if there
exists a real analyti modi ation of X su h that h Æ is analyti .
(3) Two germs of real analyti fun tions f; g : (Rd ; 0) ! (R; 0) are blowanalyti ally equivalent if there exists a blow-analyti lo al homeomorphism
( i.e. and 1 are blow-analyti ) : (Rd ; 0) ! (Rd ; 0) su h that f = g Æ.
T.-C. Kuo proved the following hara terization of blow-analyti equivalen e.
! (R; 0)
are blow-analyti ally equivalent if and only if there exist a lo al homeomorphism ,
two real analyti modi ations f ; g and a real analyti isomorphism su h that
Proposition 7.4. Two germs of real analyti fun tions f; g : (R d ; 0)
1. BLOW-NASH EQUIVALENCE
Mf ; f 1 (0)
/
81
Mg ; g 1 (0)
f
g
(Rd ; 0) M
MMM
MMM
f MMM&
(R; 0)
/ (R d ; 0)
q
qq
qqgq
q
q
qx q
is ommutative.
Remark 7.5.
(1) Another relation lose to blow-analyti equivalen e, alled Blow-analyti
equivalen e, is also studied ( f. [10℄). It is de ned in a similar way as blowanalyti equivalen e, with the di eren e that the real analyti modi ations
are required to be ompositions of blowings-up along smooth enters. Up to
now, it is not known whether Blow-analyti equivalen e is an equivalen e
relation or not as soon as d > 2.
(2) The di eren es between the de nition of blow-analyti equivalen e in the
sense of proposition 7.4 and blow-Nash equivalen e are the following : blowNash equivalen e makes sense only in the ase of Nash fun tion germs, and
moreover in the de nition of blow-Nash equivalen e, we ask :
{ the modi ations to be algebrai and their ja obian determinant to
be in normal rossings,
{ the isomorphism upstairs to be Nash,
{ to preserve the ja obian determinant orders of the modi ations.
In theorem 7.6 below, we give some arguments whi h tend to make us
believe in the fa t that the se ond point might be removed when studing
Nash germs. But the rst and the third ones are essential.
Note however that di erent de nitions of a blow-analyti homeomorphism have o ured sin e the original arti le of T.-C. Kuo [22℄ appeared,
and notably T. Fukui, T.-C. Kuo and L. Paunes u propose in [11℄ a de nition loser to our one. A tually, they de ne a blow-analyti isomorphism
to be a homeomorphism su h that there exists (as in proposition 7.4) an
analyti isomorphism upstairs whi h is moreover an isomorphism between
the riti al lo i of the modi ations.
(3) When trying to adapt the proof of T.-C. Kuo of the fa t that blow-analyti
equivalen e is an equivalen e relation to the ase of blow-Nash equivalen e,
the problem is the transitivity property. It omes from the mixing of algebrai and analyti data whi h does not enable to keep algebrai modi ations.
1.2. Properties. First, note that two real analyti fun tion germs that are
analyti ally equivalent are, of ourse, blow-analyti ally equivalent ! Note also that
two Nash germs that are analyti ally equivalent are also Nash equivalent, as proven
by M. Shiota [30℄, and therefore blow-Nash equivalent.
Now, the question of moduli is a natural question when one studies an equivalen e relation of germs. Although blow-Nash equivalen e is not proven to be an
equivalen e relation, this issue is still relevant.
82
7. AN INVARIANT OF THE BLOW-NASH EQUIVALENCE
The following theorem gives a result in this dire tion. It states that, for a Nash
parametrized family of Nash germs with an isolated singularity whi h admits an
algebrai resolution of singularities, blow-analyti equivalen e implies blow-Nash
equivalen e. A tually we prove more, namely there is a nite number of blow-Nash
equivalen e lasses, and they oin ide with the blow-analyti ones.
Theorem 7.6. Let F : (R d ; 0) P ! (R; 0) be Nash, where P is a Nash
set di eomorphi to an open simplex in an Eu lidean spa e. Assume that F (:; p) :
(Rd ; 0) ! R has an isolated singularity at 0 for ea h p 2 P , and assume moreover
that F admits an algebrai resolution of singularities.
Then the family F (:; p), for p 2 P , onsists of a nite number of blow-Nash
equivalen e lasses. Moreover ea h of these lasses is blow-analyti ally trivial.
Remark 7.7.
(1) By an algebrai resolution of singularities for F , we mean a nite omposition : M ! Rd P of blowings-up with smooth enters su h that is
an isomorphism outside 0 P , and F Æ is in normal rossing.
Note that in the parti ular ase where F is algebrai , su h a resolution
exists by Hironaka's Desingularization theorem [16℄.
(2) The proof of theorem 7.6, whi h is postponed to se tion 3, is inspired by
the main result of [22℄ where T.-C. Kuo proved the niteness of the number
of blow-analyti equivalen e lasses for an analyti ally parametrized family
of isolated singularities. However, the key argument of integration along a
ve tor eld does no longer apply in the Nash ategory, and we have re ourse
to other te hniques to solve this question.
The following parti ular ase, whi h is a onsequen e of the proof of theorem 7.6,
will help us in lassifying the blow-Nash type of Brieskorn polynomials (see se tion
2).
Corollary 7.8. Let ft : (R d ; 0) ! (R; 0), t 2 I , with I an interval of R,
be a Nash parametrized family of weighted homogeneous polynomials of the same
weight with an isolated singularity at the origin. Then the family fft gt2I is blowNash trivial.
Remark 7.9. The fa t that su h a family is blow-analyti ally trivial was already
known [10℄.
Proof. It is well-known [10℄ that, in that ase, one an nd a tori modi ation
: (M; E ) ! (Rd ; 0) of (Rd ; 0) su h that id ful ls the assumptions of proposition
7.21.
Example 7.10. Consider the Nash fun tion germs xp + y kp and xp
(R2 ; 0)
ykp from
to (R; 0). They are blow-Nash equivalent. Indeed,
2t kp
1 t2 k(p 1)
pxy
+
y ; t 2 [ 1; 1℄;
ft (x; y) = xp +
2
1+t
1 + t2
is a weighted homogeneous polynomial of weight (k; 1) with an isolated singularity
at the origin for ea h t 2 [ 1; 1℄, and therefore orollary 7.8 implies that f 1 and f1
are blow-Nash equivalent.
1. BLOW-NASH EQUIVALENCE
83
A diÆ ult issue with this kind of relations is to nd invariants, and a tually for
the blow-analyti equivalen e, only the Fukui invariants [17℄ and the zeta fun tions of
S. Koike and A. Parusinski [20℄, de ned with the Euler hara teristi with ompa t
supports, are known. Anyway, the both ombined enable to lassify the blow-analyti
type of Brieskorn polynomials of two variables, and to perform almost ompletely
the lassi ation for the three variables ase [20℄.
Note that our zeta fun tions generalize the Fukui invariants A(f ), whi h asign
to f : (Rd ; 0) ! (R; 0) the possible orders of series f Æ where is a real analyti
ar at the origin. A tually
A(f ) = ford f Æ ; : (R; 0) ! (Rd ; 0) is analyti g;
and there exist also Fukui invariants with sign de ned by
A+ (f ) = ford f Æ ; : (R; 0) ! (Rd ; 0) is analyti and f Æ (t) nonnegativeg
A (f ) = ford f Æ ; : (R; 0) ! (Rd ; 0) is analyti and f Æ (t) nonpositiveg
where nonnegative (resp. nonpositive) means that f Æ (t) 0 (resp. 0) in a
positive half neighbourhood [0; [.
Proposition 7.11. The Fukui invariants are the nonzero exponents of the naive
zeta fun tion Zf (T ). Similarly, the Fukui invariants with sign A are the nonzero
exponents of the zeta fun tions with sign Zf (T ).
Proof. It suÆ es to noti e that
n 2 A(f ) () n (f ) 6= ;
n 2 A (f ) () n (f ) 6= ;
and to remember that (X ) 6= 0 () X 6= ; (remark 5.12).
The following result is the main one of this se tion. Its proof is dire tly inspired
by the theorem 4.5 of [20℄, and is a dire t onsequen e of lemma 6.12 (see se tion 2
of hapter 6).
Theorem 7.12. The naive zeta fun tion Zf (T ) and the zeta fun tions with sign
Zf (T ) of a germ of Nash fun tions are invariants of the blow-Nash equivalen e.
Proof. Let f; g : (Rd ; 0) ! (R; 0) be two blow-Nash equivalent Nash fun tion germs.
By de nition, there exist modi ations
f : Mf ; f 1 (0) ! (Rd ; 0) and g : Mg ; g 1 (0) ! (Rd ; 0);
and a Nash isomorphism : Mf ; f 1 (0) ! Mg ; g 1 (0) as in de nition 7.1.
Then the assumptions of propositions 6.3 and 6.6 are satis ed.
Now, it suÆ es to prove that the expression of the zeta fun tions given by the
Denef & Loeser formulae oin ide. But is invariant under Nash isomorphisms by
theorem 5.17, and moreover preserves
-the multipli ities of f Æ f and g Æ g , be ause it is an isomorphism,
-the multipli ities of the ja obians of f and g along the omponents of the
ex eptional divisors, by de nition of blow-Nash equivalen e.
Therefore the zeta fun tions of f and g oin ide.
84
7. AN INVARIANT OF THE BLOW-NASH EQUIVALENCE
Remark 7.13. Our zeta fun tions are also invariants for the analyti equivalen e
of real analyti germs. A tually, the onstru tible sets n and n asso iated with
two analyti ally equivalent fun tion germs f; g are isomorphi . Indeed, let h be a
lo al analyti isomorphism su h that f = g Æ h. Then,
n (f ) ! n (g)
7 ! n(h( ))
is an algebrai isomorphism be ause, after trun ation at the level of the spa e of
ar s, the lo al analyti isomorphism be omes algebrai . Therefore the naive zeta
fun tions of f and g oin ide. The proof in the ase with sign is similar.
2. Appli ation to Brieskorn polynomials
We apply our zeta fun tions to sket h the lassi ation of two variables Brieskorn
polynomials under blow-Nash equivalen e, and to give examples in three variables.
Brieskorn polynomials in two or three variables are polynomials of the type
"p xp + "q yq (+"r z r ); p q r 2 N ; "p ; "q ; "r 2 f1g:
Remark that if p = 1, then "p x + "q yq (+"r z r ) is Nash isomorphi to x. Therefore
we will restri t our attention to the ase p 2.
A tually, the lassi ation under blow-analyti equivalen e has been done ompletely in the two variables ase, and almost ompletely in the three variables ase
in [20℄ using zeta fun tions de ned with Euler hara teristi with ompa t supports
and the Fukui invariants (see [20℄, theorem 7.3 ; for the Fukui invariants, see [17℄).
There exists only one ase that an not be de ided, and the following example shows
that we an dis uss it under blow-Nash equivalen e. However this it is not suÆ ient
to on lude for blow-analyti equivalen e.
Example 7.14. Let fp;k be the Brieskorn polynomial de ned by
fp;k = (xp + ykp + z kp ); p even; k 2 N :
We are going to prove that for xed p and di erent k, two su h polynomials are
not blow-Nash equivalent. But we do not know whether they are blow-analyti ally
equivalent or not.
In order to do this, we al ulate dire tly the naive zeta fun tion of fp;k . For n 2 N ,
we have to ompute (n ). First, it is lear that n = ; when n is not a multiple of
p. If n is a multiple of p, write n = p(mk + r) where mk + r represents the eu lidean
division of np by k. If 2 Ln , put = (a1 t + + an tn ; b1 t + + bntn ; 1 t + + n tn).
Then if r 6= 0, the rst non zero term of f Æ is given by the rst omponent of
, hen e n equals
f ; amk+r 6= 0; a1 = = amk+r 1 = b1 = = bm = 1 = = m = 0g:
In the ase where r = 0, the three omponents of play a part, and n equals
f ; (amk ; bm ; m ) 6= 0; a1 = = amk 1 = b1 = = bm 1 = 1 = = m 1 = 0g:
Therefore
(p 1)(mk+r)
p(mk+r) m 2
n ' R 3 R (p 1)mk (Rpmk m 2 ) if r 6= 0
(R ) R
(R
)
if r = 0
2. APPLICATION TO BRIESKORN POLYNOMIALS
85
hen e the oeÆ ient of T n is
8
< (u 1)u (mk+r) 2m if n = p(mk + r ); 0 < r < k
3
n
(n )u = (u3 1)u mk 2m
if n = pmk
:
0
otherwise.
Therefore the zeta fun tion of fp;k looks like
Zf = (u 1) u 1 T p + u 2 T 2p + + u (k 1) T (k 1)p + (u3 1)u k 2 T kp
p;k
+(u 1) u
+ u
2) T 2kp + :
(k+3) T (k+1)p + u (k+4) T (k+2)p +
(2k+1) T (2k 1)p +(u3 1)u 2(k
Now it suÆ es to note that, for p xed and k < k0 the pk- oeÆ ient of Zf
3
(u 1)u k 2 whereas the one of Zf 0 is (u 1)u k .
p;k
is
p;k
Remark 7.15. The ase of two variables Brieskorn polynomials have been dealt
with in [20℄, using their zeta fun tions and the Fukui invariants. A tually the only
ase where the equivalen e lass of Brieskorn polynomials of two variables an not
be distinguished using only their zeta fun tions, and whi h requires the use of the
Fukui invariants, is the following : fk (x; y) = (xk + yk ), k 2 even. Remark
that we have seen in example 6.5 that for k 6= k0 the naive zeta fun tions Zf and
Zf 0 are di erent, therefore our zeta fun tion distinguishes this ase, for blow-Nash
equivalen e.
A tually, one an say more. Indeed, for two variables Brieskorn polynomials the
naive zeta fun tion determines the exponents p and q.
Proposition 7.16. Let g = xp y q be a two variables Brieskorn polynomial.
Then thePexponents p and q are uniquely determined by the naive zeta fun tion
Zg (T ) = n1 gn T n . More pre isely
p = minfn; gn 6= 0g
P
and if l = minfn; gn 6= an un g, where n1 an T n denotes the naive zeta fun tion
of xp, then
k
k
q=
l 1 if p is odd, p divides l 1 and gl 6= (u 1)uk for any k 2 N
l
otherwise.
Proof. The hara terization of p is lear. Now, if p - q, then q = l and gq = (u 1)uk
for some k 2 N .
If q = kp for some k 2 N , then gkp = (fap bkp )g)u2kp k 1 and thus
gkp = akpukp () fap bkp 6= 0g = u(u 1) () p odd:
In that ase gkp+1 = (u 1)2 u2kp k 2, so l = kp + 1.
Therefore if p is even then q = l, and if p is odd either q = l or q = l 1. More
pre isely, if gl = (u 1)uk then p - q and q = l, whereas if gl = (u 1)2 uk , then
pjl 1 and q = l 1.
To nd the signs in front of xp and yq , the naive zeta fun tion is not suÆ ient
as shown by the following example.
86
7. AN INVARIANT OF THE BLOW-NASH EQUIVALENCE
Fig. 1. Resolution tree of x3 y 4 .
Z−
E(3;2)
Z+
E(8;5)
E(4;3)
Example 7.17. Let f : (R 2 ; 0)
E(12;7)
! ( ; 0) be de ned by f(x; y) = x3 y4. One
R
an solve the singularities of f by a su ession of four blowings-up. The resolution
tree of this modi ation is drawn in gure 1, where Z denotes the stri t transform
of f , and E (N; ) denote an irredu ible omponent of the ex eptional divisor su h
that multE f Æ = N and 1 + multE ja = . The Denef & Loeser formulae imply
that Zf+ is equal to Zf , but f+ and f are not blow-Nash equivalent (they are not
even blow-analyti ally equivalent, see [20℄, theorem 6.1 for example).
A tually the zeta fun tions with sign enable to dis uss the signs for two variables Brieskorn polynomials ex ept in one ase. The following proposition details
the possibilities. Note that (x; y) ! (x; y) gives an a tion on the blow-Nash
lasses, hen e when the power p (or q) is odd, the orresponding sign an not be
determined. By onvention, in the ase where p = q and the signs are opposite, we
onsider xp yp rather than xp + yp.
P
Proposition 7.18. Let Zg (T ) = n1 gn T n be the zeta fun tions with sign of
"p xp + "q yq , with "p ; "q 2 f1g. If p is even, then
if gp+ 6= 0
"p = +11 otherwise,
and if q is even, but not multiple of an odd p, then
if gp = 0
"q = +11 otherwise.
The remaining ase is when p is odd and q = kp with k even. But in example
7.10 we proved that in that ase xp + ykp and xp ykp are blow-Nash equivalent,
therefore we have proved the following :
3. PROOF OF THEOREM 7.6
87
Proposition 7.19. For two Brieskorn polynomials in two variables, the three
following statements are equivalent :
{ they are blow-Nash equivalent,
{ they are blow-analyti ally equivalent,
{ their naive zeta fun tion and zeta fun tions with sign oin ide.
3. Proof of theorem 7.6
For the onvenien e of the reader, let us re all that, by an algebrai resolution of
singularities for F , we mean a nite omposition : M ! Rd P of blowings-up
su h that is an isomorphism outside 0 P , and F Æ is in normal rossing.
Moreover, let us re all the statement of the theorem :
Let F : (Rd ; 0) P ! (R; 0) be a Nash mapping, where P is a Nash set
di eomorphi to an open simplex in an Eu lidean spa e. Assume that, for ea h
p 2 P , the map F (:; p) : (Rd ; 0) ! R has an isolated singularity at 0, and assume
moreover that F admits an algebrai resolution of singularities. Then this family
onsists of a nite number of blow-Nash equivalen e lasses. Moreover ea h of these
lasses is blow-analyti ally trivial.
T.-C. Kuo [22℄ proved the niteness of the number of blow-analyti equivalen e
lasses for an analyti ally parametrized family of real analyti germs with an isolated
singularity. In that setting, he used integration along ve tor elds to onstru t a
trivialization of a modi ation of the zero set of the family. Unfortunately, in the
Nash situation, this eÆ ient method is forbidden for we go out from the Nash world...
Fortunately T. Fukui, S. Koike and M. Shiota have given an e e tive tool to
show Nash triviality : the Nash Isotopy Lemma [12℄. It gives a trivialization of
Nash submanifolds, possibly with boundary, with normal rossings, and also to their
arbitrary interse tions :
Theorem 7.20 (Nash Isotopy Lemma). Let M be a Nash manifold possibly
with boundary and N1 ; : : : ; Nk be Nash submanifolds of M possibly with boundary
whi h together with N0 = M are normal rossing. Assume that Ni N0 for
i 2 f1; : : : ; kg. Let P be a Nash manifold di eomorphi to an open simplex in an
Eu lidean spa e, and ! : M ! P be a proper onto Nash submersion su h that the
restri tions of !
! : Ni1 \ \ Ni ! P;
for 0 i1 < < is k, are also proper onto submersions.
Then there exists a Nash isomorphism
: (M ; N1 ; : : : ; Nk ) ! M \ ! 1 (0); N1 \ ! 1 (0); : : : ; Nk \ ! 1 (0) P
su h that ! Æ 1 : M \ ! 1 (0) P ! P is the anoni al proje tion.
This result does not repla e totally the integration along ve tor elds be ause
it works just at the level of manifolds, and not of fun tions. Therefore it enables us
to obtain a Nash trivialization of the zero sets of our Nash parametrized family of
Nash germs, but not of the nonzero levels of the fun tions of the family. Now, in
order to show the Nash triviality, we have re ourse here to orthogonal proje tions
between levels of fun tions ( f. lemma 7.23). And the point is that, if this te hnique
s
88
7. AN INVARIANT OF THE BLOW-NASH EQUIVALENCE
does not allow us to keep a Nash isomorphism, it gives us a blow-Nash isomorphism
whi h is suÆ ient for our matter !
Note that the proof of the blow-Nash property of the trivialization, whi h requires some te hni al omputations, is just based on the impli it fun tion theorem
whi h makes sense in the Nash ategory.
Proof of theorem 7.6. Following T.-C. Kuo proof [22℄, we an subdivide P into
a nite number of Nash sets P 0 , di eomorphi to open simpli es in an Eu lidean
spa e, su h that there exists an algebrai modi ation : M ! Rd P 0 whi h
satis es :
{ is an isomorphism outside 0 P 0 ,
{ F Æ is in normal rossing,
{ if r : Rd P 0 ! P 0 denotes the anoni al proje tion, then r Æ maps the
anoni al strata of (F Æ ) 1 (0) submersively onto P 0 .
Now, on the one hand, the proof of T.-C. Kuo gives that two germs F (:; p) and
F (:; q) with p; q 2 P 0 are blow-analyti ally equivalent. On the other hand, the Nash
Isotopy Lemma, proved by T. Fukui, S. Koike and M. Shiota in [12℄, enables to apply
proposition 7.21 below, and thus to prove that two su h germs are also blow-Nash
equivalent.
So, thanks to the Nash Isotopy Lemma, we are lead to study the parti ular
ase where F : (Rd ; 0) P ! R admits, as a resolution of singularities, a produ t
idP , where : (M; E ) ! (Rd ; 0) is a proper birational morphism outside 0 and
F Æ ( idP ) is in normal rossing. More pre isely, we mean that, in that ase, there
exist lo al systems of parameters (x1 ; : : : ; xd ) entered on a point in E P su h that
F Æ(
idP )(x1 ; : : : ; xd ; p) = up(x)
d
Y
i=1
xri ;
i
where up is a nonvanishing Nash fun tion.
In that setting, we have the following triviality result :
Proposition 7.21. Let F : (R d ; 0) P
! R be a Nash mapping, where P is
a onne ted Nash set and assume that fp = F (:; p) : (Rd ; 0) ! R has an isolated
singularity at 0 for ea h p 2 P . Assume moreover that there exists a proper birational
morphism : (M; E ) ! (Rd ; 0), with E the ex eptional divisor, su h that the
produ t idP is a resolution for F . Then the family F (:; p) onsists of a unique
blow-Nash equivalen e lass.
Remark 7.22. Note that the ex eptional divisor E is ompa t, whi h is ru ial
in the proof.
We prove the proposition in several steps. By assumption we dispose of a trivialization (the identity map) of the zero sets of F (:; p) for p 2 P . But there is
no han e that this trivialization (in fa t, the identity of M ) respe ts the levels of
F (:; p). A tually we are going to proje t the trivialisation of the zero level in order
to for e it to trivialize also the others levels.
We show rst that we an de ne lo ally the proje tion dire tly on M , without
blowing-up. Note Fp instead of F (:; p).
3. PROOF OF THEOREM 7.6
89
Lemma 7.23. Take x 2 M and p0 2 P . Then there exists x > 0 su h that the
orthogonal proje tion (x; p) of x onto the level fFp = Fp0 (x)g is well de ned for
jp p0j < x.
Proof. If x 2 E , then put (x; p) = x. Now, if x 2= E , then Fp (x) 6= 0 for all p in P .
But for 6= 0, there exists a Nash tubular neighbourhood ( f. [4℄, orollary 8.9.5) of
the level fFp0 = g. Therefore for p suÆ iently losed to p0 , x belongs to the Nash
tubular neighbourhood of fFp0 = Fp (x)g, and then (x; p) is de ned as the unique
orthogonal proje tion of x onto fFp0 = Fp (x)g.
Remark 7.24.
(1) Note that, in the proof, the level fFp0 = Fp (x)g moves when p varies. But
the fa t that Fp (x) 6= 0 for all p in P assures us that the width of the
di erent Nash tubular neighbourhoods does not tend to zero when p tends
to p0 .
(2) This proje tion is of Nash type outside E P by property of Nash tubular
neighbourhoods.
(3) Su h a onstru tion seems not to be so easy to perform in a global situation
be ause the width of the Nash tubular neighbourhood fFp0 = g, for a xed
fun tion Fp0 , tends to 0 as tends to 0. Therefore it does not seem to be
reasonable to hope for a global stri ly positive .
Now let us perform the omputation of the proje tion in the lo al ase. Note
that M , as a real algebrai variety, is aÆne, and so one an assume M RN . Note
d = dim M . Take x0 2 E M RN , and p0 2 P . For simpli ity assume that
p0 = 0. There exists a system of parameters entered at x0 su h that :
{ M is lo ally the graph (z1 ; : : : ; zN d ) = G(x1 ; : : : ; xd ) of a Nash fun tion G,
{ extending in the trivial way F Æ ( idP ) to a fun tion on RN , then F Æ ( idP )
is of the form
F Æ(
idP )(x; z; p) = up(x)
d
Y
i=1
fi (x)r ;
i
where up is a Nash fun tion in (x; p) whi h does not vanish and f1 ; : : : ; fd are
normal rossings fun tions.
For (x; p) 2 M P , the orthogonal proje tion (x; p) = X satis es :
8
<
x = X + rX F drX G
zi = Z i + d
if i 2 f1; : : : ; N dg
:
F (x; z; 0) = F (X; Z; p);
where rX G (respe tively rX F ) denotes the gradient ve tor of G (respe tively F )
at X .
Lemma 7.25. is a blow-Nash isomorphism from a neighbourhood of (0; 0) in
d
d
R P to a neighbourhood of (0; 0) in R P .
Proof. Let us prove the result in two steps :
(1) is blow-Nash,
(3)
90
7. AN INVARIANT OF THE BLOW-NASH EQUIVALENCE
Fig. 2. Lo al situation.
levels of F
M : z=G(x)
x
F
X
X*
z
n
X
x1 ,...,xd
(2) is a blow-Nash isomorphism.
First step. In order to prove the lemma, it suÆ es to show that the appli ation
(X; p)
/
(X + (X; p)rX F
d(X; p)rX G; p);
whi h is by (3) the inverse of , is a blow-Nash isomorphism between two neighbourhoods of (0; 0) in Rd P . Therefore we need more informations on and d !
But and d are given by impli it formulae, so we have in mind to apply the impliit fun tion theorem, whi h makes sense in the Nash setting ( f. [4℄). However the
omputation downstairs shows us that we an not apply dire tly this theorem, and
a tually we need to separate the divisors given by fi = 0 by several blowings-up, for
ea h i su h that ri 6= 0.
To begin with, let us expli it the members of the last equation of system (3) in
oordinates. The left one is
(4)
F (x; z; 0) = u0 (X +
rX F drX G)
and the right one is
F (X; Z; p) = up (X )
d
Y
i=1
d
Y
i=1
fi X +
ri
rX F drX G
fi(X )r :
i
Remark that the Taylor Formula applied to fi implies that
fi (X + H ) = fi (X )+ < rX fi ; H > +hi;X (H );
where hi is a Nash fun tion su h that
khi;X (H )k kkH k2
on some neighbourhood of 0, for some positive onstant k.
3. PROOF OF THEOREM 7.6
91
Therefore fi (X + rX F drX G) is equal to
fi (X ) + < rX fi; rX F > d < rX fi ; rX G > +hi;X ( rX F drX G):
Now, we separate the divisors fi = 0 by a su ession of d blowings-up with respe t
to ideal sheaves generated su essively by ea h fi . By symmetry, it suÆ es to perform
the omputation in one hart, and therefore one an assume this modi ation to
be given in the hart U by
(f1 ; f2 ; : : : ; fd ) = (Y1 ; Y1 Y2 ; : : : ; Y1 Y2 Yd ):
Let us denote respe tively by 0 ; d0 the modi ed form of ; d. Put
ei =
e(Y; p)
and
d
X
V=
i=1
ri (
j =i
d
Y
=
de(Y; p) =
d
X
i=1
rj ;
Yie
i
2
0;
d0
Qd
i=1 Yi e(Y; p)
d
Y
j =i+1
Yj )r(Y ) fi :
Remark that V an not vanish sin e the fi form a system of oordinates.
Then, after the modi ation, the term rX F is transformed in
d
Y
upe(
due to the lassi al formula
i=1
Yi )V + e(
d
Y
i=1
Yi )2 r(Y ) up ;
d
rF = rup + X
rf
r i:
F
up i=1 i fi
As a onsequen e fi(X + rX F drX G) be omes :
d
Y
Y1 Yi + e
where
i = up < r(Y ) fi ; V > +e(
+
d
Y
i=1
Yi i ;
Yi)2 < r(Y ) fi; r(Y ) up > de < r(Y ) fi; r(Y ) G >
i=1
Q
d
hi;X e( i=1 Yi )
Qd
i=1 Yi ) (Y ) up
Qd
e
i=1 Yi
r
V +(
der(Y ) G
Therefore equation (4) is transformed to
(5)
r
X
k=1
k
e
X
d
Y
j1 ++jd =k i=1
ji 2f1;:::;ri g
ji j j1 ++j
Y
ri i i
i
i
1
e Y );
= v(e; d;
:
92
7. AN INVARIANT OF THE BLOW-NASH EQUIVALENCE
with
up (Y )
eY) =
v(e; d;
Qd
u0 (Y ) + e( i=1 Yi ) V
1:
der(Y ) G
So e and de are given impli itly by the following equations :
(6)
e Y1 ; p)
E1 (e; d;
r
X
=
k=1
e
d Y
ji
X
k
ri
j1 ++jd =k i=1
ji 2f1;:::;ri g
ji Yij1 ++j
i
i
1
eY) =0
v(e; d;
and
(7)
e Y1 ; p) = de
E2 (e; d;
G (Y ) + (
d
Y
i=1
Yi )eV
de (Y ) G
r
G Æ (Y ) = 0;
where we onsider Y2 ; : : : ; Yd as parameters. Now we an apply the impli it fun tion
theorem. First, remark that E1 (0; 0; 0; 0) = 0 and E2 (0; 0; 0; 0) = 0. Now, let us
expli it the oeÆ ient of e in equation (6). This oeÆ ient is
d
X
i=1
ri (
d
Y
j =i+1
Yj )i a;
e Y ), thus the oeÆ ient of e equals
where a is the ontribution oming from v(e; d;
d
Y
upkV k2 + (
i=1
Yi ) < V; r(Y ) up > de < V; r(Y ) G >
Q
Qd
Qd
Pd
d
Y
)
h
r
(
j =i+1 j i;X e( i=1 Yi ) V + ( i=1 Yi ) (Y ) up
i=1 i
+
Qd
e
i=1 Yi
r
der(Y ) G
a:
Note that a tends to zero as (Y1 ; p) tends to (0; 0).
Now, it is easy to ompute the ja obian matrix of (E1 ; E2 ) with respe t to the
variables (e; de) at the point (0; 0; 0; 0). The result is :
e
e
; 0; 0; 0)
; 0; 0; 0)
E1
(0
E2
(0
e
e
; 0; 0; 0)
; 0; 0; 0)
E1
(0
d
E2
(0
d
!
=
up (0)kV k2 0
0
1
;
whi h is an invertible matrix be ause up and V do not vanish. Therefore e and de are
de ned and analyti in a neighbourhood of (0; 0) in M P .
Now, let us ome ba k to . For a xed p, write p (:) instead of (:; p). Then p
is de ned in a neighbourhood V (Rd ) of 0 in Rd , and in restri tion to a neighbourhood
of Y1 = 0 in the hart U , one has
p
d
Y
d
Y
d
Y
i=1
i=1
i=1
Æ (Y ) = Y1 + e( Yi)W1 ; Y1 Y2 + e( Yi)W2 ; : : : ; Y1 Y2 Yd + e( Yi)Wd ;
where W is the ve tor W = V
der(Y ) G:
3. PROOF OF THEOREM 7.6
Then
p
lifts in a fun tion
ep
:
V (U ) _
f_ _/
p
V (U )
V( d )
R
/
p
V( d )
between neighbourhoods of Y1 = 0 in U , with
d
Y
93
Q
R
ep
being given in oordinates by
1 + e( di=3 Yi )W2
1 + eWd :
= Y1 1 + e( Yi)W1 ; Y2
; : : : ; Yd
Qd
1 + eYd Wd 1
1 + e( i=2 Yi )W1
i=2
Note that the denominators an not vanish in a neighbourhood of (Y1 ; p) = (0; 0)
be ause e is small for (Y1 ; p) suÆ iently small.
ep (Y )
Se ond step. It suÆ es to prove that the ja obian determinant of e is non
zero for Y1 and p small be ause e is a bije tion. Indeed, e is a bije tion outside E
be ause so is , and restri ted to E , it is just the identity.
e , for i 2 f2; : : : ; dg, vanish when (Y ; p) = (0; 0). Therefore,
Note that e and Y
1
evaluated at (Y1 ; p) = (0; 0),
ei
ei
= 1;
=0
Yi
Yj
if i 6= j and j 6= 1. So the ja obian determinant of e equals 1 at (Y1 ; p) = (0; 0) and
thus is nonzero for Y1 and p small.
i
Remark 7.26. The proje tion onstru ted is lo ally the minimal one be ause it
is trivial if x 2 E or p = 0.
Now, we have to prove that these proje tions glue together.
Lemma 7.27. The proje tions of lemma 7.25 glue together.
Proof. Cover E by a nite number of neighbourhoods of the kind of lemma 7.25 using
the ompa tness of E . We an assume that the interse tions two by two of these
neighbourhoods are onne ted. Then, we dispose of a nite number of proje tions
i de ned on neighbourhoods of the form V i B (p0 ; i ), with i > 0. Denote by the minimal of the rays i and by U the union of the neighbourhoods. We are going
to prove that the proje tions i glue together on U B (p0 ; ).
Assume that there exists a point x in the interse tion V1 \ V2 . Then, on a neighbourhood of (x; p0 ) in M P , the three proje tions 1 ; 2 and the one of lemma 7.23
oin ide by minimality. Therefore the analyti fun tions 1 and 2 , whi h oin ide
on a nonisolated set of points, are equal on the onne ted set V1 \ V2 .
The proof of proposition 7.21 follows now easily from these lemmas. A tually,
it suÆ es to prove that two germs Fp and Fq are blow-Nash equivalent for p and q
suÆ iently losed, and the lemmas give a relevant blow-Nash isomorphism in that
ase.
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