Annales de l’institut Fourier (AIF)

Silverberg, Alice

Zarhin, Yuri G.

Semistable reduction and torsion subgroups of abelian varieties

Achab, Dehbia

ANNALES DE L’INSTITUT FOURIER
DEHBIA ACHAB
Représentations des algèbres de rang 2 et
fonctions zêta associées
Annales de l’institut Fourier, tome 45, no 2 (1995), p. 437-451
<http://www.numdam.org/item?id=AIF_1995__45_2_437_0>
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Ann. Inst. Fourier, Grenoble
45, 2 (1995), 437-451
REPRESENTATIONS DES ALGÈBRES
DE RANG 2
ET FONCTIONS ZÊTA ASSOCIÉES
par Dehbia ACHAB
0. Introduction.
Dans un travail précédent (cf. [l], [2]), nous avons défini et étudié
les fonctions zêta des représentations des algèbres de Jordan euclidiennes
déployées, pour lesquelles nous avons démontré la convergence dans un
demi-plan, établi un prolongement analytique et une équation fonctionnelle
scalaire.
De manière plus précise, soient V une algèbre de Jordan euclidienne
simple de rang m et de dimension n, E un espace euclidien de dimension
N, (/> une représentation auto-adjointe de V dans £', îî le cône symétrique
associé à Y, et G(Q.) le groupe des automorphismes du cône, c'est-à-dire
G(îî) = {g e GL{V) | g(Sï) = fî}.
Soit Q la forme quadratique vectorielle associée à <^, déterminée par
<Q(0,a;>=(^)$|0 vçe£;,Vrcey.
Nous supposons que les hypothèses suivantes sont vérifiées :
(ffi) : V, E et (j) sont définis sur Q,
Mots-clés : Algèbres de Jordan - Algèbres de Clifford - Groupe arithmétique - Fonctions
zêta.
Classification math : 20.
438 DEHBIA ACHAB
(^2) La Q-structure VQ de V est déployée, c'est-à-dire qu'il existe
un système complet de m idempotents primitifs deux à deux orthogonaux
dans VQ.
Soit F(())) le groupe algébrique défini par
F((/>) ={hç GL(E) | 3p(h) ç G(î2) | Q(hç) = p(h)Q(^ e E}.
Alors p : F(0) —> G(îî) est un morphisme de groupes algébriques
dont l'image est notée G{(J)) (cf. [l], chapitre 1). L étant un réseau dans
EQ, soit Fo le sous-groupe arithmétique de F(0) défini par
Fo = {7 € F((/>) | 7(L) = L}.
La fonction zêta CL associée à la représentation (j) et au réseau L a
été définie (dans [l], chapitre 1) par
000= ^ det(Q(l))-8 sçC,
içr^L'
où £ ' = = { ( ç £ | det(Q(Q) ^ 0}.
Si Fo est un sous-groupe d'indice fini de Fo, on peut aussi considérer
la fonction zêta
000= ^ det(Q(Q)-5 ^ e C .
tëro^
Les séries zêta ci-dessus convergent absolument pour Re(s) > —
(cf. [l], chapitre 1), admettent un prolongement analytique en tant que
fonctions méromorphes sur C, et vérifient une équation fonctionnelle (cf.
[l], chapitre 2).
Ces fonctions zêta ont déjà été étudiées dans les cas où V =
Sym(m,R) (fonction zêta de Kœcher, cf. [6]) et V = Herm(m,H) (cf. [7]).
L'objet de cet article est l'étude explicite du cas d'une algèbre de
Jordan de rang 2 qui donne lieu à de nouvelles fonctions zêta. Nous allons
notamment décrire les groupes F(0) et G(<^).
REPRÉSENTATIONS DES ALGÈBRES DE JORDAN DE RANG 2 439
1. Préliminaires.
a) Algèbres de Clifford.
Soit U un espace vectoriel réel de dimension finie n muni d'une forme
quadratique non dégénérée F, soit ci, 63,. . . , en une base orthogonale pour
F. On désigne par T(U) l'algèbre tensorielle de (7, et par Z l'idéal de T(U)
engendré par les éléments de la forme x 0 x — F(x).
L'algèbre quotient C(F) = T(U)/I s'appelle l'algèbre de Clifford
de F.
On a une injection canonique de U dans C(F). On sait que l'algèbre
de Clifford C(F) est l'algèbre associative engendrée par ei , . . . , en, sujette
aux relations
CjCk + CkCj =0 1 < j, k <: n j ^ k
ej = F(ej) • 1 Vj, 1 < j < n.
Pour une partie H de {1,2, . . . , n}, on pose
en = e/ii ... e^ où H = {/ i i , . . . , hq | hi < ... < hq}
avec 60 = 1. On démontre que les 271 éléments CH forment une base de
C(F) (comme espace vectoriel réel).
Dans C(F), on a trois involutions :
i) / définie par : Cy ^ —ey 1 < v < n, appelée automorphisme
principal.
ii) * définie par : e^ ... e^ i—^ e^ ... e^, appelée antiautomorphisme
principal.
iii) "définie par : a = (a')* = (û*)7 Va ç C(F)^ appelée involution
principale.
n
Les éléments de C(F) de la forme x = ^ a^Ci (où eo = 1) sont
1=0
appelés vecteurs.
Dans toute la suite on suppose -F définie positive et on s'intéresse aux
algèbres de Clifford associées à F et —F,
^=C(F), Cn=C(-F).
440 DEHBIA ACHAB
Remarquons que
i) C^ est l'algèbre associative engendrée par e i , . . . ,eyi sujette aux
relations
eJ = 1 1 < j <, n et ejCk + e^- =0, fc 7^ j .
ii) C^" est l'algèbre associative engendrée par /i , . . . , /nî sujette aux
relations
^=-1 Kj<netfjfk^-fkfj=^ k^ j .
L'espace des vecteurs de C^ sera noté Vn.
Pour tout entier 1 < p < n, on notera Vp l'espace des vecteurs de
l'algèbre de Clifford C^ = C(Fp), où Fp est la restriction de F au sous-
espace Up de Î7, qui est engendré par ei,. . . , Cp.
Un élément a de Cn est dit transformateur si pour tout x dans Vn,
il existe î/ dans Vn tel que arr = ya!. Tout vecteur est transformateur
et tout produit de transformateurs est un transformateur. Si a est un
transformateur, alors le nombre N(a) = ad est un réel positif et N(a) = 0
si et seulement si a = 0. De plus, tout transformateur a de C^ se décompose
comme suit en produit de vecteurs
a = aiû2 • • • un où dp ç. Vp pour tout p, 1 < p <, n
(cf. [8]).
b) Algèbres de Jordan euclidiennes de rang 2.
U étant comme ci-dessus, soit Vn = R (B U et notons -B la forme
bilinéaire associée à -F. Sur Vn, on considère les produits suivants :
(À, u) o (/^, v) = (À^LA + B(n, -y), \v + /^);
< (À, n); (/A, v) >= À/A -h B(îA, î;).
Munie de ces deux produits, Vn est une algèbre de Jordan euclidienne
de rang 2, de dimension n + 1 dont l'unité est e = (1,0) (cf. [3], [5]).
REPRÉSENTATIONS DES ALGÈBRES DE JORDAN DE RANG 2 441
2. Construction d'une représentation de Vn dans C^.
Pour tout ç- = ^H^HÎH dans C^, on associe canoniquement
l'élément $+ = ̂ H^CH de C^.
On peut alors réaliser l'algèbre de Cliffbrd Cn comme l'espace E des
matrices
î-fe £) c•-f'-e''"-
(C^_i étant la sous-algèbre de C^ engendrée par /i,..., fn-i)-
L'isomorphisme linéaire entre C^ et E est défini par
Ï : E -^ C^
^(^ f[)-a++^e»•
L'algèbre de Jordan V = Vn est l'espace des vecteurs de C^, c'est-à-dire
n
les x de la forme x = ̂  a^ei. Pour tout rr dans V, notons A l'élément de
i=0
E de la forme
. _ / XQ Xi 4- 3;2/1 + . . . + Xnfn-l \
\Xl- X^fl - ... - Xnfn-l Xo )
la correspondance a; i-^ est un isomorphisme.
Soit (f> l'application définie par
(j) : V -^ End(E)
x i—^ (f>(x) : ç ^—> x^
où ^$ est le produit matriciel des éléments de E.
2.1. PROPOSITION. — ( / ) est une représentation de Palgèbre de Jordan
V dans E.
Preuve. — II est clair que (j> est linéaire. Soient a;, y ç. V, vérifions que
^y) = JW^O/) +00/)<^)).
Cela revient à montrer que
xy= l.(xy-^yx)
442 DEHBIA ACHAB
n n
ce qui est vrai car, si x = (xo, u) et y == (î/o, zQ où u == ̂  a;,e^ et v = ̂  î/,e,,
on a i=l ^=l
/ n \
xy = ( S awtî a:oî; + î/ot& )
\i=0 )
et alors
^ 1 (_ , _\
xy = 2 ̂ 2/ 2 / /
,fa- ^\
V^ a'J
n
où a- = ̂  Xiyi, f3- = (a;o2/i + VoXi) + ... + (a;o2/n + yoXn)fn-i'
i=0
(Le produit des coefficients des matrices de E étant celui de Cyi_i.)
D
Pour tout ? = ( a/ ^/ ) ' a ^ / 3 e ^n-i» on P°se
^ - ^ - ( a y}^^ ~{f3 a ' ) '
On vérifie facilement que pour $,77 € .E, on a ($77) = yyç et alors
l'application ç i—^ $ est un antiautomorphisme de E.
Pour tout $ = E^ejf, ou ç = E^/H on note Re(Q = $0.
On considère sur E le produit scalaire
(ç |r7)=|ReTY(^).
Vérifions que c'est effectivement un produit scalaire. Il est clair que
c'est une forme bilinéaire. Elle est symétrique car
Re(fIY(^)) = Re(Tr(^)) = Re(Tr(^)).
Elle est définie positive car
(ç|0=Re(aa+/^)
^^(-l^'^et^^^l)^^11^
REPRÉSENTATIONS DES ALGÈBRES DE JORDAN DE RANG 2 443
donc
- |H|(|H|+1)
ÎH = (-1) 2 fH
__ __ 1 .ffl f 1 Jîl-H)
si a = Y^OLHÎH est dans C^_i, alors a = ̂ ajï(-l) 2 y^ et alors
Re(aa)=^a2H. D
2.2. PROPOSITION. — Là représentation (f) est âutoadjointe pour ce
produit scalaire.
Preuve. — Montrons d'abord que pour ç, 77 € £', on a
Re(T^))=Re(T^O);
en effet, si $ = ( a ^ ) , ^ = ( // î ) 5 alors
Tr(^) = aÀ + /ÎAA' + /3^ + a'X1 et Tr(^) = Xa + ̂ /3' + ̂ '/î + ÀW.
Il ne reste plus qu'à montrer que Re(Àa) = Re(aA),VA,a G C^_^ ce
qui est vrai car si a = ̂  C^H/H et A = ̂  Ajy/jf, alors
Re(aA) = ̂ a^Ajf = Re(Aa).
Montrons maintenant que la représentation est auto-adjointe, en effet
(<^)$ 1 ri) = iReTï(^) = JRe^^^)
=jRefIY($(i77))=(ç |^)77). D
Revenons au produit scalaire sur V : pour x ^ y ç . V o u 8 L < x ^ y > =
n
^ XiVi. On vérifie facilement que
i=o
< x, y >= (x \ y) pour tout rc, y € V.
Soit Q la forme quadratique vectorielle associée à (j> ; Q : E —>• V est
déterminée par
< Q(Q,^ >= (^)$ | QVa; e v,$ e £?.
444 DEHBIA ACHAB
Or
< Q(^x >== (Ç(0 | x) = ̂ ReTr(Q(^)x)
et
(<^)$ 1 0 = JReTY(^) = JReTY(^).
Ceci ne nous permet pas de conclure que Ç($) = $$ pour tout $ dans
E car çç n'est pas toujours un vecteur de C^".
Notons Vect(77) la partie vecteur pour tout rj dans C^ et soit p la
projection de E sur V définie par :
Pour tout $ = ( , , ) dans -E', on lui associe p($) dans V tel que
?($)=
/ Re(a) Vect(/î)\
YVect(^) Re(a) )
alors
Q(0==p(rô.
3. Le groupe Fyi.
Dans ce qui suit, nous explicitons le sous-groupe arithmétique associé
à (f> et à un réseau L de E.
Nous commençons d'abord par décrire les groupes FÇcf)) et (?(</>)
définis dans l'introduction.
3.1. PROPOSITION. — L7 ensemble Fn des matrices
( , , ) , a,/? transformateurs dans C^_i tels que
ISa' ç Vn-i et N(a) - N(f3) ̂  0
est un groupe. Il est engendré par les (f>{x), x inversible dans V.
Preuve. — Remarquons d'abord que si /3a' 6 Vn-i, alors son conjugué
Û//3 e Vn-i et alors l'image de ce dernier par /, oft' ç Vn-i-
REPRÉSENTATIONS DES ALGÈBRES DE JORDAN DE RANG 2 445
n
D'autre part, si x = ^ xiCi est dans Vrn alors x est inversible si et
1=0
seulement si
q(x)=x2o-xï-...-x2,^0.
Soit x ç. Vn inversible, alors x appartient à Fn^ en effet
x = ( , ' ) telle que a = a;o et /3 = x\ + a?2ei + ... + a^en-i.
II est clair que /3a* ç. Vn-i- De plus
AT'(a) - N(f3) = ̂  - x\ - ... - ̂  ̂  0 car a; est inversible.
Soient maintenant rr, î/ € Y inversibles et montrons que xy € Fn ; en
effet,
_ fxo x\fyo y\ ( a /3\
^x .oAy yJ=^ -'>
tels que
x = a;i + ... 4- Xnfn-i, oi .= xoyo + xy, /3 = a:oy + 2/ox.
Vérifions que a/3* 6 Vn-i î en effet
a/3* = x^yoy + a;o^x + a;oxyy + 2/oxyx,
il est clair que les trois premiers termes sont dans Vn-i, quant au dernier,
il existe z dans Vn-i tel que î/oxyx = yoz-^x et appartient donc à Vn-i-
De plus, on montre que
N{a) - N(f3) = (^ - N(x))(yï - N{y) = q{x)q(y) + 0.
Nous allons maintenant montrer que si $ = ( ^ , j € Fn^x € Vn
inversible, alors xç ç Fn. En effet,
/^o x \ / a ( 3 \ _ f > ^\
^-{x x^ ) \ l 3 1 a')~\^ \ 1 )
tels que À = rcoa + x/3' et ^ = a;o/3 + XQ!'.
^ DEHBIA ACHAB
II est clair que ̂  et À sont des transformateurs de C__,, montrons que
fzX* çVn-i. En effet,
^À* = a /̂?û'* + a;oJV(/3)x + xoxN(a) + xa'ftx
les trois premiers termes sont dans Vn.i car /3a* et x le sont, quant au
dernier terme, comme a'0 est dans Vn_i, alors il existe y dans V»_i tel
que xa'/îx = N(x)y et appartient donc à Vn-i.
De plus, en utilisant le fait que aff = (3d' et /3'a = a'0, on montre
que
N(\) - N(^) = (^ - N(x))(N(a) - N(/3)) = q(x)(N(a) - N(/3)) ̂  0.
Nous avons donc montré que le groupe engendré par les (f>(x),x
inversible dans V, est contenu dans Fn. Il reste à montrer que tout ç de Fn
s'écrit comme produit d'un nombre fini de x, x inversible dans V.
Pour cela, nous allons utiliser le fait que tout transformateur a de
€„_! se décompose a = a^z... a^-i où dp ç Vp, pour tout l^p^n-1.
On a : a = ai... a^_i et ^ = ̂ i.../3^_i tels que ap,/3p ç Vp. Donc
a = aan-i et (3 = b/3n-i où a,b e î^ (T^ désignant l'ensemble des
transformateurs de €„).
^ Comme N(a) - N((3) ^ 0, alors N{a) et 7v(/3) ne sont pas simul-
tanément nuls.
1) si N(a) / 0, comme N(a) = N(a)N(an,i) alors N(a) et
N(an-i.) sont non nuls. De plus,
( a ( 3 \ = ( a ov^-i o \( 1 ^^
{(3' a ' ) {O a)[ 0 a^)\J_ ^
\N(Q)ap 1 ,
et f 0 °\=(^ ^ (^-2 0 \
\0 a ' } \0 aj--\ 0 &n,J'
On a donc montré, dans le cas où N(a) ̂  0, que l'élément ( a, /3\\
de Fn est dans le groupe engendré par les (f>(x),x € V, inversible.
2) Si N(a) = 0, alors a = 0 et notre matrice s'écrit ( ° 13\ avec
AT(/î)^O.Ona v/3 0/
( 0 0 \ ( 0 b\(^_, 0 \ /o b\ _
l/?' o;-(,y oj(, o ^_J avec [b' oj6^-1-
REPRÉSENTATIONS DES ALGÈBRES DE JORDAN DE RANG 2 447
Ce qui termine la preuve de la proposition. D
Le groupe Fn opère sur E par multiplication à gauche
$^^ V^eFn, $eE.
On a
\fg e Fn, 3 p(g) e G(^) tel que Q(^) = p(<y)Q(0, V$ e E
îî désignant le cône symétrique associé à l'algèbre de Jordan Y, c'est-à-dire
le cône de Lorentz et G(îî) étant le groupe des automorphismes du cône
défini dans l'introduction.
Pour g e Fn,p(g) opère sur V par
x ̂  gxg pour tout x dans Y;
en effet, pour tout x dans V et tout ç dans £', on a
(W) 1 x) = ̂ ReTr(Q(^)x) = JRel̂ )̂
{gW9 1 ^) = |ReT^Ç(0^) = jRefIY(Ç(î)^)
== JReTY^^) = ̂ ReTr(g^gx)
ce qui implique que (Q(g^) \ x) = {gQ{Ç)g \ x) et alors Ç(^) = gQ(Ç)g.
Notons Gn le sous-groupe de Fn formé des g = ( ^, , ) tels que
^V(a) - JV(/3) = 1. On sait, (cf. [8]), que le groupe Gn est un revêtement à
deux feuillets du groupe orthochrone 50o(l,n) et p(Gn) = S0o(\^n).
Pour tout g = ( a ' ] dans Fn, notons A(p) = N(a) - N(l3). Si
Fn désigne le sous-groupe de Fn formé des g tels que A(^) > 0, alors tout g
dans F^" s'écrit g = A(^) 2 ^i où g\ est dans Gn. Par suite, si on note p(g)
la transformation de Y qui à a; dans V associe y dans V tel que y = gxg,
on montre que pour tout g dans F^p{g) appartient à G(fî), car
G(îî) = R:̂  x An+i où An+i = 5'0o(l,n) U /i5'0o(l,n)
448 DEHBIA ACHAB
h étant la matrice diagonale d'ordre n 4-1 dont les n premiers termes sont
égaux à 1 et le dernier à -1. Remarquons que h conserve le cône de Lorentz,
et Det(fc) = -1.
La composante connexe G(fî)° de Pidentité de G(îî) s'écrit
GW° =R4. x6'0o(l,n).
Nous avons donc
p(F^)=GW°
et d'après la théorie générale (cf. [1] chapitre 1), on a montré que si FÇcf))0
est la composante connexe de l'identité de F((I)) (pour la topologie usuelle),
alors
p(FW°) == GW°
donc
p(F^)=p(F(0)°).
Nous avons déjà vu que pour tout g dans F^, il existe p(g) dans
G(î2), opérant sur V comme ci-dessus, tel que
Q(^) = PW{Ç) Pour tout $ dans E.
Le groupe F^ est donc contenu dans le groupe F((f>).
3.2. PROPOSITION. — Le noyau Ker(p) du morphisme p : F((J)) —>
G(îî), est un sous-groupe du groupe orthogonal 0(E) et F((f))° s'écrit
F{(t>)0 = F^K où K est un sous-groupe fermé de Ker(p).
Par suite, si L est un réseau de E, alors le sous-groupe Fo de F^
défini par
r o = { 7 e ^ l 7 W = ^ }
est d'indice fini dans Fo = {7 e F((f)) \ ̂ (L) = L}.
Preuve. — Soit g tel que p{g) = id, alors Q(g^) = Ç($) pour tout ç
dans E, donc
ReTr(g^g) == ReTr($0
REPRÉSENTATIONS DES ALGÈBRES DE JORDAN DE RANG 2 449
ce qui implique
{9^ 1 9^) = ($ 1 0 pour tout $ dans j&
<7 est donc bien dans le groupe orthogonal 0(E).
Comme p(F((f))0) = p(F^), alors les quotients F(ç'))°/F(^)0 nKer(^)
et F^/F^ nKer(/?) sont isomorphes. Par suite, comme le noyau Ker(p) est
compact, alors Fo est d'indice fini dans le sous-groupe
Fi = {7 6 F(0)° | 7W = ̂ }.
De plus, comme F((j>) est un groupe algébrique (cf. [1] chapitre
1), alors, d'après un théorème de Whitney, il admet un nombre fini de
composantes connexes pour la topologie usuelle, il s'ensuit que F((f))° est
d'indice fini dans F((p) et alors Fi (donc Fo) est d'indice fini dans Fo. En
effet, notons /i i , . . . , hk un système de représentants de F((^)/F(0)0 et soit
7ero.
Si 7 est dans F((f))° alors 7 ç Fi et alors 7Fi = Fi.
Si 7 est dans un certain hiF((j))° alors il existe 7^ G F{(I))° tel que
7 = fa^^. Alors
Fo/Ti = {Fi, fai7iri,.... hk^i}',
en effet, remarquons d'abord que les 7^ vérifient 7i(2y) = /^(L). Soit
7Fi ç Fo/ri tel que 7 ne soit pas dans Fi, alors il existe 1 < i < k et 7^
dans F((f))° tels que 7 = h^ et on a
7z(L)=7^)=\-1^)
donc 7^-17^ € F(()))° et 7^~17^(L) = L et alors 7^-17^ est dans Fi, ce qui
entraîne 7^1 = 7^1. D'où
7Fi = hi^Ti = /in î
et Fo/ri est fini. D
Nous en déduisons que Fo est un sous-groupe arithmétique de F ((/>).
450 DEHBIA ACHAB
4. Fonction zêta de la représentation 0.
Prenons le réseau M = ^ Ze^ dans l'algèbre de Clifford
H(='P({l,..,n})
Cy{' et considérons son image L = ^"^(M) dans -E. Nous obtenons un réseau
dans E.
Nous avons vu ci-dessus, que la forme quadratique vectorielle Q
associée à (j) est telle que pour tout ç dans -E, Q($) est en quelque sorte la
partie-vecteur de ç$.
Plus précisément, si ç = ( , , ), alors
^- / Re(aa+/^) Vect(c^'+/3a')\
^vç/ Y Vec^a/î7 + /îa7) Re(aa + /3/3) /
Si on note det le déterminant de l'algèbre de Jordan Y, qui est donné
par
det(a;) = x^ — x^ — ... — x^
alors
det(Q(Q) = Re(aa + ̂ )2 - JV[Vect(a^ + /îa')] = ||ç||4 - ̂ (e^ | Q2.
2=1
Finalement, la fonction zêta associée à la représentation (j) et au réseau
L est donnée par
W= ^ W-i^lO2]-5 ^ c.
fefoYL7 ^::=l
n
Soit VQ la Q-structure de V définie par VQ = ̂  Qe^. La représentation 0
2=0
est définie sur Q et on a le théorème suivant :
4.1. THÉORÈME. — La série CL^) converge absolument pour Re(s) >
2n-2, elle admet un prolongement analytique en tant que fonction méro-
morphe sur C et vérifie V équation fonctionnelle
C.(2- - .) = vol(L)^-1-- ̂ ^-^W
où r^ est la fonction gamma du cône de Lorentz.
REPRÉSENTATIONS DES ALGÈBRES DE JORDAN DE RANG 2 451
Preuve. — La convergence est une conséquence du théorème 1 de [l],
chapitre 1 (on remarquera que dans ce cas, VQ est déployée car la forme
quadratique
\ _ 2 2 2
X ) — XQ — X-^ — . . . — X^çC
admet des solutions rationnelles non triviales, cf [l], chapitre 4). Quant
au prolongement analytique et l'équation fonctionnelle, ils découlent du
théorème 2 de [l], chapitre 2.
BIBLIOGRAPHIE
[1] D. ACHAB, Fonction zêta des représentations des algèbres de Jordan, thèse
Université Paris 6, 1993.
[2] D. ACHAB, Fonction zêta d'une représentation d' algèbre de Jordan, C.R.A.S,
Paris, t. 316, Série 1 (1993), 977-982.
[3] J.L. CLERC, Représentations d'une algèbre de Jordan, polynômes invariants et
harmoniques de Stiefel, J. reine angew. Math., 423 (1992), 47-71.
[4] J. FARAUT et A. KORANYI, Analysis on symmetric cônes (preprint).
[5] J. FARAUT et G. TRAVAGLINI, Generalized Bessel Functions, Journal of Funct.
Analysis, 71, No. 1 (1987).
[6] M. KŒCHER, Uber Dirichlet-Reihen mit Funktionalgleichung, J. reine angew.
Math., 192 (1953), 1-23.
[7] A. KRIEG, Kœcher-Maass- Séries for Modular Forms of Quaternions, Manuscripta
Math., 66 (1990), 431-451.
[8] R. TAKAHASHI, Série discrète pour les groupes de Lorentz S0o(2n, 1), Colloque
sur les fonctions sphériques et la théorie des groupes, Nancy, 1971.
[9] P.L. WATERMAN, Mobius Transformations in Several Dimensions, Adv. Math.,
101 (1993), 87-113.
Manuscrit reçu le 9 mai 1994.
Dehbia ACHAB,
Université de Paris VI
Mathématiques
4, place Jussieu
75252 Paris Cedex 05 (France).


Représentations des algèbres de rang 2 et fonctions zêta associées

Nakajima, Haruhisa

Equidimensional actions of algebraic tori

Youssfi, Abdellah

ANNALES DE L’INSTITUT FOURIER
ABDELLAH YOUSSFI
Regularity properties of commutators and
BMO-Triebel-Lizorkin spaces
Annales de l’institut Fourier, tome 45, no 3 (1995), p. 795-807
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Ann. Inst. Fourier, Grenoble
45, 3 (1995), 795-807
REGULARITY PROPERTIES OF COMMUTATORS
AND BMO-TRIEBEL-LIZORKIN SPACES
by Abdellah YOUSSFI
1. Introduction.
Let b be a locally integrable function and 1 < p < +00. It is a well
known fact that the commutators {[b,Rk])i<,k<,n are bounded on L^R71)
if and only if b € BMO^ where (Rk)i<,k<n are the Riesz transforms in
the n-dimensional euclidean space W1. This result is due to Coifman-
Rochberg-Weiss [5], and extends to the Hardy space 7Y1 in several variables
certain well known factorization theorems on the unit disk. In [4] this result
gives the regularity of various nonlinear quantities (like the jacobian, "div-
curl"...) identified by the compensated compactness theory.
Another interest of commutators comes from the pseudo-differential
calculus and the theory of singular integral operators (see [7]). In particular
in the case when b is Lipschitz, the commutators ([b^Rk\)k are bounded
from L2 to the homogeneous Sobolev spaces H1. This result is due to
Calderon [3] and has been generalized to the commutator [&, T] when T is
a reasonable Calderon-Zygmund operator.
Our purpose here is the study of the intermediate case, in particular
we give necessary and sufficient conditions for the boundedness of ([6, Rk}k
from Lp into the Besov space B^P for 1 < p < +00 and 0 < s < 1.
Key words : Commutators - Besov spaces - BMO.
Math. classification : 42B - 46E - 47B.
796 ABDELLAH YOUSSFI
Notice that our result is similar to the Murray's theorem [8] when the
Riesz transforms Rk is replaced by the Riesz potential operator of order s
and p = 2.
The paper is organized as follows. In Section 2 we give the statements
of our results. In section 3 we establish some results which are related to the
boundedness of the paraproduct. In section 4 we prove our main results.
In the sequel, C will denote a constant which may differ at each
appearance, possibly depending on the dimension or other parameters. The
symbols / will stand for the Fourier transform of / and / for inverse Fourier
transform of /. We also use the notations :
V^) = space of C'°°-functions with compact support, P^R71) its dual.
^(R71) = the space of Schwartz test functions.
(S^R71) = space of tempered distributions.
[s] = the greatest integer smaller than or equal to s and s* = s — [s].
2. Definitions and statements of the main results.
Let (p C S^W) be supported in the ball |̂ | <, 1 and satisfy <^($) = 1
for |$| < ^. The function
^(0=^/2)-^)
is (7°°, supported in
{ J < 1 $ 1 < 2 }
and satisfies the identity
][>(2-^)=1
jez
for $ 7^ 0. We denote by Aj and Sj the convolution operators with symbols
^y^"-7^) and y^"-^) respectively.
For s e R, 1 < p <. +00 and 1 <: q <: +00 the homogeneous Besov
space is defined by
imiB^-lE^ii^ii^
jez
with the usual modification if q = +00.
REGULARITY PROPERTIES OF COMMUTATORS 797
For s € M, 1 ̂  p < +00 and 1 < g ^ +00, the homogeneous Triebel-
Lizorkin space F^'9 is defined by
11 /11^ -lllE2^!^!9]'!
jez p
with usual modification if q = +00. To give the definition of the BMO-
Triebel-Lizorkin space I%9, let us first recall the definition of Carleson
measures. We shall say that a sequence of positive Borel measures (^j)jez
is a Carleson measure in R71 x Z if there exists a positive constante C > 0
such that
^(B)<^C\B\
j>k
for all k G Z and all euclidean balls B with radius 2"^, where \B\ is the
Lebesgue measure of B. The norm of the Carleson measure v = (vj)j^z is
given by
\M\=^p{—^Wv i^
where the supremum is taken over all A; € Z and all balls B with radius
2-^.
The homogeneous BMO-Triebel-Lizorkin space F^1 (1 < q < +00) is
the space of all distributions b for which the sequence (2s:)q\^j(b)(x)\qdx)j
is a Carleson measure (see [6]). The norm of b in F^ is given by
n6!!^—811? 1-1 E I^ y^wxwdx
1 1 ^ > k J B
where the supremum is taken over all k G Z and all balls B with radius
2~k. For q = +00, we set F^°° = B^00. In the inhomogeneous case, the
BMO-Triebel-Lizorkin spaces were studied by different methods in [12].
When 9 = 2 , the space F^2 is the Sobolev space (1 <p< +00) and the
space F^2 is, modulo polynomials, the BMO space. More generally, F^2
is, modulo polynomials, the BMO-Sobolev space considered by Strichartz[ii].
Note that the spaces F^ and B^ consist of distributions modulo
polynomials. We consider now the operator
7Z(/)=^A,(/).
jez
798 ABDELLAH YOUSSFI
Notice that, when / e ^(R"), / = 7Z(/) in ^'(M71). But this equality does
not hold for / is a polynomial.
Let s C M, 1 < p, q < +00 and let m = [s - n] and denote by A5'9
either F^ or B^. Then for all / € A;'9 one can show that
/=$>,(/)
j'ez
modulo polynomials of degree less than or equal to m. In particular
U(f) € S^R^/P^ where S'^/Prn is the space of distributions modulo
polynomials of degre less than or equal to m. Moreover, K is an embedding
from Ay to S'^/Prn' For / e A;'9, we set
\\f\\n^}=\\f\\A^
The space 7^(A^9) is isomorphic to A^9 and invariant by dilations and
translations. Throughout the paper we identify A8^ with K(A8^). For the
study of these realizations the reader is referred to [2]. In the particular
case where 0 < 5 < - , A = F and q = 2, we have
P
^(^^(^(r1))
where
WW-j^-.-iy
is the Riesz potential.
The paraproduct of J.-M. Bony between two functions /, g is defined
by
^J)=7r,(/)=^A,(^-3(/).
JCZ
It is a well known fact that, for b € ̂ °°, TT^ is bounded on L2 if and only
if b € F^2 = BMO.
Our main results are the following.
THEOREM 1. — Let s € R, b G B^°° and 1 < p < +00.
1) If be F^f, then 7^ is bounded from L^ into B8^.
2) IfTTb is bounded from B^ into B^, then b € F^.
Notice that B^ c F^ C LP for 1 ̂  q < 2, then we obtain the
following.
REGULARITY PROPERTIES OF COMMUTATORS 799
COROLLARY 1. — Let s € M, b C B^°°, 1 <p< +00 and 1 < q < 2.
Then b G F^ if and only if the operator TTb is bounded from F0^ into B8^.
THEOREM 2. — Let 0 < s < l , l < p < +00, q = inf(p, 2) and b be
a locally integrable function.
1) If b C ^p, then the commutators ([b^Rk])i<k<n o-re bounded
from F^ into B^P.
2) If the commutators ([b^Rk])i<k<n are bounded from B^ into
B^P, then b € F^P.
COROLLARY 2. — Let 0 < s < 1, 1 < p < +00, 1 < q < inf(p,2)
and b be a locally integrable function. Then b € F^ if and only if the
commutators ([b^Rk])i<k^n 0're bounded from F^ into B^.
3. Preparatory results.
The connection betwen the spaces F^ and B^ is given by the
following.
PROPOSITION 1. — Let 1 < p, q < +00 and s € R. Then
1) F^CB^ ifp<q;
2) B^CF^ ifq<p.
The spaces F^ and B^ are independent of the choice of ^. This is
due to the following lemma [13], [14], which will be needed herein later.
LEMMA 1. — Let s C R, 1 < p,q ^ +00 and 7 > 1. For any
sequence (/j)j'ez of functions such that for each j, fj is supported in
{^~1^3 < l^ l < 72 '̂}, we have
IIE^L^^IE^II^II^
j'ez p j-cz
and for 1 <: p < +00,
llE^IL^OE^w]^
j'ez p j-ez p
where C = C(n, p, 9,5,7).
800 ABDELLAH YOUSSFI
For s > 0 we can replace {7~12J ^ |$| < 72-7} by the ball {|^| ^ 72-7'}.
LEMMA 2. — Let s > 0, 1 < p, g ^ +00 and 7 > 1. For any sequence
(/j)jez of functions such that for each j, fj is supported in {|$| <; 72^}, we
have
lE^^E2^!^
"jez "^ S-ez J
where C = (7(n,p,^,s,7).
Now we consider the boundedness of the paraproduct operator on
Besov spaces. One of the most useful characterizations of the boundedness
of the paraproduct on Besov spaces comes from the following.
PROPOSITION 2. — Let b e B^°, s e R, 5i e M and 1 ̂  p, ̂ , ̂  <
+00, gi < ^2- Then
1) TTfc is bounded from B^'91 to ̂ +sl'92 if and only if there exists
C > 0 such that
[E^^^n^w^^a)!!?]^ < ^ii/iia3^
jez p
for an f C B^.
2) The commutator [TT;,,^] is bounded from B^1'91 to B^31^2 if
and only if there exists a constant C > 0 such that
[E^^^II^W-^K5^^))!!?]^ < ^11/11^—
J€Z
for an /C^1 '9 1 .
Proof. — Proposition 2 is due to the author [15] for the case 5 = 0
and «i > 0. To obtain extension for the general case we observe that
^^^We^00.
j'ez
Using this remark we are led to establish Proposition 2 for s = 0 and «i < 0.
Indeed, by Peetre's Theorem ([9], p. 155-158) one has that, for 5i < 0,
[^y^ll^-sWII^ <G||/||^.
j'ez
Hence we obtain the "only if part" ; for the "if part" we use the Lemma 1.
REGULARITY PROPERTIES OF COMMUTATORS 801
Remarks. — 1) Notice that for si < 0 the operator TT;) is bounded
from B^ to ̂ +sl'92 if and only if b e B^.
2) In Proposition 2, one can replace Rk by polynomials of the Riesz
transforms.
The first approach in dealing with the commutators consists of
Lemmas 3 and 5 below. The Lemma 3 is classical and due to Calderon
[3].
LEMMA 3. — Let h be a differentiable function and suppose that
r\
-—h € L°° for i = 1, ...,n. Then the commutator [h^Rk] is bounded from
dxi
L^ into the Sobolev space F^2 (1 < p < +00) and
11[^W)11^ < c?[f: 1^11 J.H/llp.
The following lemma related to the Bernstein's inequality, is classical
and will be used in the proof of Lemma 5 below.
LEMMA 4. — Let 7 > 1 and 1 <: p < +00, then for all R > 0 and
each function f such that f is supported in
{^7-1^1^7}
we have
ciRm? < E|̂ i|| ^ ^n/iip
where Ci = (7(7,p), i == 1 ,2 .
LEMMA 5. — Let b € B^°°, 1 < p < +00, 1 <, q < +00, 0 < s < 1,
N € N and Si e] — 5, +oo[, then the operator
T(f)=^[S^N(b),R^(f))
3
is bounded from B!p1^ to B^81^.
Proof.— Since the Fourier transform of [^+^v(6),J?fe](Aj(/)) is
supported in the ball {$, |$| < 2•7+J?V+2} and s + s-i > 0, then by Lemma 2,
it suffices to show that
^S^^^II^.^W.^KA^/))!!^ < c\\f\\^
3
802 ABDELLAH YOUSSFI
and it is enough to prove that
\\[S^N(b)^R^(fmp < G2-^||A,(/)||^
But
J'+N
S^{b)=S^(b)+ ^ A,(6)
^=J-2
and
||[A^)^](A,(/))||p <, C72-^||&||^oc||A,(/)||^.
Thus we are led to show that
||[^-3W^](A,(/))||p < C2-^||A,(/)||^
To do so, we observe that the Fourier transform of [Sj-sW,Rk](/^j{f)) is
supported in
{2J-2 < |^| < 2^2}.
By virtue of Lemma 4, we obtain
||[5,-3(6),^](A,(/))|^<C2-^||^([^,_3(&),^](A,(/)))HL^j-aw^^j^j
Indeed
1 1 ^ < G2 ^^^ii^—u^-s^^ /cj^^
._- ' i '~"^"i i ip
^-(^•-3(6)) ^ < C2^-5)19^
so that by Lemma 3 we see that
—([^_3(6),^](A,(/)))|| ^ ^^-^IIA.^IIp.
L/J/^ Up
This completes the proof.
We recall now the definition of the maximal function related to a
sequence of measurable functions. For a sequence of measurable functions
F = (/j)jez m W1, the maximal function F* is defined by
F^x) = sup{|^Q/)| /\x -y\< 2-^j e Z}.
The Carleson measure and the maximal function are related by the
following classical lemma (see [1]).
LEMMA 6. — If (^)jez is a Carleson measure in R71 x Z and
p c]0,+oo[, then
^/'I^^I^^^CIII^III.IIF*!^
1-cy. J
REGULARITY PROPERTIES OF COMMUTATORS 803
4. Proof of the main results.
4.1. Proof of Theorem 1.
To prove "part 1)'\ notice that F^P C %°°. By Lemma 1 we only
need show that
[^2^||A,(&)^-3(/)||^ <C||/||,
jez
for all f eS^).
Denote by F = (Sjf)j and observe that F*(x) ^ Cf^x) where /*(a;)
is the Hardy-littlewood maximal function. By virtue of Lemma 6 we obtain
that
[^2^||A,(6)5,-3(/)||^ ^ C\\r\\, < C\\f\\,.
jez
Next we prove "part 2)". Proposition 2 implies that
[^2^||A,(^-3(/)|d' <C\\f\\^.
j'ez
Let B = (a;o,2-^) (^ e Z) and let / € ^(R71) be such that f(x) = 1 for
\x\ < 1. Set
fB{x)=f(2\x-xo)).
For j > t one has that
sj<<fB){x) > C > 0 for all x <E B.
Thus
^2^ /* |A,(&)(^)|^ < C||/B||^o,i < C|B|,
j>^ JB p
from which the theorem follows.
4.2. Proof of Theorem 2.
To prove "part 1)", we write
[b.Rk](f)=[^Rk}{f)^Ak{f)
where
^(/)=E^+2W,^](A,/).
j'ez
804 ABDELLAH YOUSSFI
By virtue of Lemma 5, Ak is bounded from B^ to B^ ; in particular, Ak is
bounded from ̂ 'lnf^2) fo ̂ 'p. On the other hand, Theorem 1 guarantees
that TTb is bounded from ^'mf^'2) fo ̂ 'p. Since the Riesz transforms are
bounded on B^ and on F^'9, it follows that the commutators ([b,Rk})k
are bounded from p^^ into B^P.
To prove "part 2)", we use the following lemma.
LEMMA 7. — Let 0 < s < 1, 1 < p < +00 and 6 be a locally
integrable function. If the commutators ([b,Rk])i<k^n are bounded from
By toB^P^thenbeB^.
Let us first assume Lemma 7 and prove "part 2)". As in [5] we shall
make use of spherical harmonics. Let (Ym)m be an orthogonal basis for
the space of spherical harmonics of degree n. There exists a finite sequence
(Ym)m (see [10], p. 137-145) such that {Ym)m are homogeneous polynomials
of degree n; i.e. Ym(x) are of the form
Ym{x) = ̂  C^
H=n
and satisfy ^(^(^
2^ M2n - G"-
m ' '
In particular the operator Tm of kernel
Ym(x-y)
\x - y^
is a polynomial in the Riesz transforms and
Ym(x-y)= ^ C^mX^.
I^H/3|=n
Suppose that the commutators ([b,Rk})k are bounded from B°'1 to B8^.
By virtue of Lemma 7, we see that b e %°°. Lemma 5 implies that the
operator
Ak(f)=^[S^(b)^R^f)
jez
is bounded from B^ to B^. Hence the commutators ([jTb,Rk})k are
bounded from B^51 to B^. Indeed, a little computing shows that
[^RkRk'} = [^Rk}Rk'+Rk[^Rk'}'
Hence [^b'>Tm} is bounded from B^1 to ̂ 'p.
REGULARITY PROPERTIES OF COMMUTATORS 805
For £ e Z and a ball B with radius 2~^, we prove that b satisfies
^2^ f \^(b)(x)\Pdx<C\B\.
j>e J B
Let / e ^(R71) be such that f(x) > C > 0 for |a;| < 1, / supported in
{1^1 <^} and
ff(x)dx=l.
For any ball B = B(xo, 2-^) ( £ £ Z), we set
/B^)=2"^(2^a--a;o)).
Since, for j > ^, the Fourier transform of Aj(&)/5 is supported in
{y-2 ^ |$| < 2^2},
it follows that
f^(b)(y)fB(y)dy=0.
We write
AjW(a-)^(a;) = f(Mx)^(b)(x)f^y) - f^(x)Aj(b)(y)f^y))dy
_ y ly (x — v}\2
-E / -^——^-(/B(.")A,(&)(a-)/,(y)-/,(^)A,(6)(y)/,(y))dym J ^ v\
=E E ^^/^—^(^^A.w^/^y)
m H+|/3|=n J 1 '/1
-^(^)A,(6)(^)^(2/))^
where f^(x) = {x -XQ^f^x). Then we get
^W(x)f,(x)=^ ^ Ca,^^(^)[A,(6)(^)r^(^)(^)
m |a|+|^|=n
-r^(A,(6)/f)(^)].
But
( \^{b)(x)\Pdx < 02-^ ( \^(b)(x)f^x)\^dx.
J B J B
Therefore,
f \^(b)(x)\Pdx]^ C2-"^ ^ ll/:lloo||A,(6)r^(/f)[/,B m |a|+|/3|=n
-r^(A,(^)||p.
806 ABDELLAH YOUSSFI
On the other hand, SJ-^B) = f^ for j > £, so that by Proposition 2 we
see that
[^2^||A,(6)r^(/f)-r^(A,(6)^)||^ ^ CH/^II^.
3>l J
Since |a| + \{3\ = n, ||/^||oo ^ C'2("-H^ and
ll/fll^i ^ C2^"-^l-?),
it follows that
^sjp [ \^(b)(x)\Pdx < C\B\.
j>e J B
This completes the proof.
4.3. Proof of Lemma 7. — We need to prove that
l|A,(6)||,o<C2-<
For t > 0 and u e R71, we set
bt,u(x) = b(u --)
and denote by ||[^,n,^]||s,p the norm of the bounded operator [^,n,^]
from B^ into B^. Next we write
^j(b)(x)=<b^^^> .
Let / € <S(ir) be such that J f{x)dx = 1 and write
W = [Wy)f{z) - {-W(z)f{y)dz
because
I ^(z)dz=0.
Using now the spherical harmonics, we obtain
^ = E E c^^mf - (-1)^7^)
m |a|+|/3|=n
where ^(.r) = ̂ ^(.r) and f^(x) == x^f{x). Hence
<^^>= -(-1)^ ^ G,,^ < [^^(/̂ r >,
m |a|+|^[=^
showing that
1< ̂ ^ >\ < C^ ^ l̂ ^m|||[̂ ,.,̂ ]||̂ ||̂ ||̂ ,i||̂ ||̂ ,,,.
7^ |o'|+[^|=n p P7
To complete the proof it is enough to notice that
ll[^,.,^]||^<2-^||[6,r^]||^.
REGULARITY PROPERTIES OF COMMUTATORS 807
BIBLIOGRAPHY
[1] G. BOURDAUD, Analyse fonctionnelle dans Pespace euclidien, Pub. Math. Paris
VII, 23 (1987).
[2] G. BOURDAUD, Realisations des espaces de Besov homogenes, Arkiv for Math., 26
(1988), 41-54.
[3] A.P. CALDERON, Commutators of singular integral operators, Proc. Nat. Acad.
ScL, 53 (1965), 1092-1099.
[4] R. COIFMAN, P.L. LIONS, Y. MEYER and S. SEMMES, Compensated compactness
and Hardy spaces, J. Math Pures et AppL, 72 (1993), 247-286.
[5] R. COIFMAN, R. ROCHBERG and G. WEISS, Factorization theorems for Hardy
spaces in several variables, Ann. of Math., 103 (1976), 611-635.
[6] M. FRAZIER, and B. JAWERTH, A discrete transform and applications to distribu-
tion spaces, J. Funct. Anal., 93 (1990), 34-170.
[7] Y. MEYER, Ondelettes et operateurs. Tome II, Hermann, 1990.
[8] M.A.M. MURRAY, Commutateurs with fractional differentiation and BMO-
Sobolev spaces, Indiana Univ. Math. J., 34 (1985), 205-215.
[9] J. PEETRE, New thoughts on Besov spaces, Duke Univ. Math. Series I Durham,
North Carolina, 1976.
[10] E. STEIN and G. WEISS, Introduction to Fourier Analysis on Euclidean spaces,
Princeton Univ. Press, Princeton, 1971.
[11] R.S. STRICHARTZ, Bounded mean oscillations and Sobolev spaces, Indiana Univ.
Math. J., 29 (1980), 539-558.
[12] H. TRIEBEL, Theory of function spaces (Leipzig 1983).
[13] H. TRIEBEL, Theory of function spaces II, Basel-Boston-Berlin, Birkhauser, 1992.
[14] A. YOUSSFI, Localisation des espaces de Lizorkin—Triebel homogenes, Math. Na-
chr., 147 (1990), 107-121.
[15] A. YOUSSFI, Commutators on Besov spaces and factorization of the paraproduct,
Bull. Sc. Math., 119 (1995), 157-186.
Manuscrit recu Ie 14 fevrier 1994,
accepte Ie 16 decembre 1994.
Abdellah YOUSSFI,
Universite Marne—La—Vallee
Equipe d'Analyse et de Maths Appliquees
2, Rue de la Butte Verte
93166 Noisy-Le-Grand Cedex (France).


Regularity properties of commutators and $BMO$-Triebel-Lizorkin spaces

Guès, Olivier

Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites

Candel, Alberto

Gómez-Mont, X.

ANNALES DE L’INSTITUT FOURIER
ALBERTO CANDEL
X. GÓMEZ-MONT
Uniformization of the leaves of a rational vector field
Annales de l’institut Fourier, tome 45, no 4 (1995), p. 1123-1133
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Ann. Inst. Fourier, Grenoble
45, 4 (1995), 1123-1133
UNIFORMIZATION OF THE LEAVES
OF A RATIONAL VECTOR FIELD
by A. CANDEI/*) and X. GOMEZ-MONT^**)
In the present paper we prove uniformization theorems for one dimen-
sional holomorphic foliations with hyperbolic linearizable singularities of a
compact complex manifold M. Each leaf of such a foliation is a Riemann
surface. If they are hyperbolic (i.e., conformally covered by the unit disc),
each has a canonical metric of constant curvature —1, compatible with the
complex structure. The uniformization function T] ([7], [11]) compares this
canonical metric with a metric induced by restriction to the leaves of a
Hermitian metric on the ambient manifold M. The main results are :
THEOREM 2.2. — Let M be a compact complex manifold, F a holo-
morphic foliation by curves with linearizable hyperbolic singularities and
with hyperbolic leaves. Then the uniformization function rj is continuous
on M — Sing(^7) and has a continuous extension to M as rj(Smg(J^)) = 0.
THEOREM 2.3. — Let T be a holomorphic foliation by curves with
hyperbolic singularities in CP(2) without leaves with algebraic closure,
then the uniformization function T] is continuous on CP(2) with 77 = 0 on
the singular set ofT.
This last theorem may be applied to the generic foliation in CP(2).
Note that the statement of this theorem contains the fact that if a
holomorphic foliation of CP(2) (with hyperbolic singularities) has an
euclidean leaf then it also has an algebraic leaf. It would be interesting
(*) Research supported by NSF Grant DMS-9100383.
(**) Research supported by CONACYT 3398-E9307 and Guggenheim Foundation.
Key words : Holomorphic foliations — Uniformization.
Math. classification : 58F23 - 32L30.
1124 A. CANDEL, X. GOMEZ-MONT
to know if the uniformization map is continuous when there are algebraic
leaves, and also to understand how it varies with the foliation.
We would like to mention that some of the problems discussed here
are posed as open questions in the article [6] by D. Cerveau.
Two recent papers discussing uniformization of leaves of holomorphic
foliations are [8] and [9], by Glutsuk and Lins Neto, respectively. The
method in those papers appears to be a global one, rather than the local
approach of this one. We thank the referee for this reference.
1. Uniformization of the local linear hyperbolic foliation.
Let A" C C71 be the product of the unit disc A in C provided with
the norm |(ai,... ,an)| = "^^'d^jl)? where [ | is the usual norm in the
complex numbers C. A diagonal vector field
x=Z^
in C71 such that any pair of distinct eigenvalues A^, \k are linearly indepen-
dent over the real numbers will be called a linear hyperbolic vector field.
Let J^x be the holomorphic foliation by curves defined by X in A71. It has
an isolated singularity at 0. Given a point p = (pi,... ,pn) in A71 — {0},
the solution of the differential equation d z / d t = X(z) with initial condition
z(Q) = p is given by
(1.1) z(T) = (piexp(Air),...,pnexp(Anr)).
In the T-plane, there is associated to p a set of real lines {A^}!^
formed by those points T such that |^(r)| = 1. In parametric form the
lines are :
(1.2) ('__iog(——)+^)A, r e R .
\\A3\ \Pj\ /
The line A^ has direction vector i\j and the distance from A^ to the
origin is
-i^108^)-
LEAVES OF A RATIONAL VECTOR FIELD 1125
If pj tends to 0, the corresponding lines are parallel, but the distance
to the origin is increasing to infinity. If pj == 0, the corresponding line
disappears. Conversely, the set of lines determine the absolute values \pj \.
n
Denote by P{p) the connected component of C — |j A.3 containing 0. This
j'=i
set V(p) is a convex (and hence simply connected) domain with boundary
a polygon formed by line segments parallel to i\j. Not all sides appear in
given polygon, reflecting the fact that the corresponding coordinate pj is
comparatively small (i.e. A3 is far from 0).
Let P be the open subset of [A71—^}] x C formed by points (p, r) such
that T belongs to the polygon P(p). The projection TT : [A71 - {0}] x C —^
A71 — {0} is a locally trivial fibre bundle with simply connected fibres.
Recall that a uniformization of a Riemann surface S is a holomorphic
map from one of the three simply connected Riemann surfaces to S which
is also a covering map. It follows from the uniformization theorem for
Riemann surfaces [2] that there is a map
v : [A71 - {0}] x A -^ P C [A71 - {0}] x C
over A71 — {0} such that v(p, — ) : p x A —> p x P{p) is the unifomization map
ofP(p), normalized by the conditions v(p, 0) = (p, 0) and (Qv/Qz) (p, 0) > 0.
We claim that v is a homeomorphism. To obtain an expression for
z?, recall that the Schwarz-Christoffel formula [I], p. 228, gives an explicit
parametrization by a disc A of a polygon of n sides as a function of the
form :
(1.3) C r T](w - Wk^dw + C'
^ k=i
where f3k are the exterior angles of the polygon, the Wk are points in the
unit circle and C and C' are constants. If V(p) consists of n sides, then
continuous coordinates for the polygons with parallel sides are the points
{wk(p)} on the unit circle and the constant C(p) in the expression (1.3),
since the exterior angles are fixed [I], p. 228. The normalization of the
uniformizing map implies that C' = 0 and
n
C=r(p)]^(-Wk(p)fk,
k=l
1126 A. CANDEL, X. G6MEZ-MONT
where r(p) is a strictly positive real number that depends continuously on
p. In conclusion, we obtain the expression
/ n \ ^ f c ^ n
(1.4) ?;(p,w) =r(p) n(-Wfc(p) / Jpw-w^p))-^.
U=l / Jo k=l
If w^ and wje tend to the point w^, then the corresponding polygons
have one side which becomes smaller and smaller until it disappears. We
also have
(w - Wj)~^(w - Wk)~^ -^ (w - wi)-^^.
Since the exterior angles of the polygon have the additivity property
(3j +/3fc = A? the integrand in (1.4) is continuous if the polygon has n sides,
or in the limiting case when it has m sides but the remaining n—m lines pass
through the vertices of the corresponding polygon. For an arbitrary point
p in A71 — {0} we can increase the components of those coordinates whose
line K3? does not form part of the polygon P(p) until the corresponding line
passes through a vertex of P(p) to obtain a point q. Then P(q) = P(p) and
so for expression (1.4) corresponding to the uniformization ofp we use the
expression for q. This shows that v is a homeomorphism.
The function z(T) in (1.1) restricted to P(p) is a parametrization
of the leaf Cp C A71 - {0} of Fx passing through p. It is a bijective
correspondence, except if only one coordinate of p is non-vanishing, in
which case the leaf is a punctured disk and the map is the universal covering
map.
The function
u: [A71 - {0}] x A —— [A71 - {0}]
defined by
u(p, w) = (^p,i(w),.. . . Up,n(w)) = (pi exp(Ai-y(p, w)) , . . . ,pn exp(A^(p, w))
is then the expression of the uniformization of the leaves of Fx m A" — {0}.
If s is a positive number and Up^ is a coordinate of a uniformization, then it
is identically zero (if the leaf is contained in zjc == 0) or it is non-vanishing.
In the first case (up^)8 := 0 and otherwise (u8 ^ = exp(s\og(up^))' We
will also use the notation {up}8 = ((^p,i)8 , . . . , (up^n)8) '• A —> A71 - {0}.
LEMMA 1.1. — For p C A71 - {0}, s a positive real number, Up the
uniformization of the leaf Cp, then the function (Up)8 is a uniformization
of the leaf Cps.
LEAVES OF A RATIONAL VECTOR FIELD 1127
Proof. — The distance I3 of the lines K3 to 0 is given by
—7—7 log ( -—- }. Therefore the distance from A^s to 0 satisfies :l^-l Vbjl/
(1.5) ^ = ——— log (——} = -s—— log f 1 ) = ̂ .
1^-1 \hjr/ 1^-1 \hjl/ '
This means that P(p8) = sP{p) and by the uniqueness of the uniformizing
map we have ^(p^w) = sv(p,w). Thus
u^p8, w) = pj exp(Xjv(ps,w))
=psjexp(\jsv{p,w))
= (p^ exp(A^(p, w)))8 = Uj(p, w)8.
PROPOSITION 1.2. — Let {pm} be a sequence of points in A71 — {0}
converging to 0, and Up^ : A —> A71 — {0} be the uniformizing maps of
Cp^, then :
(1) {up^} converges uniformly in compact sets of A to 0.
(2) lim (9n/9w)(p^0)=0.
m—>^oo
Proof. — The family {up^} is a normal family, so by taking a
subsequence, assume that it converges uniformly in compact sets to a
function Uoo. Let Sm = -,——— and qm = (pm)8'"1- The points qm have
logbml
norm 1/2 and lim Sm = 0.
771—»00
By Lemma 1.1 the uniformization of jCg^ is (up^}sm. By taking
a subsequence, we may assume that lim {up^)sm := u^ uniformly in
771—>00
compact sets.
From this convergence we obtain that for every R € (0,1), every e > 0
there exist R' € (0,1), N > 0 such that if |w| ^ R, m > N then
(1.6) ^(w)5771! ^ R\ |<,(w)| ^ fi7, In^w)5- - <,(w)| < e.
Hence we obtain
\Um(w)\ ̂ (R^.
Since Sm tends to zero, we obtain that Um converges uniformly to 0 on
|w| ^ R. Hence Uoo = 0. This proves part (1).
1128 A. CANDEL, X. GOMEZ-MONT
To prove part (2), note that from the continuity of the parametriza-
tion (1.3) as a function of the polygon it follows that lim r(qm) =
m—*-oo
r(i/(0)) := ro. This means
Ve > 0 3M such that m > M => \r(qm) — ^ol < €.
But r(qm) = ^((pm)8771) = Smr(pm) due to (1.5). Hence \Smr{pm) -TQ\ < e.
Multiplying this inequality by \pm\/Sm and doing some manipulations, one
obtains :
\Pm\r(p^ ^ (ro + e)1^ = (ro + e)^!10^^0.
Sm log(J)
Hence
(1.7) lim \pm\r(pn,) = 0.
m—^oo
From (1.4), the definition oiu and some elementary calculations one obtains
r\
^(?, 0) = r(p)(Aipi, .... \nPn)'
Using (1.7) one obtains (2).
Let gp be the 'metric' on the tangent bundle to Tx on A71 — {0}
made up by putting together the metrics of constant curvature —1 on
each leaf of the foliation, and let QL be the metric on the same bundle
obtained by restricting a (Hermitian) metric on A71. Since both metrics
are conformal on each leaf, we have g^ = rjgp for a positive function on
A71 - {0}, smooth on each leaf, which is called the uniformization map of
FX' Then we have rj(p) = |(<9n/<9w)(p,0)|, and so Proposition 1.2 has the
following consequence :
COROLLARY 1.3. — Let T] : A71 - {0} —> (0, oo) be the uniformization
map of a linear hyperbolic vector field. Then rj is continuous and has a
continuous extension to 0 with rf(0) = 0.
LEAVES OF A RATIONAL VECTOR FIELD 1129
2. Uniformization of foliations
with linearizable hyperbolic singularities.
Let M be a compact complex manifold with a Hermitian metric, T
a holomorphic foliation by curves with isolated singularities, such that at
each singular point the foliation is described by a linear hyperbolic vector
field. The leaves of the foliation are the leaves in M — Sing(.F) (we do
not attach a singular point to any separatrix passing through it). Assume
further that all the leaves of F in M — Sing(^7) are conformally covered
by the unit disc (i.e. hyperbolic) and let T] : M — {0} —»• (0, oo) be the
uniformization function comparing the complete hyperbolic metric on the
leaves g p with the metric g^ induced on the leaves by the Hermitian metric
on M '. gL = rjgp.
PROPOSITION 2.1. — Let M be a compact complex manifold, F a
holomorphic foliation by curves with linearizable hyperbolic singularities
and with hyperbolic leaves. If{um} is a family of uniformizations of leaves
ofF converging uniformly on compact sets to u, then u is a uniformization
of a leaf or a constant in Sing (.F).
Proof. — Assume first that u(0) = g, a singular point of T. Let U be
a neighborhood of q with coordinates such that F is described by a linear
hyperbolic vector field as in section 1. The foliation T restricted to U will
be denoted by Q. Let Vm '' A —>• U be the uniformization of the leaf of Q
with Vrn(P) = ^m(O)' Since the leaves of Q are contained in the leaves of T^
and due to the explicit nature of <?, one sees that there is a conformal map
(holomorphic and injective) 0m '- A —> A such that 0^(0) == 0 and
(2.1) Vm=UmOOim'
Since {vm} and {c^m} ^T^ normal families, by taking a subsequence, one
may suppose that {v-m} converges uniformly to v = 0 (by Proposition 1.2)
and 0m to a : A —> A satisfying v = u o a.
Let R < 1 be such that the image by u of the closed disc {z € A; \z\ ^
R} is contained in U. By uniform convergence this also holds for Umi for
m sufficiently large (so assume it is true for every m). Let (3 : A —> A
be defined by {3{z} = Rz. Since the image of (3 is contained in the image
of Oyn, and 0m is a conformal map, there exists ^rn '- A —^ A such that
1130 A. CANDEL, X. GOMEZ-MONT
/3 = dm o 7m- Taking derivatives at 0 we obtain
a"(o)=^))•
By Schwarz's lemma we have |7m(0)| < 1, as 7 ,̂ cannot be a surjection.
Hence
(2.2) ^<K(0) |^1
and so |o;'(0)| ^ R. This implies that a is non-constant. Since v = no a we
conclude that IA takes the value q in a neighbourhood of 0, and consequently
u is identically q by analytic continuation.
Now assume that u(ft) does not belong to Sing(J'). Then the image
of u contains no singular points of F, for otherwise u would be identically
this singular point by the proceeding argument. Thus the image of u is
contained in the leaf Cq. We claim that u is a uniformization of Cq. Let u
be a uniformization of Cq with u(0) = q. Then
(2.3) |<(0)| - |n'(0)| ^ [^(O)].
The argument used to prove the lower semicontinuity of T] (see [7], [II],
[5]) can be applied here to show that we may transport approximations of
u to nearby leaves. It then follows that a strict inequality in (2.3) would
contradict that Um are uniformizations. Therefore |ZA'(O)| = ^'(O)! and so
u = u as we wanted.
THEOREM 2.2. — Let M be a compact complex manifold with a Her-
mitian metric, F a holomorphic foliation by curves with linearizable hy-
perbolic singularities and with hyperbolic leaves. Then the uniformization
function rj is continuous on M - Sing(^7) and has a continuous extension
to M as rj(Smg(^)) = 0.
Proof. — Note that the argument to prove the lower semicontinuity
°^ r] (t'7]? [11]? [5]) stm works here because it reduces to a local question
around a compact piece of a leaf.
The argument in Theorem 4.8 of [5], which is based on [3], shows that
r] is bounded above, for otherwise one could prove that there is a leaf of
F which is not hyperbolic, contradicting the hypothesis. This implies that
the family of uniformizations is equicontinuous, and hence it is a normal
family. Therefore, by Proposition 2.1, r] is continuous on M - Sing(J').
LEAVES OF A RATIONAL VECTOR FIELD 1131
To analyze the behaviour of rj near the critical points, we take
sequences and coordinates as in the proof of Proposition 2.1. By (2.1) and
(2.2) we obtain
(2.4) A|<(0)| < |<(0)| ^ |<(0)|.
By part (2) of Proposition 1.2 we have lim ^(0) = 0 and using (2.4) we
771—>00
obtain that lim z^(0) = 0.
771—>00
Hence 77 extends continuously to the singular points as 77 = 0 there
and the theorem is proved.
In case M is the projective plane CP(2), the condition of hyperbolic
leaves can be somehow relaxed. We have :
THEOREM 2.3. — Let T be a holomorphic foliation by curves with
hyperbolic singularities in CP(2) and with no leaf with algebraic clo-
sure. Then the uniformization function rj is continuous on CP(2) with
77(Sing(^)) = 0.
Proof. — First we show that there is no invariant transverse measure
for F in CP(2) — Sing(^'). Near the singular points, the transverse measure
has support only at the separatrices, since there is a hyperbolic holonomy
on each separatrix. The hypothesis of no algebraic leaves implies that
the measure cannot be a Dirac delta on a separatrix. So, the transverse
measure has support on the complement of the basin of attraction of the
critical points. This set (the support of the measure) is a Riemann surface
lamination and it is shown in [4] that there are no invariant transverse
measures there.
The proof of the theorem would be complete if we show that every
leaf is hyperbolic. Suppose there is an euclidean leaf and let B{n) be
the averaging sequence it carries ([5]; Ahlfors5 lemma). The sequence of
currents in CP(2),
w = Area(B(n)) ^(n)0
converges to a closed current T in CP(2). This is a non-trivial current.
Indeed, if uj denotes the Kaehler form of projective space, then
Tn^)=T^)=l
by Wirtinger's theorem.
1132 A. CANDEL, X. G6MEZ-MONT
Although the foliated manifold we are working on is not compact, the
foliation current T still defines a measure on compact discs transverse to F
in CP(2) - Sing(^), which is invariant by holonomy (see for example the
proof of the Riesz representation theorem in [10]). In the non-compact case
it is not automatic that T is nontrivial (as a current acting on forms with
compact support in CP(2) — Sing^)), and that is what we show next.
Let C be a projective line in CP(2) not containing any of the singularities.
Then the Poincare dual of C may be represented by the Kaehler form a;,
but also by a closed form a with support contained in a small tubular
neighborhood of C. Since a; and a are cohomologous, T(a) = T(u)) = 1.
Acknowledgments. — The first author would like to thank IAS and CIMAT
for their hospitality, and S. Zakeri for pointing out an oversight in one of
the proofs.
BIBLIOGRAPHY
[1] L. AHLFORS, Complex Analysis, MacGraw Hill, New York, 1966.
[2] L. AHLFORS, Conformal Invariants, MacGraw Hill, New York, 1973.
[3] R. BRODY, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc., 235
(1978), 213-219.
[4] C. CAMACHO, A. LINS, P. SAD, Minimal sets of foliations on complex projective
spaces, Publ. Math. IHES, 68 (1989), 187-203.
[5] A. CANDEL, Uniformization of surface laminations, Ann. Scient. EC. Norm. Sup.,
26 (1993), 489-516.
[6] D. CERVEAU, Equations different ielles algebriques : Remarques et problemes,
J. Fac. Sci. Univ. Tokyo, 36 (1989), 665-680.
[7] E. GHYS, Gauss-Bonnet Theorem for 2-dimensional foliations, J. of Funct. Anal.,
77 (1988), 51-59.
[8] A.A. GLUTSUK, The hyperbolicity of phase curves of a generic polynomial vector
field in C71, Functional Analysis and its applications, 28 (1994), 77-84.
[9] A. LINS, Simultaneous uniformization for the leaves of projective foliations by
curves, to appear in Bol. Soc. Brasileira.
LEAVES OF A RATIONAL VECTOR FIELD 1133
[10] C. MOORE & C. SCHOCHET, Global analysis of foliated spaces, Springer-Verlag,
New York, 1988.
[11] A. VERJOVSKY, A uniformization theorem for holomorphic foliations., Contemp.
Math., 58(111) (1987), 233-253.
Manuscrit recu Ie 3 octobre 1994,
accepte Ie 16 mars 1995.
A. CANDEL,
The University of Chicago
Dept. of Mathematics
Chicago, IL 60637 (USA).
e-mail .'candel@math.uchicago.edu
&
X. GOMEZ-MONT,
CIMAT
AP 402
Guanajuato 36000 (Mexico).
e-maii ;gomezm@mailer. main.conacyt. mx


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