On the existence of Levi Foliations

Abstracts

Let L <img src="http:/img/fbpe/aabc/v73n1/0059c.gif"> <img src="http:/img/fbpe/aabc/v73n1/0059c2.gif"> be a real 3 dimensional analytic variety. For each regular point p <img src="http:/img/fbpe/aabc/v73n1/0059e.gif"> L there exists a unique complex line l p on the space tangent to L at p. When the field of complex line p <img ALIGN="MIDDLE" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img4.gif" ALT="$\displaystyle \mapsto$"> l p is completely integrable, we say that L is Levi variety. More generally; let L <img src="http:/img/fbpe/aabc/v73n1/0059c.gif"> M be a real subvariety in an holomorphic complex variety M. If there exists a real 2 dimensional integrable distribution on L which is invariant by the holomorphic structure J induced by M, we say that L is a Levi variety. We shall prove: Theorem. Let <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img5.gif" ALT="$ \cal {L}$"> be a Levi foliation and let <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$"> be the induced holomorphic foliation. Then, <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$"> admits a Liouvillian first integral. In other words, if <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img5.gif" ALT="$ \cal {L}$"> is a 3 dimensional analytic foliation such that the induced complex distribution defines an holomorphic foliation <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$">; that is, if <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img5.gif" ALT="$ \cal {L}$"> is a Levi foliation; then <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$"> admits a Liouvillian first integral--a function which can be constructed by the composition of rational functions, exponentiation, integration, and algebraic functions (Singer 1992). For example, if f is an holomorphic function and if theta is real a 1-form on <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img8.gif" ALT="$ \mathbb {R}$">; then the pull-back of theta by f defines a Levi foliation <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img5.gif" ALT="$ \cal {L}$"> : f*theta = 0 which is tangent to the holomorphic foliation <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$"> : df = 0. This problem was proposed by D. Cerveau in a meeting (see Fernandez 1997).

Levi foliations; holomorphic foliations; singularities; Levi varieties

Seja L <FONT FACE="Symbol">Ì</FONT> <img src="http:/img/fbpe/aabc/73n1/0059c2.gif"> uma variedade real de dimensão 3. Para todo ponto regular p <FONT FACE="Symbol">Î</FONT> L existe uma única reta complexa l p no espaço tangente à L em p. Quando o campo de linhas complexas p <img ALIGN="MIDDLE" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img4.gif" ALT="$\displaystyle \mapsto$"> l p é completamente integrável, dizemos que L é uma variedade de Levi. Mais geralmente, seja L <FONT FACE="Symbol">Ì</FONT> M uma subvariedade real em uma variedade analítica complexa. Se existe uma distribuição real integrável de dimensão 2 em L que é invariante pela estrutura holomorfa J induzida pela variedade complexa M, dizemos que L é uma variedade de Levi. Vamos provar: Teorema. Seja <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img5.gif" ALT="$ \cal {L}$"> uma folheação de Levi e seja <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$"> a folheação holomorfa induzida. Então <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$"> tem integral primeira Liouvilliana. Em outras palavras, se <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img5.gif" ALT="$ \cal {L}$"> é uma folheação real de dimensão 3 tal que a folheação holomorfa induzida define uma folheação holomorfa <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$">; isto é, se <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img5.gif" ALT="$ \cal {L}$"> é uma folheação de Levi; então <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$"> admite uma integral primeira Liouvilliana - uma função que pode ser construida por composição de funções rationais, exponenciações, integrações e funções racionais (Singer 1992). Por exemplo, se f é uma função holomorfa e se teta é uma 1-forma real em <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img8.gif" ALT="$ \mathbb {R}$">²; então o pull-back de teta por f define uma folheação de Levi: <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img5.gif" ALT="$ \cal {L}$"> : f*teta = 0 a qual é tangente a folheação holomorfa <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$"> : df = 0. Este problema foi proposto por D. Cerveau em uma reunião (Fernandez 1997).

folheações de Levi; folheações holomorfas; singularidades; variedades de Levi

On The Existence of Levi Foliations

RENATA N. OSTWALD

Instituto de Matemática Pura e Aplicada - IMPA, Est. Dona Castorina 110

22460-320 Rio de Janeiro, Brasil

Manuscript received on September 20, 2000; accepted for publication on December 6, 2000;

presented by CÉSAR CAMACHO

Abstract

Let L be a real 3 dimensional analytic variety. For each regular point p L there exists a unique complex line l_p on the space tangent to L at p. When the field of complex line

p l _p

is completely integrable, we say that L is Levi variety. More generally; let L M be a real subvariety in an holomorphic complex variety M. If there exists a real 2 dimensional integrable distribution on L which is invariant by the holomorphic structure J induced by M, we say that L is a Levi variety. We shall prove:

Theorem. Let be a Levi foliation and let be the induced holomorphic foliation. Then, admits a Liouvillian first integral.

In other words, if is a 3 dimensional analytic foliation such that the induced complex distribution defines an holomorphic foliation ; that is, if is a Levi foliation; then admits a Liouvillian first integral--a function which can be constructed by the composition of rational functions, exponentiation, integration, and algebraic functions (Singer 1992). For example, if f is an holomorphic function and if q is real a 1-form on ; then the pull-back of q by f defines a Levi foliation : f^*q = 0 which is tangent to the holomorphic foliation : df = 0.

This problem was proposed by D. Cerveau in a meeting (see Fernandez 1997).

Key words: Levi foliations, holomorphic foliations, singularities, Levi varieties.

ANNOUNCEMENT

Let be a Levi foliation and let be the holomorphic foliation tangent to . Note that if h in an holomorphic function such that is h-invariant ( h^* = ); then is also h-invariant ( h^* = ). We shall mainly use that property in order to prove

THEOREM. Let be a Levi foliation and let be the induced holomorphic foliation. Then admits a Liouvillian first integral.

We proceed as follows:

We first show that if is a Levi foliation, there exists analytic real functions g₁, g₂ such that: if G = g₁ + ig₂, then the Levi foliation is defined by

: + G = 0,

where w is an holomorphic 1-form so that w = 0 defines the holomorphic foliation tangent to the Levi foliation . We then verify that; if ^* is the holomorphic foliation obtained from after a finite number of blow-ups, there exists a Levi foliation ^* tangent to ^*. Therefore, by Seidenberg Theorem (Seindenberg 1968), we analyse the foliation ^* for which all singularities are reduced.

Let D denote the divisor obtained on the process of reducing the singularity and let D_j denote the irreducible curves with normal crossings such that D = È D_j. We consider the induced Levi foliation on sections transversal to the holomorphic foliation through each component D_j of the divisor. We show that the holomorphic diffeomorfisms for which the Levi foliation is invariant must satisfy an equation on one variable of the type

;

(*)

We can then find an holomorphic coordinate system y on the section such that

F(y) = .

We refer to such coordinate system as a normalizable coordinate system. We verify that it is unique up to homographies.

If either l ¹ 0 or k = 0, then t = 1 for all solutions h of the differential equation (*). Furthermore, if k = 0, then the group of solutions of the differential equation is a linear group. On both cases we have an abelian group for the group of solutions of (*). We can already conclude:

THEOREM A. Let p be a singularity of the foliation

: = l xdy + ydx + {higher order terms} = 0 l Î ^* - .

Suppose there exists a Levi foliation tangent to . Then the singularity is analytically equivalent to a linear singularity.

PROOF. For if there exists a Levi foliation, the holonomy associated to the singularity must satisfy an equation as (*). If so, the order of F at 0 cannot be but 1; that is, k=0. The holonomy is linearizable; as a result, so is the singularity (Mattei & Moussu 1980).

We still have to consider the case l = 0. There are solutions for which t ¹ 1, (h'(0))^k =

. These solutions are necessarily linearizable, but not those for which t = 1. The latter, though, also determine an abelian group. We shall then describe the abelian group of solutions of (*) for t = 1, k > 0.

We can take an holomorphic coordinate system (x, y) such that the group of solutions of the differential equation is in normalizable coordinate system on each transversal section x = cte.

For an holomorphic vector field X, let exp X denote its exponential application, that is, its flow for t = 1:

exp((z))(z) = z + f₁(z) + f₂(z) + f₃(z) +...

satisfying

If h is a diffeomorfism which satisfies

h'(z) =

then the k-th interate of h; h^k, is tangent to the identity. There exists m such that h^k is the exponential of the vector field:

Y = 2pi

;

that is

h^k(w) = exp(2pi

)(w).

Consequently

h(w) = exp(2pi

)(w); = 1.

If

X = x + yf (x, y)

is the vector field which defines the holomorphic foliation; then the holonomy application is defined by

exp 2piX.

We have found two linear independent vector fields--X, Y that define h. Therefore; they commute:

[X, Y] = 0 .

We can describe X to be so as to satisfy the commutability condition. We then show the local result:

THEOREM B. Let p be a singularity of the foliation

: = l xdy + ydx + {higher order terms} = 0,l Î .

Suppose there exists a Levi foliation tangent to . Then the singularity is normalizable in the sense of Martinet and Ramis (1982), Martinet and Ramis (1983). In particular, w admits an analytic integrating factor.

PROOF. If l

- , the singularity is linerizable by Poincare's Theorem. If l - , we have proved (Theorem A) that is also a linerizable singularity. Thus, we have to prove the result for l ; since the singularity is a reduced one, l

₊. Let

be the vector field whose exponential application determines the holonomy application on x₀. If there are two invariant curves through the singularity, then the vector field that defines the holomorphic distribution can be written as x + yf (x, y). By solving the commutability condition [X, Y] = 0 :

Let f (x, y) = f (x, 0) + g(x, y), then f must be as to satisfy

which leads us to

f (x, y) = + (x).

The foliation on the punctured neighborhood is defined by the following 1-form

= xdy + y

- (x)

dx

or still by

Necessarily has an holomorphic extension through 0 and has either an holomorphic or a meromorphic extension through 0. If it were meromorphic, the singularity would not be a reduced one, contradicting our hypotheses. The extension is then an holomorphic one. We have then a normal form for either cases:

If

^*, we have a saddle-node; if

- ^* and let p be the order of the zero of f at 0, we have a ressonant singularity.

If there is only one invariant curve through the singularity; the singularity is a saddle-node and the invariant curve is y = 0. Therefore the vector field that defines the holomorphic distribution can be written as X = (x + h(y)) + yf (x, y), f (0) = 0. The holonomy is defined by the exponential application of the vector field . The commutability condition [X, Y] = 0 implies that

[X, Y] = d.Y

X .

By solving the equation just above, we obtain that must be an holomorphic function which contradicts f (0) = 0.

Following, we prove results that will allow us to relate the first integrals obtained on the neighborhood of each component D_j.

THEOREM C. Let p be a singularity of the foliation : w = 0 and

w = fdF isanholomorphic1 - form

where F is a Liouvillian function and f is an holomorphic integrating factor of w . There exists a Levi foliation defined by

: (fdF) + f ().

Furthermore, if p is not a linearizable ressonant singularity, then any other Levi foliation must be of the type:

: l (fdF) + f ().

Note that (l F) is a first integral of the Levi foliation . We can then show:

COROLLARY. Let p be a singularity of the holomorphic foliation : w = 0. Let F_j be Liouvillian functions and let f_j be holomorphic functions such that

w = f_jdF_j.

Suppose there exists a Levi foliation tangent to and suppose that (F₁ ),(F₂ ) are first integrals of . Then:

PROOF. Follows from dF_i = dF_j and d (F_i + Ù d (F_j + ) = 0.

We are then able to show:

THEOREM D. Let be an holomorphic foliation and be a Levi foliation tangent to . Suppose all singularities lie on an irredutible curve S; which is -invariant. Then admits a Liouvillian first integral I defined on a neighborhood of S. Furthermore, d (I + ) defines a Levi foliation tangent to .

PROOF. To show the existence of a Liouvillian first integral of it is enough to show the existence of a Liouvillian first integral of the reduced foliation ^*. Let D = È D_j be the divisor obtained on the process of reducing the singularities. Let us fix a transversal section of ^* through D_j. Since there exists a Levi foliation tangent to ^*, there exists a normal coordinate system on the section so that the holonomy applications determined by the singularities on D_j satisfy (*). For each D_j, we then find an holomorphic vector field Z_j that defines the foliation ^* in a neighborhood of the divisor. Let Y be the holomorphic vector on each transversal section which defines the holonomies. To find Z_j, all we have to do is solve the equation

[Z_j, Y] = 0.

The vector field Z_j allows us to describe a Liouvillian first integral of the holomorphic foliation on a neighborhood of each irreduceble component D_j of the divisor D = È D_j obtained on the resolution of the singularity. Let F_j be a Liouvillian first integral of the holomorphic foliation ^* on a neighborhood of the D_j such that (F_j) is a first integral of ^*. By Theorem b, for each

p Î D _i

D _j

we have

= .

Therefore

= {}.

is a well defined closed 1-form. Thus

I = exp

is a Liouvillian first integral of the holomorphic foliation

^* and there is a Levi foliation d (I + ) = 0; The Theorem is thereby proved.

RESUMO

Seja L uma variedade real de dimensão 3. Para todo ponto regular p L existe uma única reta complexa l_p no espaço tangente à L em p. Quando o campo de linhas complexas

p l _p

é completamente integrável, dizemos que L é uma variedade de Levi. Mais geralmente, seja L M uma subvariedade real em uma variedade analítica complexa. Se existe uma distribuição real integrável de dimensão 2 em L que é invariante pela estrutura holomorfa J induzida pela variedade complexa M, dizemos que L é uma variedade de Levi. Vamos provar:

Teorema. Seja uma folheação de Levi e seja a folheação holomorfa induzida. Então tem integral primeira Liouvilliana.

Em outras palavras, se é uma folheação real de dimensão 3 tal que a folheação holomorfa induzida define uma folheação holomorfa ; isto é, se é uma folheação de Levi; então admite uma integral primeira Liouvilliana - uma função que pode ser construida por composição de funções rationais, exponenciações, integrações e funções racionais (Singer 1992). Por exemplo, se f é uma função holomorfa e se q é uma 1-forma real em ; então o pull-back de q por f define uma folheação de Levi: : f^*q = 0 a qual é tangente a folheação holomorfa : df = 0.

Este problema foi proposto por D. Cerveau em uma reunião (Fernandez 1997).

Palavras-chave: folheações de Levi, folheações holomorfas, singularidades, variedades de Levi.

REFERENCES

FERNANDEZ J. (editor) 1997. Ecuaciones Diferenciales: Singularidades, Universidad de Valladolid.

MARTINET J & RAMIS JP. 1982. Problèmes des Modules pour des Équations Differentielles Non Linéaires du Premier Order, Math. Inst. Hautes Études Scientifiques 55: 63-124.

MARTINET J & RAMIS JP. 1983. Classification Analytique des Équations Differentielles Non Linéaires Réssonantes du Premier Ordre, Ann Sc Éc Norm Sup, 16: 571-621.

MATTEI JF & MOUSSU R. 1980. Holonomie et Intégrales Premières, Ann Sc Norm Sup, 13.

SEINDENBERG A. 1968. Reduction of Singularities of The Differential Equation Ady = Bdx, Amer J Math., 248-269.

SINGER M. 1992. Liouvillian First Integrals of Differential Equations, Trans American Math Soc., 333(2): 673-688.

E-mail: [e-mail]

Publication Dates

• Publication in this collection
  09 Mar 2001
• Date of issue
  Mar 2001

History

• Accepted
  06 Dec 2000
• Received
  20 July 2000