## Anais da Academia Brasileira de Ciências

##
*On-line version* ISSN 1678-2690

### An. Acad. Bras. Ciênc. vol.73 no.1 Rio de Janeiro Mar. 2001

#### http://dx.doi.org/10.1590/S0001-37652001000100002

**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 *g*_{1}, *g*_{2} such that: if *G* = *g*_{1} + *ig*_{2}, 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

h'(z) = t; t Î | (*) |

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* + *f*_{1}(*z*) + *f*_{2}(*z*) + *f*_{3}(*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* = 2p*i*;

that is

*h*^{k}(*w*) = exp(2p*i*)(*w*).

Consequently

*h*(*w*) = exp(2p*i*)(*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 2p*iX*.

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 *x*_{0}. 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 _{j}dF_{j}. *

Suppose there exists a Levi foliation tangent to and suppose that (*F*_{1} ),(*F*_{2 }) 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. [ Links ]

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. [ Links ]

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

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

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

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

E-mail: ostwald@impa.br