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Print version ISSN 0001-3765

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 lp on the space tangent to L at p. When the field of complex line

p $\displaystyle \mapsto$ lp

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 $ \cal {L}$ be a Levi foliation and let $ \cal {F}$ be the induced holomorphic foliation. Then, $ \cal {F}$ admits a Liouvillian first integral.
In other words, if $ \cal {L}$ is a 3 dimensional analytic foliation such that the induced complex distribution defines an holomorphic foliation $ \cal {F}$; that is, if $ \cal {L}$ is a Levi foliation; then $ \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 q is real a 1-form on $ \mathbb {R}$; then the pull-back of q by f defines a Levi foliation $ \cal {L}$ : f*q = 0 which is tangent to the holomorphic foliation $ \cal {F}$ : 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 $ \cal {L}$ be a Levi foliation and let $ \cal {F}$ be the holomorphic foliation tangent to $ \cal {L}$. Note that if h in an holomorphic function such that $ \cal {F}$ is h-invariant ( h*$ \cal {F}$ = $ \cal {F}$); then $ \cal {L}$ is also h-invariant ( h*$ \cal {L}$ = $ \cal {L}$). We shall mainly use that property in order to prove

THEOREM. Let $ \cal {L}$ be a Levi foliation and let $ \cal {F}$ be the induced holomorphic foliation. Then $ \cal {F}$ admits a Liouvillian first integral.

We proceed as follows:

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

$\displaystyle \cal {L}$ : $\displaystyle \overline{G}$$\displaystyle \omega$ + G$\displaystyle \overline{w}$ = 0,

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

Let D denote the divisor obtained on the process of reducing the singularity and let Dj denote the irreducible curves with normal crossings such that D = È Dj. We consider the induced Levi foliation on sections transversal to the holomorphic foliation through each component Dj 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$\displaystyle {\frac{F(h)}{F}}$;    t Î $\displaystyle \mathbb {R}$ (*)

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

F(y) = $\displaystyle {\frac{y^{k+1}}{1-\lambda y^k}}$.

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

$\displaystyle \cal {F}$ : $\displaystyle \omega$ = l xdy + ydx + {higher order terms} = 0    l Î $\displaystyle \mathbb {R}$* - $\displaystyle \mathbb {Q}$ .

Suppose there exists a Levi foliation $ \cal {L}$ tangent to $ \cal {F}$. 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 = $ {\frac{1}{t}}$ $ \mathbb {R}$. 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($\displaystyle \xi$(z)$\displaystyle {\frac{\partial }{\partial z}}$)(z) = z + f1(z) + $\displaystyle {\textstyle\frac{1}{2}}$f2(z) + $\displaystyle {\frac{1}{3!}}$f3(z) +...

satisfying

\begin{displaymath*} \left\{ \begin{array}{l} f_1 = \xi ,\\ f_n = \xi f_{n-1}' . \end{array} \end{displaymath*}

If h is a diffeomorfism which satisfies

h'(z) = $\displaystyle {\frac{h^{k+1}}{1-\lambda h^k}}$$\displaystyle {\frac{1-\lambda y^k}{y^{k+1}}}$

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

Y = 2pi$\displaystyle \mu$$\displaystyle {\frac{y^{k+1}}{1-\lambda y^k}}$$\displaystyle {\frac{\partial }{\partial y}}$;

that is

hk(w) = exp(2pi$\displaystyle \mu$$\displaystyle {\frac{y^{k+1}}{1-\lambda y^k}}$$\displaystyle {\frac{\partial }{\partial y}}$)(w).

Consequently

h(w) = exp(2pi$\displaystyle {\frac{\mu}{k}}$$\displaystyle {\frac{y^{k+1}}{1-\lambda y^k}}$$\displaystyle {\frac{\partial }{\partial y}}$)($\displaystyle \epsilon$w);$\displaystyle \epsilon^{k}_{}$ = 1.

If

X = x$\displaystyle {\frac{\partial }{\partial x}}$ + yf (x, y)$\displaystyle {\frac{\partial }{\partial y}}$

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

$\displaystyle \cal {F}$ : $\displaystyle \omega$ = l xdy + ydx + {higher order terms} = 0,l Î $\displaystyle \mathbb {C}$.

Suppose there exists a Levi foliation $ \cal {L}$ tangent to $ \cal {F}$. 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 $ \mathbb {C}$ - $ \mathbb {R}$, the singularity is linerizable by Poincare's Theorem. If l $ \mathbb {R}$ - $ \mathbb {Q}$, we have proved (Theorem A) that is also a linerizable singularity. Thus, we have to prove the result for l $ \mathbb {Q}$; since the singularity is a reduced one, l $ \mathbb {Q}$+. Let

$\displaystyle {\frac{y^{k+1}}{1-\lambda\mu(x_0)^ky^k}}$

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$ {\frac{\partial }{\partial x}}$ + yf (x, y)$ {\frac{\partial }{\partial y}}$. By solving the commutability condition [X, Y] = 0 :

\begin{displaymath*} \begin{array}{ll} 0 &=\left [x\frac {\partial }{\partial ... ...)^ky^k}\biggr)\frac {\partial }{\partial y} \ . \end{array} \end{displaymath*}

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

\begin{displaymath*} \left\{ \begin{array}{lll} f(x,0) &= & \frac{\mu^/(x)x}{... ...{y}{(1-\lambda\mu^ky^k)^{\frac {1}{k}}}\biggr); \end{array} \end{displaymath*}

which leads us to

f (x, y) = $\displaystyle {\frac{\mu'(x)x}{\mu(x)}}$ + $\displaystyle \delta$(x)$\displaystyle {\frac{y^{k}}{1-\lambda \mu(x)^ky^k}}$.

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

$\displaystyle \omega$ = xdy + y$\displaystyle \biggl($$\displaystyle {\frac{\mu'(x)x}{\mu(x)}}$ - $\displaystyle \delta$(x)$\displaystyle {\frac{y^k}{1-\lambda \mu^ky^k}}$$\displaystyle \biggr)$dx

or still by

\begin{displaymath*} \begin{array}{lll} \frac {\mu}{x} \omega & = & \mu dy+y(1... ...{(\mu^ky^k)^2}+\frac {\delta}{x}dx \biggr) \ . \end{array} \end{displaymath*}

Necessarily $ \delta$ has an holomorphic extension through 0 and $ \mu^{k}_{}$ 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 $ \mu^{k}_{}$ $ \cal {O}$*, we have a saddle-node; if $ \mu^{k}_{}$ $ \cal {O}$ - $ \cal {O}$* 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))$ {\frac{\partial }{\partial x}}$ + yf (x, y)$ {\frac{\partial }{\partial y}}$, f (0) = 0. The holonomy is defined by the exponential application of the vector field $ {\frac{yf(x,y)}{x+h(y)}}$. The commutability condition [$ {\frac{x}{x+h(y)}}$X, Y] = 0 implies that

$\displaystyle {\frac{x}{x+h(y)}}$[X, Y] = $\displaystyle \biggl($d$\displaystyle {\frac{x}{x+h(y)}}$.Y$\displaystyle \biggr)$X .

By solving the equation just above, we obtain that $ {\frac{1}{f}}$ must be an holomorphic function which contradicts f (0) = 0. $ \square$

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

THEOREM C. Let p be a singularity of the foliation $ \cal {F}$ : 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

$\displaystyle \cal {L}$ : $\displaystyle \overline{f}$(fdF) + f ($\displaystyle \overline{fdF}$).

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

$\displaystyle \cal {L}$$\scriptstyle \lambda$ : l $\displaystyle \overline{f}$(fdF) + $\displaystyle \overline{\lambda}$f ($\displaystyle \overline{fdF}$).

Note that $ \Re$(l F) is a first integral of the Levi foliation $ \cal {L}$$\scriptstyle \lambda$. We can then show:

COROLLARY. Let p be a singularity of the holomorphic foliation $ \cal {F}$ : w = 0. Let Fj be Liouvillian functions and let fj be holomorphic functions such that

w = fjdFj.

Suppose there exists a Levi foliation $ \cal {L}$ tangent to $ \cal {F}$ and suppose that $ \Re$(F1 ),$ \Re$(F2 ) are first integrals of $ \cal {L}$. Then:

$\displaystyle {\frac{dF_i}{F_i}}$

PROOF. Follows from dFi = $ {\frac{f_j}{f_i}}$ dFj and d (Fi + Ù d (Fj + $ \overline{F}_{j}^{}$) = 0.

We are then able to show:

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

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

[Zj, Y] = 0.

The vector field Zj allows us to describe a Liouvillian first integral of the holomorphic foliation on a neighborhood of each irreduceble component Dj of the divisor D = È Dj obtained on the resolution of the singularity. Let Fj be a Liouvillian first integral of the holomorphic foliation $ \cal {F}$* on a neighborhood of the Dj such that $ \Re$(Fj) is a first integral of $ \cal {L}$*. By Theorem b, for each

p Î Di $\displaystyle \cap$ Dj

we have

$\displaystyle {\frac{dF_1}{F_1}}$ = $\displaystyle {\frac{dF_2}{F_2}}$.

Therefore

$\displaystyle \omega^{*}_{}$ = {$\displaystyle {\frac{dF_i}{F_i}}$}.

is a well defined closed 1-form. Thus

I = exp$\displaystyle \int$$\displaystyle \omega^{*}_{}$

is a Liouvillian first integral of the holomorphic foliation $ \cal {F}$* and there is a Levi foliation d (I + $ \overline{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 lp no espaço tangente à L em p. Quando o campo de linhas complexas

p $\displaystyle \mapsto$ lp

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 $ \cal {L}$ uma folheação de Levi e seja $ \cal {F}$ a folheação holomorfa induzida. Então $ \cal {F}$ tem integral primeira Liouvilliana.
Em outras palavras, se $ \cal {L}$ é uma folheação real de dimensão 3 tal que a folheação holomorfa induzida define uma folheação holomorfa $ \cal {F}$; isto é, se $ \cal {L}$ é uma folheação de Levi; então $ \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 q é uma 1-forma real em $ \mathbb {R}$; então o pull-back de q por f define uma folheação de Levi: $ \cal {L}$ : f*q = 0 a qual é tangente a folheação holomorfa $ \cal {F}$ : 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