On The Existence of Levi Foliations

Let L ⊂ C 2 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 �→ 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 L be a Levi foliation and let F be the induced holomorphic foliation. Then,F admits a Liouvillian first integral. In other words, if L is a 3 dimensional analytic foliation such that the induced complex distribution defines an holomorphic foliation F ; that is, if L is a Levi foliation; then 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 θ is real a 1-form on R 2 ; then the pull-back of θ by f defines a Levi foliation L : f ∗ θ = 0 which is tangent to the holomorphic foliation F : df = 0.

Let L be a Levi foliation and let F be the holomorphic foliation tangent to L. Note that if h in an holomorphic function such that F is h-invariant (h * F = F ); then L is also h-invariant (h * L = L).We shall mainly use that property in order to prove Theorem.Let L be a Levi foliation and let F be the induced holomorphic foliation.Then F admits a Liouvillian first integral.
E-mail: ostwald@impa.brAn.Acad. Bras. Ci., (2001) 73 (1) We proceed as follows: We first show that if L 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 where ω is an holomorphic 1-form so that ω = 0 defines the holomorphic foliation F tangent to the Levi foliation L. We then verify that; if F * is the holomorphic foliation obtained from F after a finite number of blow-ups, there exists a Levi foliation L * tangent to F * .Therefore, by Seidenberg Theorem (Seindenberg 1968), we analyse the foliation F * 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 = ∪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 We refer to such coordinate system as a normalizable coordinate system.We verify that it is unique up to homographies.
If either λ = 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 Suppose there exists a Levi foliation L tangent to 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 λ = 0.There are solutions for which t = 1, (h (0)) k = 1 t ∈ 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.
An. Acad. Bras. Ci., (2001) 73 (1) 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: If h is a diffeomorfism which satisfies then the k-th interate of h; h k , is tangent to the identity.There exists µ such that h k is the exponential of the vector field: Consequently is the vector field which defines the holomorphic foliation; then the holonomy application is defined by exp 2πiX.
We have found two linear independent vector fields-X, Y that define h.Therefore; they commute: 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 Suppose there exists a Levi foliation L tangent to F .Then the singularity is normalizable in the sense of Martinet and Ramis (1982), Martinet and Ramis (1983).In particular, ω admits an analytic integrating factor.
Proof.If λ ∈ C − R, the singularity is linerizable by Poincare's Theorem.If λ ∈ R − Q, we have proved (Theorem A) that is also a linerizable singularity.Thus, we have to prove the result for λ ∈ Q; since the singularity is a reduced one, λ ∈ Q + .Let ∂ ∂y 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 ∂ ∂x + yf (x, y) ∂ ∂y .By solving the commutability condition [X, Y ] = 0 : The foliation on the punctured neighborhood is defined by the following 1-form Necessarily δ has an holomorphic extension through 0 and µ 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 µ k ∈ O * , we have a saddle-node; if µ k ∈ O − 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)) ∂ ∂x + yf (x, y) ∂ ∂y , f (0) = 0.The holonomy is defined by the exponential application of the vector field x x+h(y) X = x ∂ ∂x + yf (x,y)   x+h(y) ∂ ∂y .The commutability condition .Y X .
By solving the equation just above, we obtain that 1 f 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 F : ω = 0 and ω = f dF is an holomorphic 1-form where F is a Liouvillian function and f is an holomorphic integrating factor of ω.There exists a Levi foliation defined by Furthermore, if p is not a linearizable ressonant singularity, then any other Levi foliation must be of the type: Note that (λF ) is a first integral of the Levi foliation L λ .We can then show: Corollary.Let p be a singularity of the holomorphic foliation F : ω = 0. Let F j be Liouvillian functions and let f j be holomorphic functions such that ω = f j dF j .
Suppose there exists a Levi foliation L tangent to F and suppose that (F 1 ), (F 2 ) are first integrals of L. Then: We are then able to show: Theorem D. Let F be an holomorphic foliation and L be a Levi foliation tangent to F .Suppose all singularities lie on an irredutible curve S; which is F -invariant.Then F admits a Liouvillian first integral I defined on a neighborhood of S. Furthermore, d(I + I ) defines a Levi foliation tangent to F .
Proof.To show the existence of a Liouvillian first integral of F it is enough to show the existence of a Liouvillian first integral of the reduced foliation F * .Let D = ∪D j be the divisor obtained on the process of reducing the singularities.Let us fix a transversal section of F * through D j .Since there exists a Levi foliation tangent to F * , 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 F * 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 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 = ∪D j obtained on the resolution of the singularity.Let F j be a Liouvillian first integral of the holomorphic foliation F * on a neighborhood of the D j such that (F j ) is a first integral of L * .By Theorem b, for each is a well defined closed 1-form.Thus 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 L uma folheação de Levi e seja F a folheação holomorfa induzida.Então F tem integral primeira Liouvilliana.