On deformation of foliations with a center in the projective space

Abstracts

Let <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif"> be a foliation in the projective space of dimension two with a first integral of the type <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img2.gif">, where F and G are two polynomials on an affine coordinate, <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img3.gif"> = <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img4.gif"> and g.c.d.(p, q) = 1. Let z be a nondegenerate critical point of <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img2.gif">, which is a center singularity of <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif">, and <img src="http:/img/fbpe/aabc/v73n2/ft.gif" alt="ft.gif (149 bytes)" align="middle"> be a deformation of <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif"> in the space of foliations of degree deg(<img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif">) such that its unique deformed singularity <img src="http:/img/fbpe/aabc/v73n2/zt.gif" alt="zt.gif (118 bytes)"> near z persists in being a center. We will prove that the foliation <img src="http:/img/fbpe/aabc/v73n2/ft.gif" alt="ft.gif (149 bytes)" align="middle"> has a first integral of the same type of <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif">. Using the arguments of the proof of this result we will give a lower bound for the maximum number of limit cycles of real polynomial differential equations of a fixed degree in the real plane.

Holomorphic foliation; limit cycle; center singularity

Seja <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif"> uma folheação no espaço projetivo de dimensão dois e com integral primeira do tipo <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img2.gif">, onde F e G são dois polinômios numa carta afim e <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img3.gif"> = <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img4.gif"> e g.c.d.(p, q) = 1. Seja z um ponto crítico não degenerado de <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img2.gif"> e <img src="http:/img/fbpe/aabc/v73n2/ft.gif" alt="ft.gif (149 bytes)" align="middle"> uma deformação de <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif"> no espaço das folheações de grau deg(<img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif">) tal que a singularidade deformada <img src="http:/img/fbpe/aabc/v73n2/zt.gif" alt="zt.gif (118 bytes)"> perto de z ainda é um centro. Provamos que a folheação <img src="http:/img/fbpe/aabc/v73n2/ft.gif" alt="ft.gif (149 bytes)" align="middle"> tem uma integral primeira do mesmo tipo de <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif">. Usando os argumentos da demonstração desse resultado daremos uma cota inferior para o numero máximo de ciclos limites de uma equação differential de grau fixo no plano real.

folheação holomórfica; ciclo limite; singularidade do tipo centro

On Deformation of Foliations with a Center in the Projective Space

HOSSEIN MOVASATI

IMPA - Instituto de Matemática Pura e Aplicada

Estrada Dona Castorina, 110 - 22460-320 Rio de Janeiro, RJ, Brazil.

Manuscript received on January 31, 2001; accepted for publication on February 14, 2001;

presented by A. LINS NETO

ABSTRACT

Let be a foliation in the projective space of dimension two with a first integral of the type , where F and G are two polynomials on an affine coordinate, = and g.c.d.(p, q) = 1. Let z be a nondegenerate critical point of , which is a center singularity of , and be a deformation of in the space of foliations of degree deg() such that its unique deformed singularity near z persists in being a center. We will prove that the foliation has a first integral of the same type of . Using the arguments of the proof of this result we will give a lower bound for the maximum number of limit cycles of real polynomial differential equations of a fixed degree in the real plane.

Key words: Holomorphic foliation, limit cycle, center singularity.

1. INTRODUCTION

Consider the Hamiltonian equation

in

², where H is a morse polynomial of degree n + 1 in ², i.e. the critical points of H are nondegenerate with distinct images in . The above equation has n² singularities of center type (Morse type). Let also consider the perturbation

where P and Q are two polynomials of degree at most n. Ilyashenko in (Ilyashenko 1969) proves that if (1) has a center singularity for all then it must be again Hamiltonian. The objective of this article is to generalize Ilyashenko's result to foliations in P(2) with a generic first integral of the type .

Let F and G be two polynomials of degrees a + 1 and b + 1, respectively, in ², = and g.c.d.(p, q) = 1. Consider the foliation in P(2) with the first integral

f : P(2)\({F = 0} {G = 0})

f (x, y) =

In other words is given by the 1-form pGdF - qFdG. We will denote the foliation defined by the 1-form by (). Note that the line at infinity H = P(2)\² is neither pole nor zero component of f. A center singularity of is a nondegenerate singularity of with a local Morse integral i.e., around the singularity the foliation is given by x² + y² = const. in some analytic coordinate system (x, y). Therefore in the leaves around a center singularity there exists a nontrivial cycle which is called the Lefschetz vanishing cycle.

The points of {F = 0} {G = 0} are dicritical singularities of . If F = 0 intersects G = 0 transversally, these points are called the radial singularities of .

Let be a polynomial 1-form in ². Considering as a meromorphic 1-form in P(2), it has a pole of order k along H. The degree of is defined to be k - 2. This definition of degree has a difference up to one with the maximum of the degrees of the polynomials defining (see (Lins Neto and Scárdua 1997)).

We denote by (a, b) the set of all integrable foliations of the above type, by _m(a, b) the set of = (pGdF - qFdG) (a, b) such that all singularities of are centers or radials and the image of the centers under , 0 and are distinct points in and by (2, d ) the space of foliations of degree d in P(2). It is easy to see that (a, b) (2, d ), where d = a + b. For more information about holomorphic foliations the reader is referred to (Camacho and Sad 1987) and (Lins Neto and Scárdua 1997).

PROPOSITION 1.

_m(a, b) is an open dense set in (a, b).

Let

_t

(2, d ), t (, 0) be a holomorphic deformation of the foliation = ₀ which has at least one center for all t (, 0). If we know the structure of , for example the type of singularities, holonomy groups of some leaves and etc., then what can be said about the deformed foliation _t?

The foliation has a finite number of singularities therefore, there exists a center p sing() and a sequence of centers pt_i

sing(t_i) such that pt_i

p when t_i 0. The deformed holonomy h_t(x) related to the vanishing cycle around p, is the identity for t = t_i. Since h_t is holomorphic in t, h_t is the identity for all t. In particular, p_t is a center for all t (, 0).

Given an integrable foliation (pGdF - qFdG) (a, b), let p be one of the center singularities of and let _t = (pGdF - qFdG + t + h.o.t.) be a holomorphic deformation of in (2, d ), where d = a + b, such that its unique singularity p_t near p is still a center. Our principal theorem in this article is the following:

THEOREM 1. In the above situation, if a + b > 2 then there exists an open dense subset U of (a, b) such that for all (pGdF - qFdG) U and for all deformations _t as before, _t is also an integrable foliation. More precisely, there exist polynomials F_t and G_t such that _t = (pG_tdF_t - qF_tdG_t), where F_t and G_t are holomorphic in t and F₀ = F and G₀ = G.

This theorem also says that the persistence of one center along the deformation implies the persistence of all other centers and radials. J.R. Muciño in (Muciño 1995) has also studied this problem in the case p = q = 1. However, he has not obtained the result of Theorem 1 in this case, as we will explain later in §2.

EXAMPLE. Let F and G be generic conics. The pencil F + tG = 0 contains a reducible fiber, say F + t₀G = F₁F₂, where deg(F₁) = deg(F₂) = 1. Now (FdG - GdF) = (Fd (F₁F₂) - F₁F₂dF) and this foliation admits the deformation

( + + ), 2 + + = 0

of degree two which is not integrable but has many center singularities. The reason for which Theorem 1 fails is that in this example we have reducible fibers. In fact only in the case a + b 2 the foliation

_m(a, b) has reducible fibers. In the above example Theorem 3 of §2 is not true and hence Theorem 1 is not true also. From now on we will assume that a + b > 2.

2. IDEA OF THE PROOF OF THEOREM 1

The proof of Theorem 1 lies in Theorems 2, 3, 4, and Proposition 2. First we consider the local situation of the theorem.

THEOREM 2. In the above situation, if the singularity of

_t near p persists in being a center then

= 0

for all Lefschetz vanishing cycles in the leaves around the center p, where is the tangent vector of the deformation.

DEFINITION 1. Let

(2, d ) and let and be two closed cycles contained in two leaves of . We say that the cycle is -equivalent with , if there is a continuous sequence of closed cycles , 0 t 1 such that

• is a cycle in some leaf of

  ;

• =

  and

  =

  .

A closed cycle is called a vanishing cycle if it is -equivalent with a zero cycle, which is a point p, in (P(2)\sing()) {p}, and is called a Lefschetz vanishing cycle if it is -equivalent with the singularity p of a center in (P(2)\sing()) {p}. Note that here we consider a singularity as a leaf and therefore the cycles introduced around the centers satisfy this definition.

THEOREM 3. Let (pGdF - qFdG)

_m(a, b), a + b > 2, L be a regular leaf of and be a Lefschetz vanishing cycle in L. Then the set of cycles in L which are -equivalent with in P(2)\( f^-1(c_i) {F = 0} {G = 0}) generate the first homology group of L in the field of rational numbers.

The proof of above theorem uses arguments of the article (Lamotke 1981). By definition, the leaves of (pGdF - qFdG) are the fibers of without the radial singularities. Therefore this theorem partially claims that the cycles around radial singularities are rational sums of cycles -equivalent with the cycle . In the case p = q = 1, is a Lefschetz pencil and the mentioned fact is not explicitly stated in the literature. This has not any contradiction with Hard Lefschetz Theorem (4.1.3 p.29 (Lamotke 1981)). Note that the Hard Lefschetz Theorem is for compact fibers and not fibers with deleted radial singularities. Disconsideration of the above theorem in (Muciño 1995) has caused that the author of that paper has not obtained Theorem 1 in the case p=q=1. In fact he assumes the following unnecessary hypothesis: vanishes over cycles around radial singularities. A deeper analysis of Lefschetz's argument (see (Lamotke 1981)) is needed to prove the above theorem in the case p = q = 1.

Let p be the center of in Theorem 1 which persists in being a center after the deformation of . Applying Theorem 2 to the deformation

_t : p - q + t + h.o.t.

we obtain that

(2)

for all vanishing cycles in the leaves around p and by Theorem 3 we conclude that the equality (2) holds for all closed cycles in the leaves of the foliation , where the integral is defined. Partially we obtain that the residue of on a leaf around a radial singularity is zero. The 1-form is called a relatively exact 1-form modulo the foliation (pGdF - qFdG).

THEOREM 4. Let (pGdF - qFdG)

_m(a, b). Suppose that is a polynomial 1-form in ² with deg() deg() and is relatively exact modulo . Then there is (P, Q)

_{a + 1}x _{b + 1} such that has the form

=

(3)

Now let (2, d ) be the set of foliations of degree d in P(2) with at least one center. I have learned the statement and proof of the following proposition from A. Lins Neto.

PROPOSITION 2. (2, d ) is an algebraic subset of (2, d ).

The next theorem identifies some irreducible components of (2, d ).

THEOREM 5. If a + b > 2 then (a, b) is an irreducible component of (2, d ), where d = a + b.

PROOF. Since (a, b) is parameterized by _{a + 1}x _{b + 1}, (a, b) is an irreducible variety. For any (pGdF - qFdG)

_m(a, b), we have seen in the theorems 2, 3 and 4 that

[] T

(2, d )

= pGdP - qPdG + pFdQ - QdF, (P, Q)

_{a + 1}x _{b + 1}

[] T

(a, b)

this and the fact that (a, b) (2, d ) imply that

T

(2, d )= T

(a, b),

_m(a, b)

Since

_m(a, b) is an open dense subset of (a, b), we conclude that (a, b) is an irreducible component of (2, d ).

Theorem 1 is a direct consequence of the above Theorem.

3. APPLICATION

The idea of the following definition comes from the converse version of Theorem 2.

DEFINITION 2. Let X be an irreducible component of (2, d ), () X and p be a center of . There is a coordinate (x, y) in a small neighbourhood U of p such that in this coordinate p = (0, 0) and

= gd (), = (x² + y²) + h.o.t., g(0) 0

where and g are holomorphic functions on U. Define T as the set of all 1-forms [] T

(2, d ) such that

= 0

for all Lefschetz vanishing cycles in the leaves of around p. By Theorem 2, we know that

(4)

X is called a good irreducible component of (2, d ) if for a generic choice of

X, the equality holds in (4). This definition does not depend on the choice of g.

The process of the proof of Theorem 1 shows that (a, b) is a good irreducible component of (2, d).

Now let X be an irreducible component of (2, d ), () X be a smooth point of X\((2, d) X) and p be a center singularity of . Furthermore, if is a real 1-form assume that the real foliation induced by has a real center at p

².

THEOREM 6. There is a real polynomial differential equation of degree d and with at least

N = dim(T

(2, d )/T) - 1

limit cycles in the real plane

².

We obtain this fact by a small deformation of (), where is a real 1-form. Applying the above theorem to (a, b) we have:

COROLLARY 1. There is a real polynomial differential equation of degree d and with at least

(5)

limit cycles.

RESUMO

Seja uma folheação no espaço projetivo de dimensão dois e com integral primeira do tipo , onde F e G são dois polinômios numa carta afim e = e g.c.d.(p, q) = 1. Seja z um ponto crítico não degenerado de e uma deformação de no espaço das folheações de grau deg() tal que a singularidade deformada perto de z ainda é um centro. Provamos que a folheação tem uma integral primeira do mesmo tipo de . Usando os argumentos da demonstração desse resultado daremos uma cota inferior para o numero máximo de ciclos limites de uma equação differential de grau fixo no plano real.

Palavras-chave: folheação holomórfica, ciclo limite, singularidade do tipo centro.

E-mail: [e-mail]

• CAMACHO C AND SAD P. 1987. Pontos Singulares de Equaçőes Diferenciais Analíticas, 16^o Colóquio Brasileiro de Matemática.
• ILYASHENKO YU. 1969. The Origin of Limit Cycles Under Perturbation of Equation dw/dz = -R_z/R_w, where R(z,w) is a Polynomial, Math. USSR, Sbornik, vol 7, No. 3.
• LAMOTKE K. 1981. The Topology of Complex Projective Varieties After S. Lefschetz, Topology, 20: 15-51.
• LINS NETO A AND SCÁRDUA BA. 1997. Folheaçőes Algébricas Complexas, 21^o Colóquio Brasileiro de Matemática.
• MUCIŃO JR. 1995. Deformations of holomorphic foliations having a meromorphic first integral, J reine angew Math 461: 189-219.

Publication Dates

• Publication in this collection
  08 June 2001
• Date of issue
  June 2001

History

• Accepted
  14 Feb 2001
• Received
  31 Jan 2001