On deformation of foliations with a center in the projective space

Abstract

Let F be a foliation in the projective space of dimension two with a first integral of the type G q , which is a center singularity of F, and F t be a deformation of F in the space of foliations of degree deg(F) such that its unique deformed singularity z t near z persists in being a center. We will prove that the foliation F t has a first integral of the same type of F. 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.

Center conditions: rigidity of logarithmic differential equations

“…The similar result for foliations with a first integral of the type F p G q ; degðF Þ degðGÞ ¼ q p ; g:c:d:ðp; qÞ ¼ 1 is obtained in [Mo,Mo1]. Some basic tools of this kind of generalizations for Lefschetz pencils on a manifold is worked in [Muc].…”

Abelian integrals in holomorphic foliations

“…This result is announced in [Ho1]. We can restate our main theorem as follows: Let F ∈ I(a, b) This theorem also says that the persistence of one center implies the persistence of all other centers and dicritical singularities (the points of {F = 0} ∩ {G = 0}).…”

“…Let I(a, b) be the closure of the set of the mentioned holomorphic foliations in F(d). Our main result in this section is the following: This result is announced in [Ho1]. We can restate our main theorem as follows: Let F ∈ I(a, b), p one of the center singularities of F and F t a holomorphic deformation of F in F(d), where d = a+ b, such that its unique singularity p t near p is still a center.…”

“…J. Muciño in [Mu] has classified a certain class of relatively exact 1-forms modulo a Lefschetz pencil. In a generalization of Ilyashenko's result to integrable foliations in M , I had to classify another types of relatively exact 1-forms in [Ho1].…”

“…Without the hypothesis of connectedness of f -fibers the classification of relatively exact polynomial 1-forms modulo a polynomial is done in [Bo]. In [Ho1] we have classified relatively exact 1-form in P 2 with non-invariant divisors. The forth condition is trivial for an algebraic manifold with a Lefschetz pencil in it.…”