Deformations of holomorphic foliations having a meromorphic first integral. (English) Zbl 0816.32022

The author studies singular holomorphic foliations of codimension one defined on complex projective manifolds \(M\), with \(H^ 1(M, \mathbb{C}) = 0\), and having generic meromorphic functions as first integrals (where generic means that the functions define Lefschetz pencils of hypersurfaces in \(M)\). The above foliations form in a natural way analytic subspaces of the analytic spaces of singular holomorphic foliations on \(M\). The work studies the behavior of some properties of Lefschetz pencils under deformation. The main result asserts that an infinitesimal deformation of a Lefschetz pencil (this means a tangent vector to the space of foliations) is tangent to the subspace of foliations having a Lefschetz pencil as first integral if and only if the holonomy of the infinitesimal deformation is trivial. (The result is a generalization of a theorem of Yu. S. Ilyashenko for deformation of polynomial hamiltonian foliations in \(\mathbb{C}^ 2)\). The computation of the holonomy of the infinitesimal deformation depends on the study of the abelian integrals associated to the integrals of the one-form defining the infinitesimal deformation, along the leaves of the Lefschetz pencil. If the complex dimension of \(M\) is two, as one application it is possible to give a study of the nature and number of the homotopy loops in the leaves that persist under the deformation (which extends some results of I. G. Petrovskiĭ and G. M. Landis on limit cycles of hamiltonian equations). Also is possible to study the persistence (up to first order) of algebraic leaves of Lefschetz pencils. Moreover one explicit bound for the multiplicity of the zeros of the abelian integrals is given and some dynamical applications of this fact are given (the abelian integrals in this case are rational, some explicit bounds for the case of polynomial abelian integrals were given by Yu. S. Ilyashenko, G. Petrov and P. Mardešić and the existence of a global bound was shown by A. Varchenko and A. Khovanskiǐ). If the complex dimension of \(M\) is at least three, it is shown that Lefschetz pencils give open and dense sets of some irreducible components of the space of foliations (i.e. all foliations near a Lefschetz pencil are also Lefschetz pencils), in particular the structural stability of the foliations defined by Lefschetz pencils follows; the above results were firstly given by X. Gómez-Mont and A. Lins-Neto. By the same ideas it is also proved that polynomial foliations in \(\mathbb{C}^ m\), where \(m\) is at least three, having \(k\) generic polynomial first integrals (generic means which are Morse functions and are in general position between any two of them) and where the complex dimension of the leaves is at least two give open and dense sets of some irreducible components of the space of polynomial foliations of codimension \(k\) in \(\mathbb{C}^ m\).


32S65 Singularities of holomorphic vector fields and foliations
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
57R30 Foliations in differential topology; geometric theory
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