On topological rigidity of projective foliations

Let us denote by chi(n) the space of degree n is an element of N foliations of the complex projective plane CP(2) which leave invariant the line at infinity. We prove that for each n greater than or equal to 2 there exists an open dense subset Rig(n) subset of chi(n) such that any topologically trivial analytic deformation {F-t}(t is an element of D) of an element F-0 is an element of Rig(n), with F-t is an element of chi(n), for all t is an element of D, is analytically trivial. This is an improvement of a remarkable result of Ilyashenko. Other generalizations of these results are given as well as a description of the class of nonrigid foliations.