Campos de vetores tangentes a folheações holomorfas de codimensão Um

Abstract

We say that a germ of vector field X in (...) is tangent to a holomorphic foliation defined by a germ of integrable 1-form (...). In this thesis we aim to study some geometric properties arising from this setting. We first observe that the singular set of ! is invariant by X. Thus, if the singularset of ! contains a curve, then X has a separatrix. In 1992, X. Goméz-Mont and I. Luengo presented a family of vector fields in (...) without separatrices. We prove that vector fields in this family are not tangent to foliations. Besides, we prove that if a vector field X tangent to a foliation has, in some desingularization, a singularity in the Poincaré domain, then X has a separatrix. A germ of vector field in (...) is said to be strongly non-resonant Poincaré if the linear part of X is in the Poincaré domain with strongly non-resonant eigenvaluesthat is, without non-trivial linear relations with integer coefficients. A foliation G of codimension one is complex hyperbolic if for every map (...), holomorphic and transversal to G, the two-dimensional foliation (...)G is of generalized curve type that is, there are no saddle-nodes in its desingularization. Let F be a germ of one dimensional foliation in (...) with isolated singularity at (...), having a desingularization by nondicritical punctual blow-ups such that all singularities are of strongly non-resonant Poincaré type. If a foliation G of codimension one is invariant by F with such characteristics, then G is a complex hyperbolic foliation. Finally, we considere a germ of holomorphic vector field X in (...) tangent to three independent foliations. We prove that X is tangent to a linear pencil of foliations and, therefore, to infinitely many foliations. As a consequence, the vector field X has invariant surfaces.