Campos de vetores tangentes a folheações holomorfas de codimensão Um
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 singularse...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2016 |
| País: | Brasil |
| Institución: | Universidade Federal de Minas Gerais (UFMG) |
| Repositorio: | Repositório Institucional da UFMG |
| Idioma: | portugués |
| OAI Identifier: | oai:repositorio.ufmg.br:1843/EABA-A9FJRW |
| Acceso en línea: | http://hdl.handle.net/1843/EABA-A9FJRW |
| Access Level: | acceso abierto |
| Palabra clave: | Integral primeira holomorfa separatriz feixe linear de folheações folheação hiperbólica complexa dessingularização variedade invariante Matemática Folheações (Matemática) Campos vetoriais Folheações (Matematica) Geometria integral Funções holomorficas |
| Sumario: | 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. |
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