Holomorphic Foliations: Singularities and Local Geometric Aspects

This text has been written with the aim of providing a fast introduction to the framework of holomorphic foliations with singularities. Our methodology is based in proving a couple of important results. The first is the theorem of Mattei-Moussu about existence of holomorphic first integrals for germs of holomorphic foliations. The second is the linearization theorem of Camacho-Lins Neto-Sad, for foliations in the complex projective plane. With these we address both aspects, local and global, of this interesting subject. This text is highly influenced by the author’s personal interests and it is not intended to exhaust the subject, nor to be a complete full introduction to this beautiful field. Indeed, I also recommend various other texts as, for instance, by D. Cerveau and J.-F. Mattei, Y. Ilyashenko and F. Loray (see references below). I hope these notes will be helpful to those interested in this interesting field in mathematics.