Introduction to Holomorphic Foliations

Prerequisite:
Complex Analysis (can be run in parallel)

Singularities of holomorphic vector fields. Poincaré linearization theorem.  Poincaré-Dulac theorem. Reduced singularities and reduction theorem (Seidenberg). Mattei-Moussu criterion for existence of holomorphic first integrals. Groups of germs of diffeomorphisms. Solvable groups. Nakai’s theorem. Foliations in the projective plane. Algebraic leaves. Darboux-Jouanolou theorem. Density of leaves for generic affine foliations. Inexistence of algebraic leaves for generic projective foliations.

References:
ARNOLD, V. I. – Geometrical Methods in the Theory of Ordinary Differential Equations. New York, Springer-Verlag, 1983.
CAMACHO, C., SAD. P. – Pontos Singulares de Equações Diferenciais Analíticas. 16º Colóquio Brasileiro de Matemática, Rio de Janeiro, IMPA, 1987.
GOMEZ-MONT, X., ORTÍZ-BOBADILLA, L. – Sistemas Dinámicos Holomorfos en Superfícies. México: Sociedad Matemática Mexicana, 1989.
LORAY, F. – Pseudo-groupe d’une Singularité de Feuilletage Holomorphe en Dimension Deux. Disponível em http://hal.archives-ouvertes.fr/ccsd-00016434, 2006.

* Standard program. The teacher has the autonomy to make any changes