Foliations on Complex Manifolds

Abstract

In these notes we survey some aspects of the theory of holomorphic foliations on complex manifolds. The origins of the theory go back to works of Darboux, Poincaré and Painlevé, where it was developed to study solutions of ordinary differential equations on ℂ 2 . We briefly discuss some of the early works on this theory, mostly concerned with the local behavior of the leaves near the singularities. We then move the focus from local to global properties. Birational geometry has had a great influence on the development of a global theory of holomorphic foliations. After reviewing the Enriques-Kodaira classification of projective surfaces and explaining the general philosophy of the Mininal Model Program, we explore some of their recent counterparts for foliations.