Unipotent Reduction and the Poincaré Problem

we obtain 1.1 ([4], [14]). The aim of this paper is to give some results concerning the so called Poincaré Problem. Roughly speaking, this problem asks for a numerical criteria to identify when a morphism F : L −→ TS, defining a foliation, is equal to another of the form Tf −→ TS, with f : S −→ P 1 a holomorphic map and Tf (that must be isomorphic to our original L) the sheaf of relative vector fields of f . The original statement of the problem is on P ([11], [12]), in this case f must be a rational map f : P 99K P, undefined in a finite number of points (Bezout’s Theorem). Section 2 of this paper is devoted to an explanation of how this situation can be modified to the case of a holomorphic map f : S −→ P, (S will be the blowing-up of S in the indetermination locus of the original rational map). The classical formulation of the Poincaré Problem is explained there. This expository section (which includes, moreover, several results on foliation theory used below) does not contains any original result and is intended as an effort to fill a hypothetical gap between the specialists in foliation theory and those in fibration theory. Once the bridge between both theories is constructed we work almost completely with the language of fibration theory. Section 3 studies the problem of bounding the genus of a fibration: