Cohomologie relative des applications polynomiales

Abstract

Let F be a polynomial dominating mapping from C n to C q with n>q. We study the de Rham cohomology of the fibres of F, and its relative cohomology groups. Let us fix a strictly positive weighted homogeneous degree on C [x 1,…,x n] . With the leading terms of the coordinate functions of F, we construct a fibre of F that is said to be “at infinity”. We introduce the cohomology groups of F at infinity. These groups, denoted by H k(F −1(∞)) , enable us to study all the other cohomology groups of F. For instance, if the fibre at infinity has an isolated singularity at the origin, we prove that any quasi-homogeneous basis of H n−q(F −1(∞)) provides a basis of all groups H n−q(F −1(y)) , as well as a basis of the (n−q)-th relative cohomology group of F. Moreover the dimension of all these groups is equal to a global Milnor number of F, which only depends on the leading terms of the coordinate functions of F.