6533b825fe1ef96bd12826f6

RESEARCH PRODUCT

Cohomologie relative des applications polynomiales

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Let F be a polynomial dominating mapping from Cn to Cq 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[x1,…,xn]. 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 Hk(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 Hn−q(F−1(∞)) provides a basis of all groups Hn−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.