Relative cohomology of polynomial mappings

Abstract

Let $F$ be a polynomial mappping from $\mathbb{C}^n$ to $\mathbb{C}^q$ with $n>q$. We study the De Rham cohomology of its fibres and its relative cohomology groups, by introducing a special fibre $F^{-1}(\infty)$ "at infinity" and its cohomology. Let us fix a weighted homogeneous degree on $\mathbb{C}[x_1,...,x_n]$ with strictly positive weights. The fibre at infinity is the zero set of the leading terms of the coordinate functions of $F$. We introduce the cohomology groups $H^k(F^{-1}(\infty))$ of $F$ at infinity. These groups enable us to compute all the other cohomology groups of $F$. For instance, if the fibre at infinity has an isolated singularity at the origin, we prove that every weighted homogeneous basis of $H^{n-q}(F^{-1}(\infty))$ is a basis of all the groups $H^{n-q}(F^{-1}(y))$ and also a basis a the $(n-q)^{th}$ relative cohomology group of $F$. Moreover the dimension of $H^{n-q}(F^{-1}(\infty))$ is given by a global Milnor number of $F$, which only depends on the leading terms of the coordinate functions of $F$.