Relative exactness modulo a polynomial map and algebraic $({\mathbb{C}}^p,+)$-actions

Abstract

Let $F=(f_1,...,f_q)$ be a polynomial dominating map from ${\mathbb{C}}^n$ to ${\mathbb{C}}^q$. We study the quotient ${\mathcal{T}}^1(F)$ of polynomial 1-forms that are exact along the fibres of $F$, by 1-forms of type $dR+\sum a_{i}d{f}_i$, where $R,a_1,...,a_q$ are polynomials. We prove that ${\mathcal{T}}^1(F)$ is always a torsion $\mathbb{C}[t_1,...,t_q]$-module. The we determine under which conditions on $F$ we have ${\mathcal{T}}^1(F)=0$. As an application, we study the behaviour of a class of algebraic $({\mathbb{C}}^p,+)$-actions on ${\mathbb{C}}^n$, and determine in particular when these actions are trivial.

26 pages