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 ${\cal{T}}^1(F)$ of polynomial 1-forms that are exact along the fibres of $F$, by 1-forms of type $dR+\sum a_idf_i$, where $R,a_1,...,a_q$ are polynomials. We prove that ${\cal{T}}^1(F)$ is always a torsion $\mathbb{C}[t_1,...,t_q]$-module. The we determine under which conditions on $F$ we have ${\cal{T}}^1(F)=0$. 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.