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

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Published in Bulletin de la société mathématique de France. 2003, vol. 131, no. 3, p. 373-398
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.
Stable URL https://archive-ouverte.unige.ch/unige:12293
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Deposited on : 2010-11-02

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