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Relative exactness and algebraic $$(\mathbb{C}^p,+)$$-actions. (Exactitude relative et actions algébriques de $$(\mathbb{C}^p,+)$$.) (French) Zbl 1037.14018
Summary: Let $$F$$ be a polynomial dominating map from $$\mathbb{C}^n$$ to $$\mathbb{C}^q$$. In this note we study the quotient $${\mathcal T}^1(F)$$ of polynomial 1-forms that are exact along the generic fibres of $$F$$, by 1-forms of type $$dR+\sum a_idf_i$$, where $$R$$, $$a_1,\dots,a_q$$ are all polynomials. We prove that $${\mathcal T}^1(F)$$ is always a torsion $$\mathbb{C} [t_1,\dots,t_q]$$-module. Then 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.

##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 32M17 Automorphism groups of $$\mathbb{C}^n$$ and affine manifolds 14E07 Birational automorphisms, Cremona group and generalizations
##### Keywords:
polynomial map; actions on $$\mathbb{C}^n$$
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