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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.

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
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