Relative exactness modulo a polynomial map and algebraic ( ℂ p , + ) -actions

Philippe Bonnet

Bulletin de la Société Mathématique de France (2003)

  • Volume: 131, Issue: 3, page 373-398
  • ISSN: 0037-9484

Abstract

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Let F = ( f 1 , ... , f q ) be a polynomial dominating map from ℂ n to  ℂ q . We study the quotient 𝒯 1 ( F ) of polynomial 1-forms that are exact along the generic fibres of F , by 1-forms of type d R + ∑ a i d f i , where R , a 1 , ... , a q are polynomials. We prove that 𝒯 1 ( F ) is always a torsion ℂ [ t 1 , ... , t q ] -module. Then we determine under which conditions on F we have 𝒯 1 ( F ) = 0 . As an application, we study the behaviour of a class of algebraic ( ℂ p , + ) -actions on ℂ n , and determine in particular when these actions are trivial.

How to cite

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Bonnet, Philippe. "Relative exactness modulo a polynomial map and algebraic $(\mathbb {C}^p , +)$-actions." Bulletin de la Société Mathématique de France 131.3 (2003): 373-398. <http://eudml.org/doc/272374>.

@article{Bonnet2003,
abstract = {Let $F=(f_1,\ldots ,f_q)$ 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 generic fibres of $F$, by 1-forms of type $\mathrm \{d\} R + \sum a_i \mathrm \{d\} f_i$, where $R,a_1,\ldots ,a_q$ are polynomials. We prove that $\{\mathcal \{T\}\}^1(F)$ is always a torsion $\mathbb \{C\}[t_1,\ldots ,t_q]$-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.},
author = {Bonnet, Philippe},
journal = {Bulletin de la Société Mathématique de France},
keywords = {affine geometry; relative cohomology; invariant theory},
language = {eng},
number = {3},
pages = {373-398},
publisher = {Société mathématique de France},
title = {Relative exactness modulo a polynomial map and algebraic $(\mathbb \{C\}^p , +)$-actions},
url = {http://eudml.org/doc/272374},
volume = {131},
year = {2003},
}

TY - JOUR
AU - Bonnet, Philippe
TI - Relative exactness modulo a polynomial map and algebraic $(\mathbb {C}^p , +)$-actions
JO - Bulletin de la Société Mathématique de France
PY - 2003
PB - Société mathématique de France
VL - 131
IS - 3
SP - 373
EP - 398
AB - Let $F=(f_1,\ldots ,f_q)$ 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 generic fibres of $F$, by 1-forms of type $\mathrm {d} R + \sum a_i \mathrm {d} f_i$, where $R,a_1,\ldots ,a_q$ are polynomials. We prove that ${\mathcal {T}}^1(F)$ is always a torsion $\mathbb {C}[t_1,\ldots ,t_q]$-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.
LA - eng
KW - affine geometry; relative cohomology; invariant theory
UR - http://eudml.org/doc/272374
ER -

References

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