# Un isomorphisme motivique entre deux variétés homogènes projectives sous l'action d'un groupe de type $G_2$

Abstract : In this thesis, k is a field of characteristic different from 2 and variety means separated k-scheme of finite type. We study the homogenous projective varieties $X(\alpha_1)$ and $X(\alpha_2)$ associated to each two roots of a group of type $G_2$. The first one, $X(\alpha_1)$, is a 5-dimensional projective quadric associated to a Pfister neighbour and the second one, $X(\alpha_2)$, is a Fano variety (of genus 10). They are not isomorphic as algebraic varieties but they become isomorphic as objects of the category of correspoondences (and of consequently as objects in the category of Chow motives). We establish this result whether the varieties are split or not. In the first chapter, we recall classical results about octonions and we construct a split example of such an algebra. In the second, we present our varieties and recall some essential notions about algebraic groups theory and functor of points. In the third chapter, we construct in the split case a cellular structure for $X(\alpha_2)$. This is an important step of our work. We also find the relations defining the structure product of the Chow ring of $X(\alpha_2)$. Finally, in the last chapter, we define the category of correspondences, proove a nilpotence theorem in the particular case of $X(\alpha_2)$ and establish the motivic isomorphism in generality.
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https://tel.archives-ouvertes.fr/tel-00004214
Contributor : Jean-Paul Bonnet <>
Submitted on : Monday, January 19, 2004 - 1:19:15 PM
Last modification on : Sunday, November 29, 2020 - 3:24:07 AM
Long-term archiving on: : Friday, April 2, 2010 - 8:01:51 PM

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• HAL Id : tel-00004214, version 1

### Citation

Jean-Paul Bonnet. Un isomorphisme motivique entre deux variétés homogènes projectives sous l'action d'un groupe de type $G_2$. Mathématiques [math]. Université des Sciences et Technologie de Lille - Lille I, 2003. Français. ⟨tel-00004214⟩

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