Déformations sur les variétés de Poisson et cohomologies appropriées

Abstract : In this thesis, we are interested in the tangential star products on Poisson manifolds, that is star products which restrict nicely to the symplectic leaves ? or at least to the ?generic? ones. We study in particular the tangential Poisson cohomology of regular Poisson manifolds, as a first step towards classifying the tangential star products on such manifolds. Next we give a necessary and sufficient condition under which there exist tangential star products, differential or not, on the dual g* of a nilpotent Lie algebra. We also give a cohomological existence proof of tangential differential star products on any regular Poisson manifold M, and then of tangential, differential and graded star products on the open set ? of maximal dimensional coadjoint orbits in the dual g* of a nilpotent Lie algebra. The second part concerns the formality of Kontsevich. We introduce a cohomology of Chevalley's type, which is naturally associated to the existence problem of formalities. Finally, we use the Kontsevich's local formality properties to construct tangential star products on foliated Poisson manifolds following the construction of Fedosov and Cattaneo-Felder-Tomassini. This result generalizes the existence proof of tangential star products for regular Poisson manifolds
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Angela Gammella. Déformations sur les variétés de Poisson et cohomologies appropriées. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 2001. Français. ⟨NNT : 2001METZ012S⟩. ⟨tel-01775447⟩

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