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Résolutions symplectiques et de contact de variétés de Poisson et de Jacobi

Abstract : Poisson and Jacobi structures can be singular in two ways: the structure can be singular (we then say: singularity of the first type), but the variety itself can also have singularities (we then say: singularity of the second type). In both cases, solving the singularity consists in finding a smooth object equipped a symplectic or contact structure that projects onto the singular object under consideration. Several works deal with these different types of singularities. For those of the second type, Hironaka type methods have been proposed in the framework of algebraic geometry. For those of the first type, in a framework of differential geometry, it is well known that it is possible to turn the Poisson structure and the Jacobi structure into a symplectic structure and a contact structure if we allow to double the dimension. The aim of this thesis is to give some milestones for a coherent theory of the resolution of the two types of singularities for Poisson and Jacobi varieties. We want, however, 1) not to increase the dimension and 2) to remain within the framework of differential geometry – i.e. we work with smooth functions. The first of its milestones is a negative result: we show that there are no reasonable resolutions of singularities of the first type when the singular locus is of codimension one. We also give examples that show that in codimension two such a resolution can exist. We do this for both Poisson and Jacobi structures. The last two chapters are devoted to solving the second type of singularity. We begin by suggesting a new point of view on known results on the Du Val singularity which are quotients of R^2 by finite groups of Sl (2, R). Finally, when using Du Val's symplectic resolutions, we give in the last chapter an example of a proper symplectic resolution of a singular Poisson object: the quotient of R ^ 2 by an infinite subgroup of Gl (2, R).
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Hichem Lassoued. Résolutions symplectiques et de contact de variétés de Poisson et de Jacobi. Géométrie différentielle [math.DG]. Université de Lorraine, 2019. Français. ⟨NNT : 2019LORR0211⟩. ⟨tel-02527779⟩



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