Structures de Poisson Logarithmiques : invariants cohomologiques et préquantification

Abstract : The main objective of this thesis is to propose a criteria of prequantization of singular Poisson structures with singularities carried by a free divisor of a fi nite dimensional complex manifold. For this, we start from an algebraic construction of formal logarithmic di fferentials along a fi nitely generated non trivial ideal of a commutative and unitary algebra. We introduce the concept of logarithmic Poisson algebra. Then, we show that these Poisson structures induce a new cohomological invariant, this is dow via the Lie-Rinehart algebra structure, that they induced on the module of formal logarithmic di fferentials. With the latter, we study the integrale conditions of such Poisson structures. First, we show that the Hamiltonian map of logarithmic Poisson structure extends to the module of formal logarithmic diff erential and induces a structure of Lie-Rinehart algebra on it. Furthermore, we show that its image is contained in the module of logarithmic derivations. We called logaruthmic Poisson cohomologie, the cohomologie induced by this representation. Subsequently, we show on some examples that Poisson cohomologies groups and Poisson logarithmic cohomologies groups are diff erent in general, although they coincide in the case of logsymplectic Poisson structures. We conclude with a study the prequantization conditions of all such structures by means of this cohomology.
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https://tel.archives-ouvertes.fr/tel-00985181
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Submitted on : Tuesday, April 29, 2014 - 12:31:43 PM
Last modification on : Friday, May 10, 2019 - 12:14:02 PM
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  • HAL Id : tel-00985181, version 1

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Joseph Dongho. Structures de Poisson Logarithmiques : invariants cohomologiques et préquantification. Analyse classique [math.CA]. Université d'Angers, 2012. Français. ⟨tel-00985181⟩

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