Cohomological Aspects in Semilocal Poisson Geometry

Abstract

This thesis is devoted to the study of the low-degree Poisson cohomology in the semilocal context, that is, in neighborhoods of symplectic leaves. The cohomology theory of Poisson manifolds was introduced by Lichnerowicz, together with the description of Poisson brackets in terms of bivector fields [44]. Indeed, he observed that the Jacobi identity for a Poisson bracket can be translated to its associated bivector field in terms of the Schouten-Nijenhuis bracket for multivector fields. On the other hand, by the properties of the Schouten-Nijenhuis bracket, every bivector field induces a graded derivation of degree 1 on the algebra of multivector fields, namely, the adjoint operator with respect to this bracket. It turns out that such operator is a coboundary exactly when the given bivector field is associated with a Poisson bracket. This gives rise to the Lichnerowicz-Poisson complex, and its cohomology is known as the Poisson cohomology of the Poisson manifold. From a geometric perspective, the Poisson cohomology in low degree has important interpretations. For example, the zeroth Poisson cohomology is isomorphic to the algebra of Casimir functions, being these functions the ones which are constant along the symplectic leaves of the Poisson manifold. Also, the first Poisson cohomology is the quotient of the Lie algebra of the infinitesimal automorphisms of the Poisson manifold by its ideal of Hamiltonian vector fields [44, 74]. On the other hand, the Poisson cohomology is a well-suited algebraic framework to express obstructions. For example, the unimodularity of an orientable Poisson manifold, which is the existence of a volume form invariant under every Hamiltonian flow, is controlled by the modular class, which lies in the first cohomology of the Poisson manifold [83]: the modular class vanishes if and only if such invariant volume form exists. The modular class is one of the most important Poisson cohomology classes, and is the first of the so-called characteristic classes [23]. On the other hand, the second Poisson cohomology is related to obstructions in quantization theory [63], [64, Chapter 6], and semilocal linearizability [73]. Second and higher-degree Poisson cohomology has applications, for instance, in deformation theory [21, Subsection 2.1.2]. As a cohomological theory, Poisson cohomology has many desirable properties. For example, there exists a natural morphism from the de Rham to the Poisson cohomology of the manifold. On the other hand, the Mayer-Vietoris sequence holds for Poisson cohomology, which allows to reduce the problem of global computation to smaller open sets. Moreover, the Lichnerowicz-Poisson complex admits a natural filtration whose associated spectral sequence is convergent to the Poisson cohomology [62, Section 1]. In the case of regular Poisson manifolds, there exists a recursive procedure for its computation, which was introduced first by Karasev and Vorobiev in the context of symplectic fibrations [74], and then adapted by Vaisman to any regular Poisson manifold [62, Section 2]. Also, Xu presented a method for the computation of Poisson cohomology in the regular case by means of symplectic groupoids [84].