Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

In a seminal paper "The local structure of Poisson manifold" (1983) A. Weinstein proved that for any Poisson manifold (M,P) there exists a local symplectic realization, i.e. nondegenerate Poisson manifold (M',P') and a local surjective submersion f:M'->M with f_*P'=P. Global aspects of this problem were afterwards intensively studied as they are related to the theory of symplectic and Poisson grupoids, to the integration problem of Lie algebroids, and to different quantization schemes. In this talk I will discuss a problem of local simultaneous realization of two compatible Poisson structures by means of two nondegenerate ones. Note the following essential difference between the two realization problems: there is only one local model of the nondegenerate Poisson bivector P' given by the Darboux theorem and there are many local models of bisymplectic bihamiltonian structures. So besides the problem of existence it is important to understand how many nonequivalent realizations there are in the second case.