Deformations of Poisson pencils, bihamiltonian cohomology and spectral sequences.

I will report on recent work with H. Posthuma and S. Shadrin on the computation of the bihamiltonian cohomology groups associated with pencils of Poisson brackets of hydrodynamic type. After reviewing some of the theory, I will present the following results and outline the corresponding proofs: the computation of the bihamiltonian cohomology of the KdV Poisson pencil; the generalization to the case of a general scalar Poisson pencil; the vanishing of (most of) the bihamiltonian cohomology of a semisimple Poisson pencil with n dependent variables. These results in particular imply that any infinitesimally deformed Poisson pencil can be extended to a full dispersive one.