Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed

TL;DR

This paper proves that for semisimple Poisson pencils of hydrodynamic type, the associated bihamiltonian cohomology vanishes in almost all degrees, ensuring unobstructed dispersive deformations from any infinitesimal deformation.

Contribution

It establishes the vanishing of bihamiltonian cohomology for semisimple Poisson pencils, enabling full dispersive deformations to be constructed from infinitesimal ones.

Findings

Bihamiltonian cohomology vanishes in almost all degrees.

Existence of full dispersive deformations from any infinitesimal deformation.

Unobstructed deformation theory for semisimple Poisson pencils.

Abstract

We prove that the bihamiltonian cohomology of a semisimple pencil of Poisson brackets of hydrodynamic type vanishes for almost all degrees. This implies the existence of a full dispersive deformation of a semisimple bihamiltonian structure of hydrodynamic type starting from any infinitesimal deformation.