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.
"Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed." J. Differential Geom. 108 (1) 63 - 89, 2018. https://doi.org/10.4310/jdg/1513998030