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.
Guido Carlet. Hessel Posthuma. Sergey Shadrin. "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