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Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed. (English) Zbl 1387.53105
The deformation problem of semisimple Poisson pencils of hydrodynamic type is considered. A precise formulation of this deformation problem is given in canonical coordinates.
The main open problem in the deformation theory of a semisimple Poisson pencil is the problem of extension, namely the existence of a full deformation. In [S.-Q. Liu and Y. Zhang, J. Geom. Phys. 54, No. 4, 427–453 (2005; Zbl 1079.37058)], a positive solution of the existence problem was conjectured, and also formulated in terms of vanishing of the third bi-Hamiltonian cohomology groups.
“The main result of this paper is the proof of this conjecture, i.e., the affirmative answer to the extension problem.”

53D17 Poisson manifolds; Poisson groupoids and algebroids
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