Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed

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.

💡 Research Summary

The paper addresses the long‑standing extension problem for deformations of semisimple Poisson pencils of hydrodynamic type. A Poisson pencil is a one‑parameter family of compatible Poisson brackets {·,·}_λ = {·,·}_2 – λ{·,·}_1, where the leading terms (λ‑independent) are of hydrodynamic type. Semisimplicity means that the characteristic polynomial det(g¹ – λ g²) has n distinct, non‑constant roots on a domain U ⊂ ℝⁿ; these roots serve as canonical coordinates u¹,…,uⁿ. In these coordinates the two metrics become diagonal, with entries f_i(u) and u_i f_i(u), respectively, where each f_i is a non‑vanishing smooth function.

The deformation problem asks whether a given infinitesimal deformation (i.e. a deformation truncated at order ε²) can be extended to a full formal series in the small parameter ε. Central invariants c_i(u_i), functions of a single canonical coordinate, classify infinitesimal deformations up to Miura transformations of the second kind (transformations that are the identity at order ε⁰). The main conjecture, formulated by Liu and Zhang, predicts that for semisimple pencils the obstruction groups—specifically the third bihamiltonian cohomology groups—vanish, guaranteeing unobstructed extensions for any choice of central invariants.

To prove this, the authors translate the deformation problem into a cohomological one. They work with the super‑commutative algebra
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