ar X iv : n lin / 0 10 30 52 v 1 [ nl in . S I ] 2 7 M ar 2 00 1 On deformation of Poisson manifolds of hydrodynamic type ∗

We study a class of deformations of infinite-dimensional Poisson manifolds of hydrodynamic type which are of interest in the theory of Frobenius manifolds. We prove two results. First, we show that the second cohomology group of these manifolds, in the Poisson-Lichnerowicz cohomology, is “essentially” trivial. Then, we prove a conjecture of B. Dubrovin about the triviality of homogeneous formal deformations of the above manifolds. 1 Dubrovin’s conjecture In this paper we solve a problem proposed by B. Dubrovin in the framework of the theory of Frobenius manifolds [2]. It concerns the deformations of Poisson tensors of hydrodynamic type. The challenge is to show that a large class of these deformations are trivial. In an epitomized form the problem can be stated as follows. Let M be a Poisson manifold endowed with a Poisson bivector P0 fulfilling the Jacobi condition [P0, P0] = 0 with respect to the Schouten bracket on the algebra of multivector fields on M . A deformation of P0 is a formal series Pǫ = P0 + ǫP1 + ǫ P2 + · · · in the space of bivector fields on M satisfying the Jacobi condition [Pǫ, Pǫ] = 0 (1) ∗Work sponsored by the Italian Ministry of Research under the project 40%: Geometry of Integrable Systems