Veronese Webs for Bihamiltonian Structures of Higher Corank

0 Introduction. A Cmanifold M is endowed by a Poisson pair if two linearly independent smooth bivectors c1, c2 are defined on M and cλ = λ1c1 + λ2c2 is a Poisson bivector for any λ = (λ1, λ2) ∈ R . A bihamiltonian structure J = {cλ} is the whole 2-dimensional family of bivectors. The structure J is degenerate if rank cλ < dim M,λ ∈ R. An intensive study of such objects was done by I.M.Gelfand and I.S.Zakharevich ([10], [11], [12]) in a particular case of bihamiltonian structures in general position on an odddimensional M (the corresponding Poisson pairs are necessarily degenerate: rank cλ = 2n, λ ∈ R \ {0}, if dimM = 2n+1). In [11] there was introduced a notion of a Veronese web, i.e. a 1-parameter family of 1-codimensional foliations such that the corresponding family of annihilators is represented by the Veronese curve in the cotangent space at each point. It turns out that Veronese webs form a complete system of local invariants for bihamiltonian structures of general position. More precisely, it was shown in [11] that any such structure J = {cλ} in R 2n+1 admits a local reduction to a Veronese web WJ on a (n+1)-dimensional manifold and that for any Veronese webW one can locally construct a bihamiltonian structure J(W) of general position in R with the reduction equal to W . In the real analytic case J and J(WJ) are isomorphic. The aim of this paper is to introduce a wider class of degenerate bihamiltonian structures that possess many features of the general position case and to generalize the notion