Integrable Hamiltonian systems and symmetric products of curves

Abstract

This chapter is entirely devoted to the construction and a geometric study of a big family of integrable Hamiltonian systems. The phase space is C^2d but equipped with many different Poisson structures: for each non-zero φ ∈ C[x, y] we construct (in Paragraph 2.2) a Poisson bracket {·, ·} ^φ _d which makes (C^2d, {·, ·} ^φ _d ) into an affine Poisson variety. Each of these brackets has maximal rank 2d (in particular the algebra of Casimirs is trivial) and they are all compatible. An explicit formula for all these brackets is given; they grow in complexity (i.e., degree) with φ so that only the first members are (modified) Lie-Poisson structures.