Real abelian varieties and the singularities of an integrable Hamiltonian system

Abstract

defined on a symplectic manifold ( V 2m, ¢o ) such that for the Poisson bracket { } corresponding to e , w e h a v e { H i , H j } = 0 . We assume here that H and V 2m are algebraic and that H is proper with smooth generic fibres. We are interested in the critical fibres of the moment map and in the number of connected components of the smooth fibres H l(s), se R m. This goes along the lines of a program of S. Smale to study Dynamical Systems. [8] The Arnol'd-Liouville theorem shows, for a system (1) where H is proper with generic smooth fibres, the following two facts,