Algebraic completely integrable Hamiltonian systems

Abstract

In many integrable Hamiltonian systems of interest the general level sets of the moment map are isomorphic to affine parts of Abelian varieties and the flow of the integrable vector fields is linearized by this isomorphism. These two properties lead to the definition of an algebraic completely integrable Hamiltonian system (a.c.i. system). We will discuss three quite different definitions of an a.c.i. system and extract from it a definition which is consistent with our approach to integrable Hamiltonian systems. All constructions of integrable Hamiltonian systems easily specialize to the case of a.c.i. systems, except in the case of the quotient, which requires a real proof. The definitions and these properties will be considered in Section 2.