Symplectic Singularities and Solvable Hamiltonian Mappings

Abstract

Abstract. We study singularities of smooth mappings F of lR 2 n into symplectic space (R 2 n , w) by their isotropic liftings to the corresponding symplectic tangent bundle (TJR 2 n,w) . Using the notion of local solvability of lifting as a generalized Hamiltonian system, we introduce new symplectic invariants and explain their geometric meaning. We prove that a basic local algebra of singularity is a space of generating functions of solvable isotropic mappings over F endowed with a natural Poisson structure. The global properties of this Poisson algebra of the singularity among the space of all generating functions of isotropic liftings are investigated. The solvability criterion of generalized Hamiltonian systems is a strong method for various geometric and algebraic investigations in a symplectic space. We illustrate this by explicit classification of solvable systems in codimension one.