The topology of the set of singularities for an integrable Hamiltonian system

Abstract

This paper studies the structure of the set of singu� lar points for an integrable Hamiltonian system on a compact symplectic manifold (M 2n , ω). If all singular� ities of the system are nondegenerate, then the closure of the set of singular points of rank r is a set of immersed 2rsubmanifolds in M 2n , on which a natural orientation can be introduced by means of the form ω. Thus, these submanifolds form a 2rcycle in the inte� gral homology group H 2r (M 2n ) of the phase space of the system. It is proved that this cycle is Poincare dual to the Chern class cr(TM) of the tangent bundle of the manifold M. Section 1 describes properties of nondegenerate singularities of integrable Hamiltonian systems. Sec� tion 2 presents a general construction establishing a relationship between the degeneration set of sections of a complex vector bundle and the Chern classes of this bundle. In Section 3, this construction is applied to study the topology of the set of singularities of inte� grable Hamiltonian systems. Results obtained in this paper generalize results obtained earlier by this author for the case of two degrees of freedom (see (1, 7, 8)) to the multidimensional case.