Poisson smooth structures on stratified symplectic spaces

Abstrakt

In this note we introduce the notion of a smooth structure on a stratified space and the notion of a Poisson smooth structure on a stratified symplectic space. We show that these smooth spaces possess several important properties, e.g. the existence of smooth partitions of unity.

Furthermore, under a mild condition many properties of a symplectic manifold can be extended to a symplectic stratified space provided with a smooth Poisson structure, e.g. the existence and uniqueness of a Hamiltonian flow, the isomorphism between the Brylinski-Poisson homology and the de Rham homology, the existence of a Leftschetz decomposition on a symplectic stratified space. We give many examples of stratified symplectic spaces satisfying these properties.

Klíčová slova

• $C^\infty$-ring
• stratified space
• symplectic form
• Poisson structure