Smooth structures on stratified spaces

Abstract

Motivated by the desire to quantize singular symplectic spaces we consider stratified spaces from an analytic and geometric point of view. To this end one needs an appropriate functional structure on these spaces. But unlike for manifolds such a functional structure on a stratified space is in general not intrinsically given. In this article we explain the basic notions of the theory of stratified spaces and define an appropriate concept for a so-called smooth (functional) structure on a stratified space. We explain how varieties, orbit spaces and reduced spaces of Hamiltonian group actions give rise to natural examples for stratified spaces with a smooth structure. Moreover, it is shown how smooth structures allow for the definition of geometric concepts on stratified spaces like tangent spaces, vector fields and Poisson bivectors. Finally, it is explained what to understand by the quantization of a symplectic stratified space.KeywordsOrbit SpaceOrbit TypeSmooth StructureSmooth Vector FieldPoisson BivectorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.