Smooth structures on pseudomanifolds with isolated conical singularities

Abstract

In this note we introduce the notion of a smooth structure on a conical pseudomanifold $M$ in terms of $C^{\infty }$-rings of smooth functions on $M$. For a finitely generated smooth structure $C^{\infty }(M)$ we introduce the notion of the Nash tangent bundle, the Zariski tangent bundle, the tangent bundle of $M$, and the notion of characteristic classes of $M$. We prove the vanishing of a Nash vector field at a singular point for a special class of Euclidean smooth structures on $M$. We introduce the notion of a conical symplectic form on $M$ and show that it is smooth with respect to a Euclidean smooth structure on $M$. If a conical symplectic structure is also smooth with respect to a compatible Poisson smooth structure $C^{\infty }(M)$, we show that its Brylinski-Poisson homology groups coincide with the de Rham homology groups of $M$. We show nontrivial examples of these smooth conical symplectic-Poisson pseudomanifolds.

26 pages, final version

References in corpus (1)

• Symplectic singularities

Cited by in corpus (3)

• Information geometry and sufficient statistics
• Differentiable stratified groupoids and a de Rham theorem for inertia spaces
• Diffeological statistical models, the Fisher metric and probabilistic mappings