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 ∞-rings of smooth functions on M. For a finitely generated smooth structure C ∞(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 ∞(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.