Smooth structures on pseudomanifolds with isolated conical singularities

TL;DR

This paper develops a framework for smooth structures on conical pseudomanifolds with isolated singularities, introducing tangent bundles, characteristic classes, and symplectic forms, and explores their properties and examples.

Contribution

It defines smooth structures on conical pseudomanifolds via $C^ abla$-rings, introduces tangent bundles and characteristic classes, and studies conical symplectic and Poisson structures.

Findings

Vanishing of Nash vector fields at singular points in certain smooth structures

Conical symplectic forms are smooth in Euclidean structures

Brylinski-Poisson homology matches de Rham homology for compatible structures

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.