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Smooth structures on pseudomanifolds with isolated conical singularities
Hông Vân Lê
,
Petr Somberg
,
Jiří Vanžura
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.