The symplectic leaves of the affine Poisson $C^{\infty }$-scheme associated with a singular subalgebroid

Abstract

Every singular subalgebroid, and in particular every singular foliation $\mathcal{G}$, determines a universal Poisson $C^{\infty }$-scheme $\mathrm{Spec}(\mathcal{F}(\mathcal{G}))$, which is the analogue of the linear Poisson structure on the dual of a Lie algebroid. We show that the underlying space $X_{\mathcal{F}(\mathcal{G})}$ of $\mathrm{Spec}(\mathcal{F}(\mathcal{G}))$ decomposes as a disjoint union of symplectic manifolds. Furthermore, we extend the Helffer-Nourrigat cone to a singular subalgebroid and show that it is an embedded Poisson $C^{\infty }$-subscheme of $\mathrm{Spec}(\mathcal{F}(\mathcal{G}))$.

arXiv comment27 pages. We have added an AI declaration section and fixed several typos. Comments are welcome