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

Source abstract

Every singular subalgebroid, and in particular every singular foliation $\mathscr{G}$, determines a universal Poisson $C^\infty$-scheme $\mathrm{Spec}(\mathcal{F}(\mathscr{G}))$, which is the analogue of the linear Poisson structure on the dual of a Lie algebroid. We show that the underlying space $X_{\mathcal{F}(\mathscr{G})}$ of $\mathrm{Spec}(\mathcal{F}(\mathscr{G}))$ 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 $\mathrm{Spec}(\mathcal{F}(\mathscr{G}))$.