The universal Lie $\infty$-algebroid of a singular foliation

Abstract

We associate a Lie $\infty$-algebroid to every resolution of a singular foliation, where we consider a singular foliation as a locally generated $\mathscr{O}$-submodule of vector fields on the underlying manifold closed under Lie bracket. Here $\mathscr{O}$ can be the ring of smooth, holomorphic, or real analytic functions. The choices entering the construction of this Lie $\infty$-algebroid, including the chosen underlying resolution, are unique up to homotopy and, moreover, every other Lie $\infty$-algebroid inducing the same foliation or any of its sub-foliations factorizes through it in an up-to-homotopy unique manner. We thus call it the universal Lie $\infty$-algebroid of the singular foliation. For real analytic or holomorphic singular foliations, it can be chosen, locally, to be a Lie $n$-algebroid for some finite $n$. We show that this universal structure encodes several aspects of the geometry of the leaves of a singular foliation. In particular, it contains the holonomy algebroid and groupoid of a leaf in the sense of Androulidakis and Skandalis. But even more, each leaf carries an isotropy $L_\infty$-algebra structure that is unique up to isomorphism. It extends a minimal isotropy Lie algebra, that can be associated to each leaf, by higher brackets, which give rise to additional invariants of the foliation. As a byproduct, we construct an example of a foliation generated by $r$ vector fields for which we show by these techniques that it cannot be generated by the image through the anchor map of a Lie algebroid of the minimal rank $r$.