The Universal Lie $\infty$-Algebroid of a Singular Foliation

Abstract

We consider singular foliations $\mathcal{F}$ as locally finitely generated $\mathscr{O}$-submodules of $\mathscr{O}$-derivations closed under the Lie bracket, where $\mathscr{O}$ is the ring of smooth, holomorphic, or real analytic functions on a correspondingly chosen manifold. We first collect and/or prove several results about the existence of resolutions of such an $\mathcal{F}$ in terms of sections of vector bundles. For example, these exist always on a compact smooth manifold $M$ if $\mathcal{F}$ admits real analytic generators.

We show that every complex of vector bundles $(E_\bullet,\mathrm{d})$ over $M$ providing a resolution of a given singular foliation $\mathcal{F}$ in the above sense admits the definition of brackets on its sections such that it extends these data into a Lie $\infty$-algebroid. This Lie $\infty$-algebroid, including the chosen underlying resolution, is unique up to homotopy and, moreover, every other Lie $\infty$-algebroid inducing the given $\mathcal{F}$ or any of its sub-foliations factors through it in an up-to-homotopy unique manner. We therefore call it the universal Lie $\infty$-algebroid of $\mathcal{F}$.

It encodes several aspects of the geometry of the leaves of $\mathcal{F}$. In particular, it permits us to recover the holonomy groupoid of Androulidakis and Skandalis. Moreover, each leaf carries an isotropy Lie $\infty$-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 $\mathcal{F}$ generated by $r$ vector fields for which we show by these techniques that, even locally, it cannot result from a Lie algebroid of the minimal rank $r$.