The Universal Lie (infty)-Algebroid of a Singular Foliation

摘要

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. par We show that every complex of vector bundles ((E_ullet,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}). par 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).