The universal Lie infinity-algebroid of a singular foliation
Résumé
We associate a Lie ∞-algebroid to every resolution of a singular foliation, where we consider a singular foliation as a locally generated, O-submodule of vector fields on the underlying manifold closed under Lie bracket. Here O can be the ring of smooth, holomorphic, or real analytic functions. The choices entering the construction of this Lie ∞-algebroid, including the chosen underlying resolution, are unique up to homotopy and, moreover, every other Lie ∞-algebroid inducing the same foliation or any of its subfoliations factorizes through it in an up-to-homotopy unique manner. We thus call it the universal Lie ∞-algebroid of the singular foliation. It can be chosen, locally, to be a Lie n-algebroid for real analytic or holomorphic singular foliations. 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 Lie ∞-algebra structure that is unique up to isomorphism and that extends a minimal isotropy Lie algebra that can be associated to each leaf by higher brackets containing 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.
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