The Universal Lie \infty-Algebroid of a Singular Foliation

  • Camille Laurent-Gengoux

    Institut Elie Cartan de Lorraine (UMR 7502), Université de Lorraine, 3 rue Augustin Fresnel, 57000 Metz-Technopôle, France
  • Sylvain Lavau

    Université Paris Diderot, Bâtiment Sophie Germain, 8 Place Aurélie Nemours, 75013 Paris, France
  • Thomas Strobl

    CNRS-2924, Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil, and Institut Camille Jordan, Université Claude Bernard, Lyon 1, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
The Universal Lie $\infty$-Algebroid of a Singular Foliation cover
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We consider singular foliations F\mathcal{F} as locally finitely generated O\mathscr{O}-submodules of O\mathscr{O}-derivations closed under the Lie bracket, where O\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 F\mathcal{F} in terms of sections of vector bundles. For example, these exist always on a compact smooth manifold MM if F\mathcal{F} admits real analytic generators.

We show that every complex of vector bundles (E,d)(E_\bullet,\mathrm{d}) over MM providing a resolution of a given singular foliation F\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 F\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 F\mathcal{F}.

It encodes several aspects of the geometry of the leaves of F\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 F\mathcal{F} generated by rr vector fields for which we show by these techniques that, even locally, it cannot result from a Lie algebroid of the minimal rank rr.

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Camille Laurent-Gengoux, Sylvain Lavau, Thomas Strobl, The Universal Lie \infty-Algebroid of a Singular Foliation. Doc. Math. 25 (2020), pp. 1571–1652

DOI 10.4171/DM/782