Regular Leaves of Singular Foliations
Federico Bongiorno
TL;DR
The paper develops a framework linking singular foliations to formal groupoids and infinitesimal stacks, proving that leaves through singular points are regular. By constructing residual gerbes and minimal presentations via flattening, reduction, and functorial resolutions, it extends Cerveau’s results to singular ambient spaces and provides a self-contained proof of the Zariski--Lipman conjecture for terminal varieties without relying on resolution. The approach blends formal geometry (formal groupoids, leaves, and stacks) with algebro-geometric tools (Fitting ideals, blowups, and normalisation) to obtain regularity criteria and local product structures for foliations in characteristic zero. These methods yield a robust dictionary between Lie algebroids and infinitesimal groupoids, enabling invariant theory and tangent-structure analysis on singular spaces, with broad implications for foliation theory and singularity. The results substantiate a principled pathway to study foliations on singular spaces while bypassing resolution in key contexts and offer a versatile toolkit for future extensions to more general stacks and singularities.
Abstract
We identify a class of singular algebraic foliations whose leaves through singular points retain regularity. The proof consists in showing existence of residual gerbes for certain formal stacks, which do not enjoy smooth presentations. As applications, we extend a theorem of Cerveau to the case where the ambient scheme is not smooth and we give a proof of the Zariski--Lipman conjecture for varieties with terminal singularities, which does not rely on existence of resolution of singularities.