Regular Leaves of Singular Foliations

TL;DR

This paper studies a special class of singular algebraic foliations with regular leaves through singular points, extending existing theorems and proving the Zariski--Lipman conjecture without resolution of singularities.

Contribution

It introduces a new class of singular foliations with regular leaves at singular points and extends key theorems to non-smooth schemes, including a proof of the Zariski--Lipman conjecture.

Findings

Identified a class of singular foliations with regular leaves at singular points

Extended Cerveau's theorem to non-smooth ambient schemes

Provided a proof of the Zariski--Lipman conjecture for varieties with terminal 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.