Generalized Flow-Box property for singular foliations

TL;DR

This paper introduces a generalized Flow-Box property for singular distributions and sub-varieties, providing new geometric and algebraic criteria for their transversality and foliation sections, motivated by Lie group actions.

Contribution

It extends the classical Flow-Box Theorem to singular settings and characterizes conditions for sub-varieties to serve as sections of line foliations.

Findings

Defines generalized Flow-Box property for singular distributions

Characterizes transversality conditions for sub-varieties

Provides criteria for sections of line foliations

Abstract

We introduce a notion of generalized Flow-Box property valid for general singular distributions and sub-varieties (based on a dynamical interpretation). Just as in the usual Flow-Box Theorem, we characterize geometrical and algebraic conditions of (quasi) transversality in order for an analytic sub-variety $X$ (not necessarily regular) to be a section of a line foliation. We also discuss the case of more general foliations. This study is originally motivated by a question of Jean-Francois Mattei (concerning the strengthening of a Theorem of Mattei) about the existence of local slices for a (non-compact) Lie group action.