Families of distributions and Pfaff systems under duality

TL;DR

This paper explores the duality between distributions and Pfaff systems on varieties, identifying conditions for their flat families to be equivalent and extending a theorem in the integrable case.

Contribution

It establishes criteria under which distributions and Pfaff systems have equivalent flat families, and generalizes a theorem for integrable distributions.

Findings

Conditions for equivalence of flat families of distributions and Pfaff systems.

Extension of Cukierman and Pereira's theorem to integrable distributions.

Analysis of duality in the context of singular distributions.

Abstract

A singular distribution on a non-singular variety $X$ can be defined either by a subsheaf $D$ of the tangent sheaf, or by the zeros of a subsheaf $D^0$ of the sheaf of 1-forms. Although both definitions are equivalent under mild conditions on $D$, they give rise, in general, to non-equivalent notions of flat families. In this work we investigate conditions under which both notions of flat families are equivalent. In the last sections we focus on the case where the distribution is integrable, and we use our results to generalize a theorem of Cukierman and Pereira.