Distributions, first integrals and Legendrian foliations

TL;DR

This paper investigates holomorphic distributions with separated variables, demonstrating the existence of a holomorphic submersion that reveals their non-integrability and the presence of dense leaves, linking their first integrals to those of a tangent vector field.

Contribution

It establishes a canonical form for such distributions via a holomorphic submersion and connects their dynamics to a tangent vector field, revealing their structure and integrability properties.

Findings

Existence of a holomorphic submersion for distributions with separated variables.

Distribution is non-integrable on each level of the submersion.

Presence of a tangent vector field with dense leaves in each level.

Abstract

We study germs of holomorphic distributions with "separated variables'. In codimension one, a well know example of this kind of distribution is given by the canonical contact structure on $\mathbb{P}^{2m+1}$ . Another example is the Darboux distribution, which gives the normal local form of any contact structure. Given a germ $D$ of holomorphic distribution with separated variables in $(\mathbb{C}^n,0)$, we show that there exists , for some $\kappa \in \mathbb{Z}_{\geq 0}$ related to the Taylor coefficients of $D$, a holomorphic submersion $H_{D}: (\mathbb{C}^n,0) \rightarrow (\mathbb{C}^{\kappa},0)$ such that $D$ is completely non-integrable on each level of $H_{D}$. Furthermore, we show that there exists a holomorphic vector field $Z$ tangent to $D$, such that each level of $H_{D}$ contains a leaf of $Z$ that is somewhere dense in the level. In particular, the field of meromorphic first integrals of $Z$ and that of $D$ are the same.