# Distributions, First Integrals and Legendrian Foliations

## 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

\begin{aligned} dz=(y_1dx_1-x_1dy_1)+\dots +(y_mdx_m-x_mdy_m), \end{aligned}

which defines the canonical contact structure on $${\mathbb {C}}{\mathbb {P}}^{2m+1}$$. Another example is the Darboux distribution

\begin{aligned} dz=x_1dy_1+\dots +x_mdy_m, \end{aligned}

which gives the normal local form of any contact structure. Given a germ $${\mathcal {D}}$$ of holomorphic distribution with separated variables in $$({\mathbb {C}}^n,0)$$, we show that there exists , for some $$\kappa \in {\mathbb {Z}}_{\ge 0}$$ related to the Taylor coefficients of $${\mathcal {D}}$$, a holomorphic submersion

\begin{aligned} H_{{\mathcal {D}}}:({\mathbb {C}}^n,0)\rightarrow ({\mathbb {C}}^{\kappa },0) \end{aligned}

such that $${\mathcal {D}}$$ is completely non-integrable on each level of $$H_{{\mathcal {D}}}$$. Furthermore, we show that there exists a holomorphic vector field Z tangent to $${\mathcal {D}}$$, such that each level of $$H_{{\mathcal {D}}}$$ 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 $${\mathcal {D}}$$ are the same. Between several other results, we show that the canonical contact structure on $${\mathbb {C}}{\mathbb {P}}^{2m+1}$$ supports a Legendrian holomorphic foliation whose generic leaves are dense in $${\mathbb {C}}{\mathbb {P}}^{2m+1}$$. So we obtain examples of injectively immersed Legendrian holomorphic open manifolds that are everywhere dense.

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Correspondence to Maycol Falla Luza.