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.