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.