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 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 $\kappa \in {\mathbb{Z}}_{\ge 0}$ related to the Taylor coefficients of $D$, a holomorphic submersion $H_D:({\mathbb{C}}^n,0)\to ({\mathbb{C}}^{\kappa },0)$ 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.

53 pages