Distributions, First Integrals and Legendrian Foliations.

We study germs of holomorphic distributions with "separated variables". In codimension one, a well know example of this kind of distribution is given by d z = (y 1 d x 1 - x 1 d y 1) + ⋯ + (y m d x m - x m d y m) , which defines the canonical contact structure on C P 2 m + 1 . Another example is the Darboux distribution d z = x 1 d y 1 + ⋯ + x m d y m , which gives the normal local form of any contact structure. Given a germ D of holomorphic distribution with separated variables in (C n , 0) , we show that there exists , for some κ ∈ Z ≥ 0 related to the Taylor coefficients of D , a holomorphic submersion H D : (C n , 0) → (C κ , 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. Between several other results, we show that the canonical contact structure on C P 2 m + 1 supports a Legendrian holomorphic foliation whose generic leaves are dense in C P 2 m + 1 . So we obtain examples of injectively immersed Legendrian holomorphic open manifolds that are everywhere dense. [ABSTRACT FROM AUTHOR]