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 dz=(y1dx1-x1dy1)+⋯+(ymdxm-xmdym),which defines the canonical contact structure on CP^2m+1. Another example is the Darboux distribution dz=x1dy1+⋯+xmdym,which gives the normal local form of any contact structure. Given a germ D of holomorphic distribution with separated variables in (Cⁿ, 0) , we show that there exists , for some κ∈ Z_{≥ 0} related to the Taylor coefficients of D, a holomorphic submersion HD:(Cn,0)→(Cκ,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 CP^2m+1 supports a Legendrian holomorphic foliation whose generic leaves are dense in CP^2m+1. So we obtain examples of injectively immersed Legendrian holomorphic open manifolds that are everywhere dense.