Foliations with complex leaves

Abstract

Let X be a smooth foliation whose leaves are complex manifolds of dimension n and let D be the sheaf of germs of smooth functions, holomorphic along the leaves (namely the germs of CR-functions on X). The aim of this paper is to study the ringed space ( X, D) and in particular: function theory for the algebra D( X) and cohomology with values in D. We introduce the notion of q-completeness of a foliation and we prove a general result about the embedding of a real analytic q-complete foliation. As a consequence we obtain an approximation theorem for CR-functions and an embedding theorem for real analytic Stein foliations. Using these results, we prove a vanishing theorem for the groups H 0( X, D) when X is a real analytic 1-complete foliation.