Transversally Pseudoconvex Foliations

We consider real analytic foliations X with complex leaves of transversaò dimension one and we give the notion of transversal pseudoconvexity. This amounts to require that the transverse bundle N_F to the leaves carries a metric on the fibresa having tangential pseudo-Levi form positive. This condition is of a special intererest if the foliation X is 1 complete i.e. admits a smooth exaustion function phi which is plurisubharmonic along the leaves. In this situation we prove that there exist an open neighbourhood U of X in the complexification X^ of X and a non negative smooth function u:UtoR which is plurisubharmonic in U, strongly plurisubharmonic on UX and such that X is the zero set of u. This result has many implications; every compact sublevel of u is a Stein compact and if S(X) is the algebra of smooth CR functions on $X$, the restriction map S(X)->S(X_c) has a dense image (Theorem 4.1); a transversally pseudoconvex , 1-complete, real analytic foliation X with complex leaves of dimension n properly embeds in C^{2n+3} by a CR map and the sheaf S=S_X of germ of smooth CR functions on X is cohomologically trivial