Embeddings for 3-dimensional CR-manifolds

We consider the problem of projectively embedding strictly pseudoconcave surfaces, X− containing a positive divisor, Z and affinely embedding its 3dimensional, strictly pseudoconvex boundary, M = −bX−. We show that embeddability ofM in affine space is equivalent to the embeddability ofX− or of appropriate neighborhoods of Z inside X− in projective space. Under the cohomological hypotheses: H2 comp(X−,Θ) = 0 and H (Z,NZ) = 0 these embedding properties are shown to be preserved under convergence of the complex structures in the C∞-topology. §1. Conditions for embeddability (geometric approach). Let M denote a smooth compact, strictly pseudoconvex 3-dimensional CRmanifold. Such a structure is induced on a strictly pseudoconvex, real compact hypersurface in a complex space. The CR-manifold, M is called fillable if M can be realized as the boundary of a 2-dimensional (Stein) space. From the results of H.Grauert, 1958, J.J.Kohn, 1963, H.Rossi, 1965, it follows that M is fillable iff M is embeddable by a CR-mapping into affine complex space. It is now well known that the generic CR-structure on M is not fillable or embeddable (H.Rossi,1965, A.Andreotti-Yum Tong Siu,1970, L.Nirenberg, 1974, D.Burns,1979, H.Jacobovitz, F.Treves,1982, D.Burns, C.Epstein,1990.) On the other hand L.Lempert,1995, has proved that any embeddable strictly pseudoconvex CR-manifold M can be realized as a separating hypersurface in a projective variety X. This means, that if a strongly pseudoconvex, compact 3dimensional CR-manifold M bounds a strongly pseudoconvex surface, X+ then −M also bounds a strongly pseudoconcave complex surface, X− containing a smooth holomorphic curve, Z with positive normal bundle, NZ . Suppose that the CR-manifold, −M is the boundary of a two-dimensional strictly pseudoconcave manifold, X− which contains a smooth curve, Z with positive normal bundle, NZ . It is quite possible that this assumption is valid for any strictly pseudoconvex compact CR-manifold, M. Definition. X− will be called weakly embeddable in CP if there exists a holomorphic map φ : X− → CP injective in some neighborhood of Z in X−. A