One of the most natural questions in the theory of complex and CR (i.e., CauchyRiemann) structures deals with the possibility of extending deformations of the CR structure on the boundary M o of a compact complex manifold M =M~M o to deformations of M. An equivalent formulation concerns the solvability of a first order non-linear system given by the integrability relation ~co-~[co, co] = 0 with e)eB where ~) is a C ~ form of type (0, 1) with values in the holomorphic tangent bundle of M, [, ] is the Poisson bracket and B is the space of boundary conditions which guarantee that the complex structure represented by co induces a given deformation of M o. In the strongly pseudoconvex case the results of [2] and [3] provide an indirect geometric solution to the extension problem under the assumption that the second compactly supported cohomology group with coefficients in the holomorphic tangent sheaf vanishes. This is achieved in two stages. In I-2] the corresponding non-linear system is solved by imposing certain restrictions on the boundary conditions, i.e., by considering an appropriate subset B' of B. Geometrically this means that the extension has been established for a certain class ~ of deformations of M 0. It is then proved in [3] that every sufficiently small CR structure on M o is equivalent to one belonging to 4.There are several important questions which have not been answered by the methods and results of 1-2] and [3]. The most outstanding one is the direct extension of CR deformations without going through the equivalence step in [3]. This question is also interesting from the point of view of partial differential equations alone since it concerns the solvability of the non-linear boundary-value problem (Sco-~[co, co] =0, cos B, without relying on geometric considerations to alter the boundary conditions to more managable ones (see [2] or §1 for the definition of the boundary condition (oeB). Furthermore, it is worthwhile to find out if the convexity assumptions on M 0 can be relaxed. It should be observed that the strong pseudoconvexity of M 0 is needed for the geometric solution described above only as far as the second stage is concerned. The equivalence result of 1-3] requires that every deformation of M o be also locally embedable as a real *

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