The equivalence and the embedding problems for CR-structures of any codimension

Abstract

We give a solution to the equivalence and the embedding problems for smooth CR-submanifolds of complex spaces (and, more generally, for abstract CR-manifolds) in terms of complete differential systems in jet bundles satisfied by all CR-equivalences or CR-embeddings respectively (local and global). For the equivalence problem, the manifolds are assumed to be of finite type and finitely nondegenerate. These are higher order generalizations of the corresponding nondegeneracy conditions for the Levi form. It is shown by a simple example that these nondegeneracy conditions cannot be even slightly relaxed to more general known conditions. In particular, for essentially finite hypersurfaces in $\C^2$, such a complete system may not exist in general. For the embedding problem, the source manifold is assumed to be of finite type and the embeddings to be finitely nondegenerate. Situations are given, where the last condition is automatically satisfied by all CR-embeddings.