Boundaries of singularity sets, removable singularities, and CR-invariant subsets of CR-manifolds

Let Ω be a bounded strictly pseudoconvex domain in ℂⁿ, n ≥ 3, with boundary ∂Ω, of class C². A compact subset K is called removable if any analytic function in a suitable small neighborhood of ∂Ω K extends to an analytic function in Ω. We obtain sufficient conditions for removability in geometric terms under the condition that K is contained in a generic C² -submanifold M of co-dimension one in ∂Ω. The result uses information on the global geometry of the decomposition of a CR-manifold into CR-orbits, which may be of some independent interest. The minimal obstructions for removability contained in M are compact sets K of two kinds. Either K is the boundary of a complex variety of co-dimension one in Ω or it is an exceptional minimal CR-invariant subset of M, which is a certain analog of exceptional minimal sets in co-dimension one foliations. It is shown by an example that the latter possibility may occur as a nonremovable singularity set. Further examples show that the germ of envelopes of holomorphy of neighborhoods of ∞Ω K for K ⊂ M may be multisheeted. A couple of open problems are discussed.