The canonical bundle and realizable CR hypersurfaces

The canonical bundle of a realizable CR hypersurface has closed sections. Examples are given of non-realizable hypersurfaces with closed sections and others without such sections. If however an abstract CR hypersurface of dimension 2m + 1 has m strongly independent CR functions then a closed section can be used to produce the missing function and so assures that the hypersurface is realizable. The existence of a closed section is equivalent to a condition on the range of ∂_b acting on functions. Some non-realizable CR hypersurfaces are shown to have ∂_b-cohomology groups quite different from those of realizable hypersurfaces.