The canonical bundle and realizable CR hypersurfaces (Q1070078)

The canonical bundle of an abstract CR hypersurface \(M^{2n-1}\) is the line bundle of forms of type (n,0). If the hypersurface is realizable then this bundle has a closed non-zero section. This paper considers the converse statement. First, it is shown that this converse is false in general but does hold if M is already known to be almost realizable in that there are n-1 CR functions which are independent in a strong sense. A second result shows that not every canonical bundle has a closed section. Thus there are non-realizable CR hypersurfaces with closed sections and other non-realizable CR hypersurfaces without closed sections. The paper concludes by contrasting the \({\bar \partial}_ b\)- cohomology groups for certain non-realizable CR hypersurfaces with those of the realizable CR hypersurfaces. A related paper by the same author is ''A simple example of a non-realizable hypersurface'' which will appear in the Proc. Am. Math. Soc..