
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..
Abstract manifolds and fiber bundles (category-theoretic aspects), 32F20, \({\bar \partial }_ b\)-complex, \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs, canonical bundle, CR functions, Levi operators, Sphere bundles and vector bundles in algebraic topology, CR hypersurface, 32F25, Real submanifolds in complex manifolds, \(\overline\partial\) and \(\overline\partial\)-Neumann operators
Abstract manifolds and fiber bundles (category-theoretic aspects), 32F20, \({\bar \partial }_ b\)-complex, \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs, canonical bundle, CR functions, Levi operators, Sphere bundles and vector bundles in algebraic topology, CR hypersurface, 32F25, Real submanifolds in complex manifolds, \(\overline\partial\) and \(\overline\partial\)-Neumann operators
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