Homology and cohomology in hypo-analytic structures of the hypersurface type
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The work is concerned with local exactness in the cohomology of the differential complex associated with a hypo-analytic structure on a smooth manifold. Only structures of the hypersurface type are considered, i.e., structures in which the rank of the characteristic set does not exceed one. Among them are the CR structures of real hypersurfaces in a complex manifold. The main theorem states anecessary condition for local exactness in dimensionq to hold. The condition is stated in terms of the natural homology associated with the differential complex, as inherited by the level sets of the imaginary part of an arbitrary solutionw whose differential spans the characteristic set at the central point. An intersection number, which generalizes the standard number in singular homology, is defined; the condition is that this number, applied to the intersection of the level sets ofImw with the hypersurfaceRew=0, vanish identically. In a CR structure, and in top dimension, this is shown to be equivalent to the property that the Levi form not be definite at any point—a property, that is likely to be also sufficient for local solvability.
AMS Subject ClassificationNo. 39
Key Words and PhrasesCohomology current differential form homology locally integrable system of vector fields local solvability sheaf
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