This was my (Peter Garfield's) dissertation at the University of Washington in 2001. Most of the results were (or will be) published in papers, but I tried to include more background and justification than in a typical academic paper. Please let me know if you have any questions or comments.
This dissertation studies the bigraded Rumin complex on strictly pseudoconvex CR manifolds. We prove that the cohomology of the complex reproduces Kohn-Rossi cohomology (a fact known to Akahori and Miyajima for the upper half of the complex). There is a Hodge decomposition theorem for this bigraded complex, produced by showing that some Laplacian operators on this complex are subelliptic. (Subellipticity in the upper half of the complex is originally due to Akahori, and our proof in the middle dimensions is based on a suggestion from Rumin.) This allows us to show that each Kohn-Rossi cohomology class can be represented, under the identification with cohomology of the bigraded Rumin complex, by a unique harmonic form. The maps in the Rumin complex are induced from the exterior derivatives on forms; consequently the maps are mostly first order. In the middle dimensions, however, is a map that is second order, which makes the natural Laplacians in these dimensions fourth, rather than second, order.
We use the bigraded Rumin complex to create a CR version of the Frölicher spectral sequence. Using the Hodge decomposition, we show that this new spectral sequence mostly collapses at the second step under a Kähler-like hypothesis involving the Rumin Hodge decomposition. (Normal CR manifolds, in particular, satisfy this hypothesis.) As with Kähler manifolds, there are topological consequences of this collapse: the odd Betti numbers below the middle dimensions are even, and so the even Betti numbers above the middle dimensions are also even. We explore further consequences of the Hodge decomposition by relating the collapse of the spectral sequence to two CR versions of the ddc lemma. We usually must assume either that the CR manifold is normal or (possibly a weaker assumption) that the Kähler-like hypothesis holds.
The entire thesis is available for download as a PDF file.