Holomorphic de Rham cohomology of strongly pseudoconvex CR manifolds with S1-actions

In this paper, we study the holomorphic de Rham cohomology of a compact strongly pseudoconvex CR manifold X in C-N with a transversal holomorphic S-1-action. The holomorphic de Rham cohomology is derived from the Kohn-Rossi cohomology and is particularly interesting when X is of real dimension three and the Kohn-Rossi cohomology is infinite dimensional. In Theorem A, we relate the holomorphic de Rham cohomology H-h(k)(X) to the punctured local holomorphic de Rham cohomology at the singularity in the variety V which X bounds. In case X is of real codimension three in Cn+1, we prove that H-h(n-1)(X) and H-h(n)(X) have the same dimension while all other H-h(k)(X), k > 0, vanish (Theorem B). If X is three-dimensional and V has at most rational singularities, we prove that H-h(1) (X) and H-h(2) (X) vanish (Theorem C). In case X is three-dimensional and N = 3, we obtain in Theorem D a complete characterization of the vanishing of the holomorphic de Rham cohomology of X.