Holomorphic De Rham Cohomology of Strongly Pseudoconvex CR Manifolds with S1-actions

Abstract

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 S¹-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^k_h(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 Hⁿ⁻¹_h(X) and Hⁿ_h(X) have the same dimension while all other H^k_h(X), k > 0, vanish (Theorem B). If X is three-dimensional and V has at most rational singularities, we prove that H¹_h(X) and H²_h(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.