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 ℂN 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 Hkh(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−1h(X) and Hnh(X) have the same dimension while all other Hkh(X), k > 0, vanish (Theorem B). If X is three-dimensional and V has at most rational singularities, we prove that H1h(X) and H2h(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.