Thom-Sebastiani properties of Kohn-Rossi cohomology of compact connected strongly pseudoconvex CR manifolds

Let X ₁ and X ₂ be two compact connected strongly pseudoconvex embeddable Cauchy-Riemann (CR) manifolds of dimensions 2m - 1 and 2n - 1 in ? ^m+1 and ? ⁿ⁺¹, respectively. We introduce the Thom-Sebastiani sum X = X ₁⊕X ₂ which is a new compact connected strongly pseudoconvex embeddable CR manifold of dimension 2m+2n+1 in ? ^m+n+2. Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in ? ⁿ⁺¹ for all n ≥ 2 forms a semigroup. X is said to be an irreducible element in this semigroup if X cannot be written in the form X ₁⊕X ₂. It is a natural question to determine when X is an irreducible CR manifold. We use Kohn-Rossi cohomology groups to give a necessary condition of the above question. Explicitly, we show that if X = X ₁ ⊕ X ₂, then the Kohn-Rossi cohomology of the X is the product of those Kohn-Rossi cohomology coming from X ₁ and X ₂ provided that X ₂ admits a transversal holomorphic S ¹-action.