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

Abstract

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 $\mathbb{C}^{m+1}$ and $\mathbb{C}^{n+1}$, respectively. We introduce the Thom-Sebastiani sum $X=X_1\oplus X_2$ which is a new compact connected strongly pseudoconvex embeddable CR manifold of dimension $2m+2n+1$ in $\mathbb{C}^{m+n+2}$. Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in $\mathbb{C}^{n+1}$ for all $n\geq 2$ forms a semigroup. $X$ is said to be an irreducible element in this semigroup if $X$ cannot be written in the form $X_1\oplus X_2$. 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 $X=X_1\oplus X_2$, 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.