Invariant Kohn-Rossi cohomology and obstruction to embedding of compact real \((2n-1)\)-dimensional \(CR\) manifolds in \(\mathbb{C}^ N\) (Q1915549)

Let \(X\) be a compact \(CR\) manifold of dimension \(2n - 1\), \(n \geq 3\) and \((H,J)\) be the holomorphic tangent bundle of \(x\). A smooth \(S^1\)-action on \(X\) is said to be transversal holomorphic if it preserves \(H\) and commutes with \(J\), and the vector field which generates the action is transversal to \(H\) at all points of \(X\). Following Tanaka this action defines a differential operator \(N\) which acts on the Kohn-Rossi cohomology \(H^{p,q}_{KR} (X)\). Let \(\widetilde H^{p,q}_{K,R} (x)\) be the part on which \(N\) acts trivially. The authors prove the following Theorem: If \(X\) is strongly pseudoconvex and \(CR\)-embeddable in \(\mathbb{C}^N\) then \(\widetilde H^{p,q}_{KR} (X) = 0\) for all \(1 \leq p + q \leq 2n - N - 1\).