Aneurysms of pseudoconcave \(CR\) manifolds (Q1909553)

Aneurysms are for higher codimensional \(CR\) manifolds the analogue of the bumps used by A. Andreotti and H. Grauert to prove finiteness theorems for the Dolbeault cohomology of bounded domains in a complex manifold. The study of these objects permits us to solve the Cauchy problem for cohomology classes of \(CR\) manifold \(M\) of type \((n,k)\) which is \(q\)-pseudoconcave and generically embedded in its tubular neighborhood. As an application we obtain results on the finiteness of the global \(\overline{\partial}_M\)-cohomology groups in dimension \(> n - q\) and the infinite dimensionality in dimension \(= n - q\) suitable pseudoconvexity assumptions at infinity, complementing those obtained by the authors in `Pseudoconcave \(CR\) manifolds', preprint Dip. Mat. Univ. Pisa 1.76(723), feb.(1993). When \(M\) is compact, we obtain a comparison between the \(\overline{\partial}_M\)-cohomology and Čech cohomology groups with coefficients in the sheaf of \(CR\) forms.