On the \(\overline{\partial}\)-cohomology groups of strongly \(q\)-concave manifolds (Q1917024)

Let \(E \to X\) be a holomorphic vector bundle over a complex manifold \(X\) and \(\Omega^p (E)\) the sheaf of germs of \(E\)-valued holomorphic \(p\)-forms on \(X\). The topology of the cohomology groups \(H^q (X, \Omega^p (E))\) induced by a hermitian metric on \(X\) and a fiber metric on \(E\) may be wild in general, but under the right assumptions on the topological, geometrical or analytical structure of \(X\) these spaces are well understood. In particular it is known that \(H^{n - q} (X, \Omega^p (E))\) is Hausdorff if \(X\) is a strongly \(q\)-concave manifold of dimension \(n\). For related results see \textit{A. Andreotti} and \textit{A. Kas} [Ann. Scuola Norm. sup. Pisa, Sci. fis. mat., III. Ser. 27, 187-263 (1973; Zbl 0278.32007)] or \textit{J. P. Ramis} [Ann. Scuola Norm. sup. Pisa, Sci. fis. mat., Ser. 27(1973), 933-997 (1974; Zbl 0327.32001)]. The author gives a new proof of this fact by using \(L^2\) estimator for the \( \overline\partial\)-operator with respect to a complete hermitian metric on a strongly \(q\)-concave manifold. Another application of this method yields the Hausdorffness of certain cohomology groups over the complement of the (isolated) singular set of a compact complex space.