Characterization of removable sets in strongly pseudoconvex boundaries (Q1842629)

Let \(D\) be a relatively compact domain in the complex manifold \(M\). A compact \(K \subset \partial D\) is called removable if \(\partial D \backslash K\) is \(C^ 1\)-smooth and every continuous \(CR\)-function \(f\) on \(\partial D \backslash K\) has a continuous extension \(F\) onto \(\overline D \backslash K\) which is holomorphic on \(D\). The main result of the paper is the following Theorem 1. Let \(M\) be a Stein manifold and \(D \Subset M\) be a \(C^ 2\)- smooth strictly pseudoconvex domain such that \(\overline D\) is \({\mathcal O} (M)\)-convex. Then for a compact \(K \subset \partial D\) the following two conditions are equivalent: (a) \(K\) is removable; (b) The restriction map \(Z^{n,n - 2}_{\overline \partial} (M) \to Z_{\overline \partial}^{n,n - 2} (K)\) has a dense image and \(H^{n,n - 2}_{\overline \partial} (K) = 0\). Here \(Z_{\overline \partial}^{n,n - 2} (K)\) is equipped with a locally convex inductive limit topology.