VIII Characterisation of removable singularities of CR functions on a strictly pseudoconvex boundary

Abstract

We start this chapter by giving various characterisations of the compact sets K in the boundary of a strictly pseudoconvex domain D in a Stein manifold of dimension n which have the following property: any continuous CR function on $\partial D\backslash K$ can be extended holomorphically to the whole of D. We will obtain a geometric characterisation of such sets for n = 2 and a cohomological characterisation of such sets for n ⩾ 3. Amongst other things, we prove that the suffcient cohomological condition given in Theorem 5.1 of Chapter V is necessary if the ambient manifold is Stein and the domain D is assumed strictly pseudoconvex. We end the section with a geometric characterisation of the compact sets K such that any continuous CR function defined on $\partial D\backslash K$which is orthogonal to the set of $\overline{\partial}$ -closed (n; n−1)-forms whose support does not meet K can be extended holomorphically to the whole of D. When K is empty this condition is just the hypothesis of Theorem 3.2 of Chapter IV.