A separation theorem and Serre duality for the Dolbeault cohomology

Abstract

LetX be a complex manifold with finitely many ends such that each end is either q-concave or (n−q)-convex. If $q< \tfrac{1}{2}n$ , then we prove that H^pn−q(X) is Hausdorff for all p. This is not true in general if $q \geqslant \tfrac{1}{2}n$ (Rossi’s example with n=2 and q=1). If all ends are q-concave, then this is the classical Andreotti-Vesentini separation theorem (and holds also for $q \geqslant \tfrac{1}{2}n$ ). Moreover the result was already known in the case when the q-concave ends can be ‘filled in’ (again also for $q \geqslant \tfrac{1}{2}n$ ). To prove the result we first have to study Serre duality for the case of more general families of supports (instead of the family of all closed sets and the family of all compact sets) which is the main part of the paper. At the end we give an application to the extensibility of CR-forms of bidegree (p, q) from (n−q)-convex boundaries, $q< \tfrac{1}{2}n$ .