On Serre duality
The authors prove a separation criterion for the compactly supported Dolbeault cohomology. As an application they obtain a new simple proof of an Andreotti-Vesentini separation theorem which was first obtained by G.~Henkin and J.~Leiterer by means of a rather difficult proof: Let \(X\) be an \(n\)-dimensional complex manifold which is \((n-q)\)-convex in the sense of Andreotti-Grauert, \(1\leqslant q\leqslant n-1.\) Let \(K\) be a compact subset of \(X\) which admits a basis of strictly \(q\)-convex neighborhoods. Then \(H^{p,n-q}(X\setminus K,E)\) is separated for all \(p\) and each holomorphic vector bundle \(E\) over \(X.\) Note that this result is stronger than the corresponding result of Henkin-Leiterer. It is proved that the classical Andreotti-Vesentini separation theorem for \(q\)-concave manifolds holds also for any increasing union of such manifolds.
separation criterion for the compactly supported Dolbeault cohomology
scientific article; zbMATH DE number 1436204