Compact subsets of holomorphy of a complex space (Q1092286)

The purpose of this paper is to show that certain semi-global cohomology conditions (namely \(H^ p(K,{\mathcal O})=0\) for every \(p\geq 1)\) characterize those compact subsets K of holomorphy of a complex space X satisfying \(K=S{\mathcal O}(K)\), where S\({\mathcal O}(K)\) denotes the spectrum of the holomorphic function algebra. For X a Stein manifold semi-global cohomology conditions were found by \textit{R. Harvey} and \textit{R. O. Wells} jun. [Trans. Am. Math. Soc. 136, 509-516 (1969; Zbl 0175.372)]. Examples show that, in general, such compact subsets of holomorphy in a complex space can have singularities and thus other methods are used here in the proof.