K. Diederich and T. Ohsawa [Tokyo J. Math. 21, 353-358 (1998; Zbl 0922.32010)] proved that if \(M\) is a complex manifold and \(N\) is a complex submanifold, then any locally hyperconvex Stein open subset of \(N\) is the trace of a locally hyperconvex Stein open subset of \(M\). Y. T. Siu [Invent. Math. 38, 89-100 (1976; Zbl 0343.32014)] argued that if \(Y\) is a closed complex subspace of \(X\) and \(Y\) is Stein, then \(Y\) has a Stein neighborhood. Also, it has been proved by V. Vâjâitu [C. R. Acad. Sci., Paris, Sér. I 322, 823-828 (1996; Zbl 0866.32008)] that if \(Y\) is hyperconvex, then \(Y\) has a hyperconvex neighborhood.
Using the methods of J. P. Demailly [Math. Z. 204, 283-295 (1990; Zbl 0682.32017)], the author sets up a general framework for the above theorems and generalizes Diederich and Ohsawa’s results for reduced complex spaces.