Some convexity properties of morphisms of complex spaces
The following main theorem is proved: Theorem. Let \(\pi : X \to Y\) be a proper holomorphic map of finite dimensional complex spaces with at most \(r\)-dimensional fibres. Assume that \(Y\) is \(q\)-complete (in the sense of Andreotti and Grauert). Then \(X\) is \((q+r)\)-complete. The particular case when \(Y\) is 1-complete (in our normalization this corresponds to Stein spaces) answers a question raised by Okonek [see \textit{Chr. Okonek}, Lect. Notes Math. 1194, 104-126 (1986; Zbl 0593.32011)]. As an application a characterization theorem of \(q\)- completeness for holomorphically convex spaces follows. Namely: Theorem. Let \(X\) be a finite-dimensional holomorphically convex space. Then the following statements are equivalent: 1) \(X\) is \(q\)-complete; 2) \(X\) is cohomologically \(q\)-complete; 3) \(X\) does not contain compact analytic subsets of dimension \(\geq q\).
scientific article; zbMATH DE number 704947
scientific article