Invariance of cohomological \(q\)-completeness under finite holomorphic surjections
The following two results on the invariance of cohomologically \(q\)- completeness and cohomomological \(q\)-convexity are proven: Theorem 1. Let \(\pi : X \to Y\) be a finite surjective holomorphic map of finite dimensional reduced complex spaces. Then \(X\) is cohomologically \(q\)-complete (cohomologically \(q\)-convex) iff \(Y\) is cohomologically \(q\)- complete (cohomologically \(q\)-convex). Theorem 2. Let \(\pi : X \to Y\) be a holomorphic map of reduced complex spaces with \(Y\) finite dimensional. Suppose that: (i) \(\pi\) is a Stein morphism; (ii) There is a compact (topological) group \(G\) acting continuously on \(X\) (by holomorphic transformations) which preserves the fibers of \(\pi\). (iii) Any \(G\)-invariant holomorphic function on a fibre of \(\pi\) is constant. Then \(Y\) is cohomologically \(q\)-complete (cohomologically \(q\)-convex) if \(X\) is cohomologically \(q\)-complete (cohomologically \(q\)-convex).
scientific article; zbMATH DE number 599652