A relative completeness theorem (Q1710702)

The notion of $n$-completeness of a (complex) analytic space goes back to Grauert and Andreotti, more than half a century ago. A 1-complete space is Stein and a great deal of knowledge has accumulated meanwhile. However, there are still outstanding open questions. For instance Skoda has conjectured that a holomorphic fibre bundle with Stein fiber and Stein base is 2-complete. \par The main result of this note by the author states that an analytic space $Y$ that fibers over an analytic space $X$ of dimension less than $n$ is $n$-complete provided that $Y$ admits a continuous exhaustion function that is strictly plurisubharmonic along fibers. \par Several ramifications of this theorem are included, in the best spirit and glorious tradition of complex analytic geometry.