Kähler spaces and proper open morphisms (Q1095422)

We prove among other results: (1) If \(\pi\) : \(X\to X'\) is a proper open surjective morphism of complex spaces with X Kähler and X' reduced then, if either \(\pi\) is flat or X' normal, X' is Kähler. (2) If X is a Kähler space then the Barlet space \(B_ m(X)\) of compact complex m- cycles of X is Kähler. (3) Any reduced compact complex space in Fujiki's class \({\mathcal C}\) (holomorphic image of a compact Kähler space) is bimeromorphically equivalent to a compact Kähler manifold.