**Conjecture:** Let $\pi: X\to X'$ be a proper flat surjective morphism of complex spaces.
If $X$ is Kahler, is $X'$ Kahler?


This conjecture when $X$ and $X'$ are smooth solved by Jean Varouchas from Nancy and by some assumption of geometric flatness of $\pi$ solved also by him on singular setting. 


But Is there any reference to give a solution to this conjecture?


see <https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/torus_images.pdf>


Varouchas, J.: [Stabilite de la classe des varietes kahleriennes par certains morphismes propres](http://link.springer.com/article/10.1007%2FBF01389138#page-1). Invent. Math.77(1984), no. 1, 117–127.


Varouchas, J.: [Kahler spaces and proper open morphisms](https://eudml.org/doc/164492). Math. Ann.283(1989),13–52.



A conjecture from Jean Varouchas on Kahler varieties
This problem was solved by Barlet and Varouchas in 
[this paper.](http://www.numdam.org/item?id=BSMF_1989__117_3_327_0)


The base $X'$ is assumed to be reduced (surely this is OK for you), and the fibers pure dimensional (also certainly acceptable).


