A famous theorem by Kazhdan states that a tower of coverings of a compact Riemann surface converging to the universal covering is Bergman stable. Recently, Baik, Shokrieh and Wu generalized this theorem where the universal cover is replaced with any infinite Galois cover. In this talk, we prove a generalized version of this theorem, where compact Riemann surfaces are replaced with compact Kähler manifolds. We also discuss its application to the projectivity of manifolds. This is a joint work with Jihun Yum (IBS-CCG).