Holomorphic Limits of Kahler Manifolds

Let $π:\cX\toΔ$ be a smooth family of compact complex manifolds over the unit disc, and assume that the fibers $\cX_t$ are Kähler for all $t\inΔ^*$. If $π$ is Kähler at some point $t_0\inΔ^*$, we construct a positive $d$-closed $(1,1)$-current $T$ on the central fiber $\cX_0$ such that $\widetilde{\