The holomorphic limit of Kahler manifolds

Let $\pi:\cX\to \Delta$ be a smooth family of compact complex manifolds over the unit disk $\Delta$, whose fibers $\cX_{t}$ are Kähler for all $t\in \Delta^*$.

If $\pi$ is Kähler at a point $t_0\in\Delta^*$, we prove that there exists a pseudo-effective class on the central fiber $\cX_0$ with positive volume.

Furthermore, we obtain that the holomorphic deformation limit of compact Kähler manifolds is a manifold in the Fujiki class $\mathcal{C}$ under some mild conditions.

By adapting the argument used in the Kähler setting, we prove that the holomorphic deformation limit of projective manifolds is Moishezon without any additional condition.