Holomorphic Limits of Kahler Manifolds

Abstract

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 $t\in {\Delta }^{\ast }$. If $\pi$ is Kähler at some point $t_0\in {\Delta }^{\ast }$, we construct a positive $d$-closed $(1,1)$-current $T$ on the central fiber $\cX_0$ such that ${\widetilde{\mathrm{vol}}}_n(\{T\})>0$. This implies that $\cX_0$ belongs to Fujiki's class $\mathcal{C}$ provided its upper volume is finite. We also prove that the central fiber of a smooth family whose punctured fibers are projective is Moishezon, with no additional hypothesis on the central fiber.