The holomorphic limit of Kahler manifolds

Abstract

Let $π:\cX\to Δ$ be a smooth family of compact complex manifolds over the unit disk $Δ$, whose fibers $\cX_{t}$ are Kähler for all $t\in Δ^*$. If $π$ is Kahler at a point $t_0\inΔ^*$, then we show 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. As a corollary, we prove that the holomorphic deformation limit of projective manifolds is Moishezon without any additional condition.