Deformation invariance of canonical nefness in smooth Kahler morphisms

Abstract

Let $f:X\to \Delta$ be a smooth Kähler morphism from complex manifold $X$ to the unit disc. We prove that the canonical bundle $K_{X_t}$ is nef for \emph{every} fiber as soon as it is nef for \emph{one} fiber. This answers, in arbitrary dimension and in the Kähler setting, the deformation-openness problem for non-nefness of the canonical bundle raised by Campana and Peternell.