Deformation rigidity for projective manifolds and isotriviality of smooth families

Abstract

Let $π\cln X\to Δ^m$ be a proper smooth Kähler morphism from a complex manifold $X$ to the unit polydisc ${\Delta }^m$. Suppose the fibers over the complement of a proper analytic subset are biholomorphic to a fixed projective manifold $S$. If the canonical line bundle of $S$ is semiample, then we show that all fibers over ${\Delta }^m$ are biholomorphic to $S$. As an application, we obtain that for smooth families where the canonical line bundle of the generic fiber is semiample, birational isotriviality is equivalent to isotriviality. Moreover, we establish a new Parshin-Arakelov type isotriviality criterion.