Deformation rigidity for projective manifolds and isotriviality of smooth families

Let $\pi \text{cln}X\to {\mathrm{\Delta }}^m$ be a proper smooth Kähler morphism from a complex manifold $X$ to the unit polydisc ${\mathrm{\Delta }}^m$. 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 ${\mathrm{\Delta }}^m$ 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.