Deformation rigidity for projective manifolds and isotriviality of smooth families over curves

Let π X→ Δ be a proper smooth K\"ahler morphism from a complex manifold X to the unit disc Δ. Suppose the fibers Xₜ=π⁻¹(t) are biholomorphic to S for all t≠0, where S is a given projective manifold. If the canonical line bundle of S is semiample, then we show that the central fiber X₀ is also biholomorphic to S. As an application, we obtain that, for smooth families over projective curves satisfying that the canonical line bundle of the generic fiber is semiample, birational isotriviality equals to isotriviality. Moreover, we also obtain a new Parshin-Arakelov type isotriviality criterion.