Deformation rigidity for projective manifolds and isotriviality of smooth families

Abstract

Let

be a proper smooth Kähler morphism from a complex manifold 𝑋 to the unit polydisc

Δ m \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

Δ m \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.