Under the semiample canonical bundle assumption, smooth Kähler families over the polydisc with biholomorphic fibers outside a proper analytic subset have all fibers biholomorphic to the fixed one, implying birational isotriviality equals isotriviality.
N., Duality between the pseudoeffective and the movable cone on a projective manifold, With an appendix by S
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Deformation rigidity for projective manifolds and isotriviality of smooth families
Under the semiample canonical bundle assumption, smooth Kähler families over the polydisc with biholomorphic fibers outside a proper analytic subset have all fibers biholomorphic to the fixed one, implying birational isotriviality equals isotriviality.