2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/104912Generalizing the recent result of Berndtsson, we prove the Nakano semipositivity of the direct image of relative pluricanonical systems and the direct image of relative adjoint (singular) hermitian line bundle with semipositive curvature. This gives a new proof of Kawamata's theorem on the semipositivity of the direct image of relative pluricanonical systems.Complex VariablesAlgebraic Geometry32G05text