2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/126468We prove semipositivity of the curvature of the Narashimhan-Simha metric on an arbitrary flat projective family of varieties with only canonical singularities. This also implies that the direct image of pluricanonical systems have a natural continuous hermitian strucutre with semipositive curvature curvature current in the sense of Nakano. This is an improvement of the author's preprint ``Refined semipositivity and moduli of canonical models (2005)''.24pages no figure, Reference up dated and correction on the proof of Theorem 5.5Algebraic GeometryComplex Variables14J15,14J40, 32J18text