  We 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)''.

Algebraic Geometry;Complex Variables