  We prove that the canonical bundle of any holomorphic family of compact
complex algebraic manifolds carries a singular Hermitian metric having
non-negative curvature current and such that every holomorphic section of the
canonical bundle of the central fiber is L^2 with respect to this metric. This
result was proved by Siu when the members of the family are general type, and
used to establish the deformation invariance of plurigenera in that case. In
fact, we prove our result in a more general setting of a smooth hypersurface
with not-necessarily normal trivial bundle in a manifold that is Stein outside
an analytic subvariety. Such a setting was first considered by Takayama. (In
the case of a family, the normal bundle is trivial.) The method of proof
involves an adaptation of recent work of Paun.
