  Our main goal in this article is to establish a quantitative version of the
positivity properties of twisted relative pluricanonical bundles and their
direct images. The notion of "singular Hermitian metric" on vector bundles
(together with an appropriate definition of positivity of the associated
curvature) plays a major role here, and its properties are studied via a
version of Ohsawa-Takegoshi extension theorem. Part of this article is based on
the joint work of the first named author with Bo Berndtsson, and it can be seen
as an expanded and updated version of it.
