  We show that, for any fixed weight, there is a natural system of Hodge
sheaves, whose Higgs field has no poles, arising from a flat projective family
of varieties parametrized by a regular complex base scheme, extending the
analogous classical result for smooth projective families due to Griffiths. As
an application, based on positivity of direct image sheaves, we establish a
criterion for base spaces of rational Gorenstein families to be of general
type. A key component of our arguments is centered around the construction of
derived categorical objects generalizing relative logarithmic forms for smooth
maps and their functorial properties.
