  We develop a Hodge theoretic invariant for families of projective manifolds
that measures the potential failure of an Arakelov-type inequality in higher
dimensions, one that naturally generalizes the classical Arakelov inequality
over regular quasi-projective curves. We show that for families of manifolds
with ample canonical bundle this invariant is uniformly bounded. As a
consequence we establish that such families over a base of arbitrary dimension
verify the aforementioned Arakelov inequality, answering a question of Viehweg.
