Arakelov inequalities in higher dimensions

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 satisfy the aforementioned Arakelov inequality, answering a question of Viehweg.