Arakelov type inequalities for Hodge bundles over algebraic varieties, Part 1: Hodge bundles over algebraic curves

We prove Arakelov inequalities for systems of Hodge bundles over algebraic varieties, generalizing the classical ones for families of semi-stable curves and abelian varieties. These inequalities are derived from the semi-stability of an associated Higgs bundle, a consequence of the existence of a Hermitian Yang-Mills metric.