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The Hodge numerical invariants of a variation of Hodge structure over asmooth quas--projective variety are a measure of complexity for the globaltwisting of the limit mixed Hodge structure when it degenerates. Theseinvariants appear in inequalities which they may have correction terms, calledArakelov inequalities. One may investigate the correction term to make theminto equalities, also called Arakelov equalities. We investigate numericalArakelov type (in)equilities for a family of surfaces fibered by curves. Ourmethod uses Arakelov identities in a weight 1 and also in a weight 2 variationsof Hodge structure (cf. \cite{GGK}), in a commutative triangle of fibrations.We have proposed to relate the degrees of the Hodge bundles in the twofamilies. We also compare the Fujita decomposition of Hodge bundles in thesefibrations. We examine various identities and relations between Hodge numbersand degrees of the Hodge bundles in different levels.

Hodge-Arakelov inequalities for family of surfaces fibered by curves