We study the behavior of Hodge-genera under algebraic maps. We prove that the motivic Xyc -genus satisfies the "stratified multiplicative property", which shows how to compute the invariant of the source of a morphism from its values on varieties arising from the singularities of the map. By considering morphisms to a curve, we obtain a Hodge-theoretic version of the Riemann-Hurwitz formula. We also study the monodromy contributions to the Xy-genus of a family of compact complex manifolds, and prove an Atiyah-Meyer type formula in the algebraic and analytic contexts. This formula measures the deviation from multiplicativity of the Xy-genus, and expresses the correction terms as higher-genera associated to the period map; these higher-genera are Hodge-theoretic extensions of Novikov higher-signatures to analytic and algebraic settings. Characteristic class formulae of Atiyah-Meyer type are also obtained by making use of Saito's theory of mixed Hodge modules.
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