  We study the behavior of Hodge-theoretic genera under morphisms of complex
algebraic varieties. We prove that the additive $\chi_y$-genus which arises in
the motivic context satisfies the so-called ``stratified multiplicative
property", which shows how to compute the invariant of the source of a proper
surjective morphism from its values on various varieties that arise from the
singularities of the map. By considering morphisms to a curve, we obtain a
Hodge-theoretic analogue of the Riemann-Hurwitz formula. We also study the
contribution of monodromy to the $\chi_y$-genus of a smooth projective family,
and prove an Atiyah-Meyer type formula for twisted $\chi_y$-genera. This
formula measures the deviation from multiplicativity of the $\chi_y$-genus, and
expresses the correction terms as higher-genera associated to cohomology
classes of the quotient of the total period domain by the action of the
monodromy group. By making use of Saito's theory of mixed Hodge modules, we
also obtain formulae of Atiyah-Meyer type for the corresponding Hirzebruch
characteristic classes.
