Hodge genera of algebraic varieties, II

We study the behavior of Hodge-genera under algebraic maps. We prove that the motivic ${chi^c_y}$ -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 ${chi_y}$ -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 ${chi_y}$ -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.