HODGE GENERA OF ALGEBRAIC VARIETIES, II.

Abstract. We study the behavior of Hodge-theoretic genera under morphisms of complex algebraic varieties. We prove that the additive χcy-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 an algebraic 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 χy-genus of a family of compact complex manifolds, and prove an Atiyah-Meyer type formula for twisted χy-genera, both in the algebraic and the analytic context. This formula measures the deviation from multiplicativity of the χy-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. 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. Content