Hodge genera of algebraic varieties, II

TL;DR

This paper investigates how Hodge-theoretic genera behave under morphisms of complex algebraic varieties, establishing formulas for their computation, including a Hodge-theoretic Riemann-Hurwitz analogue and Atiyah-Meyer type correction formulas.

Contribution

It introduces the stratified multiplicative property of the $hi_y$-genus and develops Atiyah-Meyer type formulas for twisted $hi_y$-genera using mixed Hodge modules.

Findings

Proved the stratified multiplicative property of the $hi_y$-genus.

Derived a Hodge-theoretic Riemann-Hurwitz formula for morphisms to curves.

Established Atiyah-Meyer type formulas for twisted $hi_y$-genera and Hirzebruch characteristic classes.

Abstract

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.