Hodge genera of algebraic varieties, II
Sylvain E. Cappell, Anatoly Libgober, Laurentiu Maxim, Julius L., Shaneson

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 -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 -genus of a smooth projective family, and prove an Atiyah-Meyer type formula for twisted -genera. This formula measures the deviation from multiplicativity of the -genus, and expresses the correction terms as higher-genera associated to cohomology classes of the quotient of the total period domain by the…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Homotopy and Cohomology in Algebraic Topology
