Hodge-theoretic Atiyah-Meyer formulae and the stratified multiplicative property

In this note we survey Hodge-theoretic formulae of Atiyah-Meyer type for genera and characteristic classes of complex algebraic varieties, and derive some new and interesting applications. We also present various extensions to the singular setting of the Chern-Hirzebruch-Serre signature formula.

💡 Research Summary

The paper provides a comprehensive survey of Atiyah‑Meyer type formulae in the context of Hodge theory, focusing on genera and characteristic classes of complex algebraic varieties, and then extends these results to singular settings. After a brief historical overview of the classical Atiyah‑Meyer signature formula—originally formulated for smooth fiber bundles—the authors introduce Saito’s theory of mixed Hodge modules (MHM) as the main technical framework. Mixed Hodge modules allow one to retain Hodge‑theoretic information (weight and Hodge filtrations) even in the presence of singularities, and they come equipped with functorial push‑forward and pull‑back operations that respect these filtrations.

Using the MHM formalism, the authors define a “Hirzebruch‑Meyer transformation” (T_{y*}) which maps the Grothendieck group of mixed Hodge modules on a variety (X) to homology classes with coefficients in (\mathbb{Q}