Characteristic classes of mixed Hodge modules

TL;DR

This paper explores the development of characteristic classes for mixed Hodge modules, linking Hodge theory with algebraic topology on singular complex algebraic spaces, and introduces functorial classes with applications to various Hodge sheaves.

Contribution

It introduces a framework for characteristic classes of mixed Hodge modules, connecting K-theoretical and homological classes, and demonstrates their functorial properties and explicit descriptions.

Findings

Defined motivic characteristic classes for mixed Hodge modules

Established functoriality under proper pushdown and products

Provided explicit descriptions for admissible variations of mixed Hodge structures

Abstract

This paper gives an introduction and overview about recent developments on the interaction of the theories of characteristic classes and mixed Hodge theory for singular spaces in the complex algebraic context. It uses M. Saito's deep theory of mixed Hodge modules as a "black box", thinking about them as "constructible or perverse sheaves of Hodge structures", having the same functorial calculus of Grothendieck functors. For the "constant Hodge sheaf", one gets the "motivic characteristic classes" of Brasselet-Schuermann-Yokura, whereas the classes of the "intersection homology Hodge sheaf" were studied by Cappell-Maxim-Shaneson. There are two versions of these characteristic classes. The K-theoretical classes capture information about the graded pieces of the filtered de Rham complex of the filtered D-module underlying a mixed Hodge module. Application of a suitable Todd class transformation then gives classes in homology. These classes are functorial for proper pushdown and exterior products, together with some other properties one would expect for a "good" theory of characteristic classes for singular spaces. For "admissible variation of mixed Hodge structures" they have an explicit classical description in terms of "logarithmic de Rham complexes". On a point space they correspond to a specialization of the Hodge polynomial of a mixed Hodge structure, which one gets by forgetting the weight filtration.