Characteristic classes of mixed Hodge modules

This paper is an extended version of an expository talk given at the workshop “Topology of Stratified Spaces” at MSRI in September 2008. It 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, Schürmann, and Yokura, whereas the classes of the “intersection homology Hodge sheaf” were studied by Cappell, Maxim, and Shaneson. The classes associated to “good” variation of mixed Hodge structures where studied in connection with understanding the monodromy action by these three authors together with Libgober, and also by the author. 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 satisfactory theory of characteristic classes for singular spaces. For “good” 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. We also indicate some relations with other subjects of the conference, like index theorems, signature, L-classes, elliptic genera and motivic characteristic classes for singular spaces.