Characteristic classes of symmetric products of complex quasi-projective varieties

TL;DR

This paper develops generating series formulas for twisted characteristic classes of symmetric products of complex quasi-projective varieties, extending known results to singular spaces and various classes.

Contribution

It introduces new generating series formulas for characteristic classes of symmetric products, applicable to singular varieties and with various coefficients, generalizing prior results.

Findings

Derived formulas for homology Hirzebruch classes of symmetric products.

Extended results to twisted homology L-classes and Todd classes.

Generalized Chern class computations for complexes of sheaves.

Abstract

We prove generating series formulae for suitable twisted characteristic classes of symmetric products of a singular complex quasi-projective variety. More concretely, we study homology Hirzebruch classes for motivic coefficients, as well as for complexes of mixed Hodge modules. As a special case, we obtain a generating series formula for the (intersection) homology Hirzebruch classes of symmetric products. In some cases, the latter yields a similar formula for twisted homology L-classes generalizing results of Hirzebruch-Zagier and Moonen. Our methods also apply to the study of Todd classes of (complexes of) coherent sheaves, as well as Chern classes of (complexes of) constructible sheaves, generalizing to arbitrary coefficients results of Moonen and resp. Ohmoto.