Characteristic classes of symmetric products of complex quasi-projective varieties

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.

Paper rewritten to also include generating series formulae for motivic homology Hirzebruch classes, Todd classes of coherent sheaves, as well as Chern classes of constructible sheaves. Some special cases of L-class formulae are also obtained. Moreover, we give a mixed-Hodge-module-free proof for the case of motivic homology Hirzebruch classes