Characteristic classes of symmetric products of complex quasi-projective varieties

Abstract

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.

Equivariant characteristic classes of external and symmetric products of varieties

“…In this paper, we obtain refined generating series formulae for equivariant characteristic classes of external and symmetric products of singular complex quasi-projective varieties, generalizing our previous results for symmetric products from [13].…”

“…Let now Z be a quasi-projective variety, and denote by Z (n) := Z n /Σ n its n-th symmetric product (i.e., the quotient of Z n by the natural permutation action of the symmetric group Σ n on n elements), with π n : Z n → Z (n) the natural projection map. The standard approach for computing invariants of the symmetric products Z (n) is to collect the respective invariants of all symmetric products in a generating series, and then compute the latter solely in terms of invariants of Z, e.g., see [13] and the references therein. In this paper, we obtain generalizations of results of [13], formulated in terms of equivariant characteristic classes of external products and resp., symmetric products of varieties.…”

“…To a given object F ∈ cat(Z) in a category as above, i.e., coherent or constructible sheaves, or mixed Hodge modules on Z (resp., morphisms f : Y → Z in the motivic context), we attach new objects as follows (see [13,27,28] for details):…”

“…The proof of Theorem 1.4 is purely formal, based on the multiplicativity and conjugacy invariance of the Lefschetz-type claracteristic classes cl * (−; g), together with the following key localization formula from [13][Lemma 3.3, Lemma 3.6, Lemma 3.10]:…”

“…For this localization formula in the context of Hirzebruch classes, it is important to work with the parameter −y and the un-normalized versions of Hirzebruch classes and their respective equivariant analogues, see [13]. In fact, formula (4) is a special case of an abstract generating series formula (37), which holds for any functor H (covariant for isomorphisms) with a compatible commutative, associative cross-product ⊠, with a unit 1 pt ∈ H(pt).…”