Twisted genera of symmetric products

Abstract

We give a new proof of formulae for the generating series of (Hodge) genera of symmetric products X (n) with coefficients, which hold for complex quasiprojective varieties X with any kind of singularities and which include many of the classical results in the literature as special cases. Important specializations of our results include generating series for extensions of Hodge numbers and Hirzebruch's χ y -genus to the singular setting and, in particular, generating series for intersection cohomology Hodge numbers and Goresky-MacPherson intersection cohomology signatures of symmetric products of complex projective varieties. Our proof applies to more general situations and is based on equivariant Künneth formulae and pre-lambda structures on the coefficient theory of a point,K 0 (A( pt)), with A( pt) a Karoubian Q-linear tensor category. Moreover, Atiyah's approach to power operations in K -theory also applies in this context toK 0 (A( pt)), giving a nice description of the important related Adams operations. This last approach also allows us to introduce very interesting coefficients on the symmetric products X (n) .

Equivariant Characteristic Classes of Singular Complex Algebraic Varieties

“…In all cases above, a G-equivariant element M is just an element in the underlying additive category (e.g., D b MHM(X)), with a G-action given by isomorphisms [MS09][Appendix]). Note that many references (e.g., [BFQ] or [CMSc]) work with the corresponding isomorphisms g * M → M defined by adjunction, which are more natural for contravariant theories such as K 0 (X) or variations of mixed Hodge structures.…”

“…, for ψ g the isomorphism induced from the action of g ∈ G. Here we use the fact that the underlying categories D b MHM(X) and D b coh (X) are Q-linear additive categories which are Karoubian by [BS, LC] (i.e., any projector has a kernel, see also [MS09]). Since P G is exact, we obtain induced functors on the Grothendieck groups:…”

“…Theorem 5.1 is an essential ingredient for obtained generating series formulae for homology Hirzebruch classes of symmetric products of singular complex quasi-projective algebraic varieties, see [CMSSY] (compare also with [MS09,MSS], where the case of Hodge numbers and Hodge-Deligne polynomials is discussed).…”

“…We follow closely the treatment from [MS09][Appendix]. For simplicity we only work in the underlying category var qp /C of complex quasi-projective algebraic varieties (whereas [MS09][Appendix] applies to any (small) category space of spaces with finite products and a terminal object).…”

“…Theorem 1.1 is used in the authors' paper [CMSSY] for obtaining generating series formulae for the homology Hirzebruch classes T y * (X (n) ) of the symmetric products X (n) := X n /Σ n of a (possibly singular) complex quasi-projective algebraic variety X (see also [MS10]). In fact, by making use of Theorem 1.3, such generating series results are formulated in [CMSSY] for characteristic classes of symmetric products of pairs (X, M), with M ∈ D b MHM(X) (compare also with [MS09,MSS], where the case of Hodge numbers and Hodge-Deligne polynomials is discussed).…”