Generating functions of orbifold Chern classes I: symmetric products

Abstract

Abstract. In this paper, for a possibly singular complex variety X, generating functions of total orbifold Chern homology classes of the symmetric products S n X are given. Those are very natural "class versions" of known generating function formulae of (generalized) orbifold Euler characteristics of S n X. The classes work covariantly for proper morphisms. We state the result more generally. Let G be a finite group and Gn the wreath product G ∼ Sn. For a G-variety X and a group A, we give a"Dey-Wohlfahrt type formula" for equivariant Chern-SchwartzMacPherson classes associated to Gn-representations of A (Theorem 1.1 and 1.2). In particular, if X is a point, this recovers a known exponential formula for counting numbers |Hom(A, Gn)|.

Equivariant characteristic classes of external and symmetric products of varieties

“…Note that the summand cl * (−; id) ∈ (H * (X)) G corresponding to the identity element of G is just the non-equivariant characteristic class, which for equivariant coefficients is invariant under the G-action by functoriality. Under the identification (2), this class also agrees (for our finite group G) with the corresponding (naive) equivariant characteristic class defined in terms of the Borel construction, e.g., for cl * = td * , this is the equivariant RiemannRoch-Transformation of Edidin-Graham [16]; and for cl * = c * , this is the equivariant Chern class transformation of Ohmoto [33,34].…”

“…In this way, we reprove and generalize some results from [35] and, resp., [41]. Moreover, in Section 4, we give another application of (37) to canonical constructible functions and orbifold-type Chern classes of symmetric products, reproving some results of Ohmoto [34].…”

“…This will make the object of future work by the authors. Results of this type for Chern classes already appeared in [34], while for degree versions see, e.g., [35,41,42,44].…”

“…Let us illustrate what these distinguished constructible functions are in the case A = Z and, resp., Z 2 (for other examples, see [34]). …”

Nonvanishing vector fields on orbifolds

“…The construction of the Γ-sectors is motivated by a construction of Tamanoi in [19] and [20] (see also [2] and [11]) which was used to define a generalized orbifold Euler characteristic of a global quotient orbifold, i.e. an orbifold that admits a presentation as M/G where M is a smooth manifold and G is a finite group acting smoothly.…”