A plethora of inertial products

Abstract

Abstract. For a smooth Deligne-Mumford stack X , we describe a large number of inertial products on K(IX ) and A * (IX ) and inertial Chern characters. We do this by developing a theory of inertial pairs. Each inertial pair determines an inertial product on K(IX ) and an inertial product on A * (IX ) and Chern character ring homomorphisms between them. We show that there are many inertial pairs; indeed, every vector bundle V on X defines two new inertial pairs. We recover, as special cases, both the orbifold products of [CR04, AGV02, FG03, JKK07, EJK10] and the virtual product of [GLS + 07].We also introduce an entirely new product we call the localized orbifold product, which is defined on K(IX ) ⊗ C.The inertial products developed in this paper are used in a subsequent paper [EJK12] to describe a theory of inertial Chern classes and power operations in inertial K-theory. These constructions provide new manifestations of mirror symmetry, in the spirit of the Hyper-Kähler Resolution Conjecture.

Inertial Chow rings of toric stacks

“…Remark 3.8. The assumption that G is diagonalizable implies that the full group G acts on X g for any g ∈ G. This simplifies Definitions 3.7 and 3.10 as compared to those found in [EJK1,EJK2].…”

“…Let G be an algebraic group acting on a scheme X. We recall the formalism of inertial products of [EJK2]. However, since our application is to toric stacks, we assume that G is diagonalizable throughout.…”

“…The basic example of an associative vector bundle is the obstruction bundle used to define the orbifold product. This, and a plethora of other associative bundles can be constructed using the logarithmic trace construction of [EJK1,EJK2].…”

“…In this paper we turn our attention to describing general intertial products on toric DM stacks. The formalism of inertial products was introducted in [EJK1] and further developed in [EJK2]. In the latter paper the authors show that if X is a smooth DM stack then any vector bundle V on X defines two inertial products V + and V − on the Chow groups of the inertia stack IX .…”

“…In Section 3 we recall the formalism of inertial products of [EJK1,EJK2]. We also recall the logarithmic trace map which is used to define the V + and V − products.…”