A plethora of inertial products

Abstract

For a smooth Deligne–Mumford stack $\mathcal{X}$, we describe a large number of inertial products on $K\left(I\mathcal{X}\right)$ and $A^{\ast }\left(I\mathcal{X}\right)$ and inertial Chern characters. We do this by developing a theory of inertial pairs. Each inertial pair determines an inertial product on $K\left(I\mathcal{X}\right)$ and an inertial product on $A^{\ast }\left(I\mathcal{X}\right)$ and Chern character ring homomorphisms between them. We show that there are many inertial pairs; indeed, every vector bundle $V$ on $\mathcal{X}$ defines two new inertial pairs. We recover, as special cases, the orbifold products considered by Chen and Ruan (2004), Abramovich, Graber and Vistoli (2002), Fantechi and Göttsche (2003), Jarvis, Kaufmann and Kimura (2007) and by the authors (2010), and the virtual product of González, Lupercio, Segovia, Uribe and Xicoténcatl (2007).

We also introduce an entirely new product we call the localized orbifold product, which is defined on $K\left(I\mathcal{X}\right)\otimes ℂ$.

The inertial products developed in this paper are used in a subsequent paper 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.