A plethora of inertial products

TL;DR

This paper develops a comprehensive theory of inertial products on the K-theory and Chow rings of smooth Deligne-Mumford stacks, unifying and extending existing orbifold product constructions and introducing new products with applications to mirror symmetry.

Contribution

It introduces a general framework of inertial pairs that generate many known and new inertial products, including a novel localized orbifold product, advancing the understanding of inertial K-theory and Chow rings.

Findings

Revealed a large class of inertial products on K(IX) and A*(IX).

Unified various existing orbifold product constructions under a common framework.

Introduced a new localized orbifold product on the complexified K-theory.

Abstract

For a smooth Deligne-Mumford stack X we describe a large number of inertial products on K(IX) and A*(IX) and corresponding 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, the orbifold products of Chen-Run, Abramovich-Graber-Vistoli, Jarvis-Kaufmann-Kimura, and Edidin-Jarvis-Kimura and the virtual product of Gonzalez-Lupercio-Segovia-Uribe-Xicotencatl. We also introduce an entirely new product we call the localized orbifold product, which is defined on the complexification of K(IX). 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-Kaehler Resolution Conjecture.