Logarithmic trace and orbifold products

TL;DR

This paper introduces a new equivariant approach to orbifold products for quotient stacks using the logarithmic trace, establishing an orbifold Chern character and an associative product on the Grothendieck ring.

Contribution

It provides a purely equivariant construction of orbifold products for quotient stacks with arbitrary linear algebraic groups, introducing the logarithmic trace and an orbifold Chern character.

Findings

Defined the logarithmic trace of an equivariant vector bundle.

Proved the orbifold Chern character induces an isomorphism with the orbifold Chow ring.

Established an associative orbifold product on the Grothendieck ring.

Abstract

We give a purely equivariant construction of orbifold products for quotient Deligne-Mumford stacks [X/G] where G is an arbitrary linear algebraic group (not necessarily finite). The key to our construction is the definition of the "logarithmic trace" of an equivariant vector bundle. We also prove that there is an orbifold Chern character homomorphism which induces an isomorphism of a canonical summand in the orbifold Grothendieck ring with the orbifold Chow ring. As an application we obtain an associative orbifold product on the Grothendieck ring of [X/G] (as opposed to its inerita stack) taken with complex coefficients.