Logarithmic trace and orbifold products

Abstract

The purpose of this article is to give a purely equivariant definition of orbifold Chow rings of quotient Deligne-Mumford stacks. This completes a program begun in [JKK] for quotients by finite groups. The key to our construction is the definition (Section 6.1) of a twisted pullback in equivariant $K$-theory, $K_G(X)\to K_G({\mathbb{I}}_G^2(X))$ taking nonnegative elements to nonnegative elements. (Here ${\mathbb{I}}_G^2(X)=\{(g_1,g_2,x)|g_{1}x=g_{2}x=x\}\subset G\times G\times X$.) The twisted pullback is defined using data about fixed loci of elements of finite order in $G$ but depends only on the underlying quotient stack (Theorem 6.3). In our theory, the twisted pullback of the class $\mathbb{T}\in K_G(X)$, corresponding to the tangent bundle to $[X/G]$, replaces the obstruction bundle of the corresponding moduli space of twisted stable maps. When $G$ is finite, the twisted pullback of the tangent bundle agrees with the class $R(m)$ given in [JKK, Definition 1.5]. However, unlike in [JKK] we need not compare our class to the class of the obstruction bundle of Fantechi and Göttsche [FG] in order to prove that it is a nonnegative integral element of $K_G({\mathbb{I}}_G^2(X))$.

We also give an equivariant description of the product on the orbifold $K$-theory of $[X/G]$. Our orbifold Riemann-Roch theorem (Theorem 7.3) states 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 show (see Theorem 8.7) that if $\mathcal{X}=[X/G]$, then there is an associative orbifold product structure on $K(\mathcal{X})\otimes \mathbb{C}$ distinct from the usual tensor product