Logarithmic Trace and Orbifold Products

The purpose of this paper 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, KG(X) → KG(I2G(X)) taking non-negative elements to non-negative elements. (Here I G (X) = {(g1, g2, x)|g1x = g2x = x} ⊂ G×G×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 T ∈ KG(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 non-negative integral element of KG(I 2 G (X)). We also give an equivariant description of the product on the orbifold Ktheory 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 (Theorem 8.7) that if X = [X/G], then there is an associative orbifold product structure on K(X ) ⊗ C distinct from the usual tensor product.