Stringy K-theory and the Chern character
ralph kaufmann
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Abstract
We construct two new G-equivariant rings: K(X, G), called the stringy K-theory of the G-variety X, and H(X, G), called the stringy cohomology of the G-variety X, for any smooth, projective variety X with an action of a finite group G. For a smooth Deligne-Mumford stack X, we also construct a new ring K orb (X) called the full orbifold K-theory of X. We show that for a global quotient X = [X/G], the ring of G-invariants K orb (X) of K(X, G) is a subalgebra of K orb ([X/G]) and is linearly isomorphic to the "orbifold K-theory" of Adem-Ruan [AR] (and hence Atiyah-Segal), but carries a different "quantum" product which respects the natural group grading. We prove that there is a ring isomorphism Ch : K(X, G) → H(X, G), which we call the stringy Chern character. We also show that there is a ring homomorphism Ch orb : K orb (X) → H • orb (X), which we call the orbifold Chern character, which induces an isomorphism Ch orb : K orb (X) → H • orb (X) when restricted to the sub-algebra K orb (X). Here H • orb (X) is the Chen-Ruan orbifold cohomology. We further show that Ch and Ch orb preserve many properties of these algebras and satisfy the Grothendieck-Riemann-Roch theorem with respect toétale maps. All of these results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds.
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