Stringy K-theory and the Chern character

Abstract

We construct two new G-equivariant rings: $\mathcal{K}(X,G)$, called the stringy K-theory of the G-variety X, and $\mathcal{H}(X,G)$, 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 $\mathcal{X}$, we also construct a new ring $\mathsf{K}_{\mathrm{orb}}(\mathcal{X})$ called the full orbifold K-theory of $\mathcal{X}$. We show that for a global quotient $\mathcal{X} = [X/G]$, the ring of G-invariants $K_{\mathrm{orb}}(\mathcal{X})$ of $\mathcal{K}(X,G)$ is a subalgebra of $\mathsf{K}_{\mathrm{orb}}([X/G])$ 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 $\mathcal{C}\mathbf{h}:\mathcal{K}(X,G)\to\mathcal{H}(X,G)$, which we call the stringy Chern character. We also show that there is a ring homomorphism $\mathfrak{C}\mathfrak{h}_\mathrm{orb}:\mathsf{K}_{\mathrm{orb}}(\mathcal{X}) \rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})$, which we call the orbifold Chern character, which induces an isomorphism $Ch_{\mathrm{orb}}:K_{\mathrm{orb}}(\mathcal{X})\rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})$ when restricted to the sub-algebra $K_{\mathrm{orb}}(\mathcal{X})$. Here $H_{\mathrm{orb}}^\bullet(\mathcal{X})$ is the Chen–Ruan orbifold cohomology. We further show that $\mathcal{C}\mathbf{h}$ and $\mathfrak{C}\mathfrak{h}_\mathrm{orb}$ preserve many properties of these algebras and satisfy the Grothendieck–Riemann–Roch theorem with respect to étale maps. All of these results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds.

We further prove that $\mathcal{H}(X,G)$ is isomorphic to Fantechi and Göttsche’s construction [FG, JKK]. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results greatly simplify the definitions of the Fantechi–Göttsche ring, Chen–Ruan orbifold cohomology, and the Abramovich–Graber–Vistoli orbifold Chow ring.

We conclude by showing that a K-theoretic version of Ruan’s Hyper-Kähler Resolution Conjecture holds for the symmetric product of a complex projective surface with trivial first Chern class.