Stringy K-theory and the Chern character

Abstract

For a finite group G acting on a smooth projective variety X, we construct two new G-equivariant rings: first the stringy K-theory of X, and second the stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct a new ring called the full orbifold K-theory of Y. For a global quotient Y=[X/G], the ring of G-invariants of the stringy K-theory of X is a subalgebra of the full orbifold K-theory of the the stack Y and is linearly isomorphic to the ``orbifold K-theory'' of Adem-Ruan (and hence Atiyah-Segal), but carries a different, ``quantum,'' product, which respects the natural group grading. We prove there is a ring isomorphism, the stringy Chern character, from stringy K-theory to stringy cohomology, and a ring homomorphism from full orbifold K-theory to Chen-Ruan orbifold cohomology. These Chern characters satisfy Grothendieck-Riemann-Roch for etale maps. We prove that stringy cohomology is isomorphic to Fantechi and Goettsche's construction. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results simplify the definitions of Fantechi-Goettsche's ring, of Chen-Ruan's orbifold cohomology, and of Abramovich-Graber-Vistoli's orbifold Chow. We conclude by showing that a K-theoretic version of Ruan's Hyper-Kaehler Resolution Conjecture holds for symmetric products. Our results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds.Comment: Exposition improved and additional details provided. To appear in Inventiones Mathematica

On the <i>K</i>-theory of Toric Stack Bundles

“…We will use the following description of the Euler class of the obstruction bundle: Proposition 3.3 (see [7], [9]). …”

“…In Section 5.2 we use the Chern character homomorphism in [9] to show that the Chern character is an ring isomorphism.…”

“…Define W v,k to be the eigenbundle of W v , where v acts by multiplication by ζ k = e 2πik/r . Following [9], we define…”

Stringy product on twisted orbifold K-theory for abelian quotients

“…We construct a stringy ring on the K-theory of the Borel construction of the inertia orbifold, similar to the stringy structure in K-theory that was defined in [JKK07]. This construction does not need rational coefficients either, and it uses the same principle of the stringy product.…”

“…Applying a calibrated Chern character map (mainly due to Jarvis-KaufmannKimura [JKK07]), we were able to prove that there is a ring homomorphism between the stringy ring on the K-theory of the inertia orbifold and the Chen-Ruan cohomology of the orbifold. We come to the conclusion that the right place to study stringy products is K-theory and not cohomology; this is due to the fact that the torsion classes in the stringy K-theory are present.…”