Let be a smooth Deligne–Mumford quotient stack. In a previous paper we constructed a class of exotic products called inertial products on , the Grothendieck group of vector bundles on the inertia stack . In this paper we develop a theory of Chern classes and compatible power operations for inertial products. When is diagonalizable these give rise to an augmented -ring structure on inertial K-theory.
One well-known inertial product is the virtual product. Our results show that for toric Deligne–Mumford stacks there is a -ring structure on inertial K-theory. As an example, we compute the -ring structure on the virtual K-theory of the weighted projective lines and . We prove that, after tensoring with , the augmentation completion of this -ring is isomorphic as a -ring to the classical K-theory of the crepant resolutions of singularities of the coarse moduli spaces of the cotangent bundles and , respectively. We interpret this as a manifestation of mirror symmetry in the spirit of the hyper-Kähler resolution conjecture.
"Chern classes and compatible power operations in inertial K-theory." Ann. K-Theory 2 (1) 73 - 130, 2017. https://doi.org/10.2140/akt.2017.2.73