Chern Classes and Compatible Power Operations in Inertial K-theory

Dan Edidin, Tyler J. Jarvis, Takashi Kimura

Published 2012-09-10, updated 2015-06-06Version 2

In a previous paper the authors constructed a class of exotic products called inertial products on the Grothendieck group of vector bundles on the inertia stack of a smooth Deligne-Mumford quotient stack. In this paper we develop a theory of Chern classes and compatible power operations for inertial products. When the group is diagonalizable these give rise to an augmented lambda-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 lambda-ring structure on inertial K-theory. As an example, we compute the lambda-ring structure on the virtual K-theory of the weighted projective lines P(1,2) and P1,3). We prove that after tensoring with C, the augmentation completion of this lambda-ring is isomorphic as a lambda-ring to the classical K-theory of the crepant resolutions of singularities of the coarse moduli spaces of the cotangent bundles T*P(1,2) and T*P(1,3), respectively. We interpret this as a manifestation of mirror symmetry in the spirit of the Hyper-Kaehler Resolution Conjecture.