
Mathematics 2012
Chern Classes and Compatible Power Operations in Inertial KtheoryAbstract: Let [X/G] be a smooth DeligneMumford quotient stack. In a previous paper the authors constructed a class of exotic products called inertial products on K(I[X/G]), the Grothendieck group of vector bundles on the inertia stack I[X/G]. In this paper we develop a theory of Chern classes and compatible power operations for inertial products. When G is diagonalizable these give rise to an augmented $\lambda$ring structure on inertial Ktheory. One wellknown inertial product is the virtual product. Our results show that for toric DeligneMumford stacks there is a $\lambda$ring structure on inertial Ktheory. As an example, we compute the $\lambda$ring structure on the virtual Ktheory of the weighted projective lines P(1,2) and P(1,3). We prove that after tensoring with C, the augmentation completion of this $\lambda$ring is isomorphic as a $\lambda$ring to the classical Ktheory 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 HyperKaehler Resolution Conjecture.
