Chern classes and compatible power operations in inertial K-theory

Abstract

Let $\mathcal{X}=\left[X∕G\right]$ be a smooth Deligne–Mumford quotient stack. In a previous paper we constructed a class of exotic products called inertial products on $K\left(I\mathcal{X}\right)$, the Grothendieck group of vector bundles on the inertia stack $I\mathcal{X}$. 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 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 $\mathbb{P}\left(1,2\right)$ and $\mathbb{P}\left(1,3\right)$. 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 ${\mathbb{T}}^{\ast }\mathbb{P}\left(1,2\right)$ and ${\mathbb{T}}^{\ast }\mathbb{P}\left(1,3\right)$, respectively. We interpret this as a manifestation of mirror symmetry in the spirit of the hyper-Kähler resolution conjecture.