Vol. 2, No. 1, 2017

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Chern classes and compatible power operations in inertial K-theory

Dan Edidin, Tyler J. Jarvis and Takashi Kimura

Vol. 2 (2017), No. 1, 73–130
DOI: 10.2140/akt.2017.2.73
Abstract

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

Keywords
quantum cohomology, inertial products, lambda rings, quantum K-theory, orbifold product, virtual product
Mathematical Subject Classification 2010
Primary: 14N35, 19L10, 53D45
Secondary: 55N15, 14H10