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2017 Chern classes and compatible power operations in inertial K-theory
Dan Edidin, Tyler Jarvis, Takashi Kimura
Ann. K-Theory 2(1): 73-130 (2017). DOI: 10.2140/akt.2017.2.73


Let X = [XG] be a smooth Deligne–Mumford quotient stack. In a previous paper we constructed a class of exotic products called inertial products on K(IX), the Grothendieck group of vector bundles on the inertia stack IX. 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 λ-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 λ-ring structure on inertial K-theory. As an example, we compute the λ-ring structure on the virtual K-theory of the weighted projective lines (1,2) and (1,3). We prove that, after tensoring with , the augmentation completion of this λ-ring is isomorphic as a λ-ring to the classical K-theory of the crepant resolutions of singularities of the coarse moduli spaces of the cotangent bundles T(1,2) and T(1,3), respectively. We interpret this as a manifestation of mirror symmetry in the spirit of the hyper-Kähler resolution conjecture.


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Dan Edidin. Tyler Jarvis. Takashi Kimura. "Chern classes and compatible power operations in inertial K-theory." Ann. K-Theory 2 (1) 73 - 130, 2017.


Received: 6 June 2015; Revised: 8 October 2015; Accepted: 23 October 2015; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1357.14068
MathSciNet: MR3599517
Digital Object Identifier: 10.2140/akt.2017.2.73

Primary: 14N35 , 19L10 , 53D45
Secondary: 14H10 , 55N15

Keywords: inertial products , lambda rings , orbifold product , quantum cohomology , quantum K-theory , virtual product

Rights: Copyright © 2017 Mathematical Sciences Publishers


Vol.2 • No. 1 • 2017
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