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 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.

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
Milestones
Received: 6 June 2015
Revised: 8 October 2015
Accepted: 23 October 2015
Published: 3 September 2016
Authors
Dan Edidin
Department of Mathematics
University of Missouri
Columbia, MO 65211
United States
Tyler J. Jarvis
Department of Mathematics
Brigham Young University
275 TMCB
Provo, UT 84602-6521
United States
Takashi Kimura
Department of Mathematics and Statistics
111 Cummington Mall
Boston University
Boston, MA 02215
United States