Annals of K-Theory

A plethora of inertial products

Dan Edidin, Tyler Jarvis, and Takashi Kimura

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Abstract

For a smooth Deligne–Mumford stack X, we describe a large number of inertial products on K(IX) and A(IX) and inertial Chern characters. We do this by developing a theory of inertial pairs. Each inertial pair determines an inertial product on K(IX) and an inertial product on A(IX) and Chern character ring homomorphisms between them. We show that there are many inertial pairs; indeed, every vector bundle V on X defines two new inertial pairs. We recover, as special cases, the orbifold products considered by Chen and Ruan (2004), Abramovich, Graber and Vistoli (2002), Fantechi and Göttsche (2003), Jarvis, Kaufmann and Kimura (2007) and by the authors (2010), and the virtual product of González, Lupercio, Segovia, Uribe and Xicoténcatl (2007).

We also introduce an entirely new product we call the localized orbifold product, which is defined on K(IX) .

The inertial products developed in this paper are used in a subsequent paper to describe a theory of inertial Chern classes and power operations in inertial K-theory. These constructions provide new manifestations of mirror symmetry, in the spirit of the hyper-Kähler resolution conjecture.

Article information

Source
Ann. K-Theory, Volume 1, Number 1 (2016), 85-108.

Dates
Received: 8 January 2015
Accepted: 26 January 2015
First available in Project Euclid: 12 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.akt/1513104170

Digital Object Identifier
doi:10.2140/akt.2016.1.85

Mathematical Reviews number (MathSciNet)
MR3514937

Zentralblatt MATH identifier
06493603

Subjects
Primary: 55N32: Orbifold cohomology 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX}
Secondary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35] 57R18: Topology and geometry of orbifolds 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45] 19L10: Riemann-Roch theorems, Chern characters 19L47: Equivariant $K$-theory [See also 55N91, 55P91, 55Q91, 55R91, 55S91] 14H10: Families, moduli (algebraic)

Keywords
quantum $K$-theory orbifold product orbifold cohomology Gromov–Witten equivariant stringy inertia Deligne–Mumford stack

Citation

Edidin, Dan; Jarvis, Tyler; Kimura, Takashi. A plethora of inertial products. Ann. K-Theory 1 (2016), no. 1, 85--108. doi:10.2140/akt.2016.1.85. https://projecteuclid.org/euclid.akt/1513104170


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