Vol. 1, No. 1, 2016

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A plethora of inertial products

Dan Edidin, Tyler J. Jarvis and Takashi Kimura

Vol. 1 (2016), No. 1, 85–108
DOI: 10.2140/akt.2016.1.85
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.

Keywords
quantum $K$-theory, orbifold, product, orbifold cohomology, Gromov–Witten, equivariant, stringy, inertia, Deligne–Mumford, stack
Mathematical Subject Classification 2010
Primary: 55N32, 55N15
Secondary: 53D45, 57R18, 14N35, 19L10, 19L47, 14H10
Milestones
Received: 8 January 2015
Accepted: 26 January 2015
Published: 31 July 2015
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
United States
Takashi Kimura
Department of Mathematics and Statistics
Boston University
111 Cummington Mall
Boston, MA 02215
United States