Translator Disclaimer
15 June 2010 Logarithmic trace and orbifold products
Dan Edidin, Tyler J. Jarvis, Takashi Kimura
Author Affiliations +
Duke Math. J. 153(3): 427-473 (15 June 2010). DOI: 10.1215/00127094-2010-028


The purpose of this article is to give a purely equivariant definition of orbifold Chow rings of quotient Deligne-Mumford stacks. This completes a program begun in [JKK] for quotients by finite groups. The key to our construction is the definition (Section 6.1) of a twisted pullback in equivariant K-theory, KG(X)KG(IG2(X)) taking nonnegative elements to nonnegative elements. (Here IG2(X)={(g1,g2,x)|g1x=g2x=x}G×G×X.) The twisted pullback is defined using data about fixed loci of elements of finite order in G but depends only on the underlying quotient stack (Theorem 6.3). In our theory, the twisted pullback of the class TKG(X), corresponding to the tangent bundle to [X/G], replaces the obstruction bundle of the corresponding moduli space of twisted stable maps. When G is finite, the twisted pullback of the tangent bundle agrees with the class R(m) given in [JKK, Definition 1.5]. However, unlike in [JKK] we need not compare our class to the class of the obstruction bundle of Fantechi and Göttsche [FG] in order to prove that it is a nonnegative integral element of KG(IG2(X)).

We also give an equivariant description of the product on the orbifold K-theory of [X/G]. Our orbifold Riemann-Roch theorem (Theorem 7.3) states that there is an orbifold Chern character homomorphism which induces an isomorphism of a canonical summand in the orbifold Grothendieck ring with the orbifold Chow ring. As an application we show (see Theorem 8.7) that if X=[X/G], then there is an associative orbifold product structure on K(X)C distinct from the usual tensor product


Download Citation

Dan Edidin. Tyler J. Jarvis. Takashi Kimura. "Logarithmic trace and orbifold products." Duke Math. J. 153 (3) 427 - 473, 15 June 2010.


Published: 15 June 2010
First available in Project Euclid: 4 June 2010

zbMATH: 1210.14066
MathSciNet: MR2667422
Digital Object Identifier: 10.1215/00127094-2010-028

Primary: 14N35
Secondary: 14L30 , 55N91

Rights: Copyright © 2010 Duke University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.153 • No. 3 • 15 June 2010
Back to Top