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

Abstract

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

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Dan Edidin. Tyler J. Jarvis. Takashi Kimura. "Logarithmic trace and orbifold products." Duke Math. J. 153 (3) 427 - 473, 15 June 2010. https://doi.org/10.1215/00127094-2010-028

Information

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

Subjects:
Primary: 14N35
Secondary: 14L30, 55N91

Rights: Copyright © 2010 Duke University Press

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Vol.153 • No. 3 • 15 June 2010
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