Duke Mathematical Journal

Logarithmic trace and orbifold products

Dan Edidin,Tyler J. Jarvis, and Takashi Kimura

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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, $K_G(X) \to K_G(\mathbb{I}_{G}^{2}(X))$ taking nonnegative elements to nonnegative elements. (Here $\mathbb{I}_{G}^{2}(X) = \{(g_1,g_2,x)|g_1x = g_2 x = x \} \subset G \times G \times X$.) 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 ${\mathbb T} \in K_G(X)$, 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 $R({\mathbf m})$ 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 $K_G(\mathbb{I}_{G}^{2}(X))$.

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 ${\mathscr X} = [X/G]$, then there is an associative orbifold product structure on $K({\mathscr X})\otimes {\mathbb C}$ distinct from the usual tensor product

Article information

Source
Duke Math. J. Volume 153, Number 3 (2010), 427-473.

Dates
First available: 4 June 2010

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1275671394

Digital Object Identifier
doi:10.1215/00127094-2010-028

Zentralblatt MATH identifier
05757676

Mathematical Reviews number (MathSciNet)
MR2667422

Subjects
Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 55N91: Equivariant homology and cohomology [See also 19L47]

Citation

Edidin, Dan; Jarvis, Tyler J.; Kimura, Takashi. Logarithmic trace and orbifold products. Duke Mathematical Journal 153 (2010), no. 3, 427--473. doi:10.1215/00127094-2010-028. http://projecteuclid.org/euclid.dmj/1275671394.


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