Duke Mathematical Journal

Logarithmic trace and orbifold products

Dan Edidin, Tyler J. Jarvis, and Takashi Kimura

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

Article information

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

First available in Project Euclid: 4 June 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]


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

Export citation


  • D. Abramovich, T. Graber, and A. Vistoli, ``Algebraic orbifold quantum products'' in Orbifolds in Mathematics and Physics, Contemp. Math. 310, Amer. Math. Soc., Providence, 2002, 1--24.
  • —, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), 1337--1398.
  • A. Adem and Y. Ruan, Twisted orbifold $K$-theory, Comm. Math. Phys. 237 (2003), 533--556.
  • M. Atiyah and G. Segal, On equivariant Euler characteristics, J. Geom. Phys. 6 (1989), 671--677.
  • A. Borel, Linear Algebraic Groups, 2nd ed., Grad. Texts in Math. 126, Springer, New York, 1991.
  • L. A. Borisov, L. Chen, and G. G. Smith, The orbifold Chow ring of toric Deligne-Mumford stacks, J. Amer. Math. Soc. 18 (2005), 193--215.
  • W. Chen and Y. Ruan, A new cohomology theory of orbifold, Comm. Math. Phys. 248 (2004), 1--31.
  • R. Dijkgraaf, V. Pasquier, and P. Roche, ``Quasi Hopf algebras, group cohomology, and orbifold models'' in Recent Advances in Field Theory (Annecy-le-Vieux, France, 1990), Nuclear Phys. B Proc. Suppl. 18B, Elsevier, Amsterdam, 1990, 60--72.
  • D. Edidin and W. Graham, Equivariant intersection theory, Invent. Math. 131 (1998), 595--634.
  • —, Riemann-Roch for equivariant Chow groups, Duke Math. J. 102 (2000), 567--594.
  • —, Nonabelian localization in equivariant $K$-theory and Riemann-Roch for quotients, Adv. Math. 198 (2005), 547--582.
  • —, Algebraic cycles and completions of equivariant $K$-theory, Duke Math. J. 144 (2008), 489--524.
  • D. Edidin, B. Hassett, A. Kresch, and A. Vistoli, Brauer groups and quotient stacks, Amer. J. Math. 123 (2001), 761--777.
  • E. Falbel and R. A. Wentworth, Eigenvalues of products of unitary matrices and Lagrangian involutions, Topology 45 (2006), 65--99.
  • B. Fantechi and L. Göttsche, Orbifold cohomology for global quotients, Duke Math. J. 117 (2003), 197--227.
  • W. Fulton, Intersection Theory, Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1984.
  • R. Goldin, T. S. Holm, and A. Knutson, Orbifold cohomology of torus quotients, Duke Math. J. 139 (2007), 89--139.
  • T. J. Jarvis, R. Kaufmann, and T. Kimura, Stringy $K$-theory and the Chern character, Invent. Math. 168 (2007), 23--81.
  • R. Kaufmann and D. Pham, The Drinfel'd double and twisting in stringy orbifold theory, Internat. J. Math. 20 (2009), 623--657.
  • S. Keel and S. Mori, Quotients by groupoids, Ann. of Math. (2) 145 (1997), 193--213.
  • B. Köck, The Grothendieck-Riemann-Roch theorem for group scheme actions, Ann. Sci. École Norm. Sup. (4) 31 (1998), 415--458.
  • G. Lusztig, ``Leading coefficients of character values of Hecke algebras'' in The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Proc. Sympos. Pure Math. 47, Part 2, Amer. Math. Soc., Providence, 1987, 235--262.
  • G. Segal, Equivariant $K$-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129--151.
  • R. W. Thomason, Equivariant resolution, linearization, and Hilbert's fourteenth problem over arbitrary base schemes, Adv. in Math. 65 (1987), 16--34.
  • B. Totaro, The resolution property for schemes and stacks, J. Reine Angew. Math. 577 (2004), 1--22.
  • G. Vezzosi and A. Vistoli, Higher algebraic $K$-theory of group actions with finite stabilizers, Duke Math. J. 113 (2002), 1--55.