Illinois Journal of Mathematics

The orbifold cohomology ring of simplicial toric stack bundles

Yunfeng Jiang

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We introduce a new quotient construction of toric Deligne–Mumford stacks. We use this new construction to define toric stack bundles which generalize the construction of toric bundles by Sankaran and Uma [Comment. Math. Helv. 78 (2003) 540–554]. The orbifold Chow ring of such toric stack bundles is computed. We show that the orbifold Chow ring of the toric stack bundle and the Chow ring of its crepant resolution are fibres of a flat family, generalizing a result of Borisov–Chen–Smith.

Article information

Illinois J. Math. Volume 52, Number 2 (2008), 493-514.

First available in Project Euclid: 23 July 2009

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Zentralblatt MATH identifier

Primary: 14A20: Generalizations (algebraic spaces, stacks) 14F40: de Rham cohomology [See also 14C30, 32C35, 32L10]


Jiang, Yunfeng. The orbifold cohomology ring of simplicial toric stack bundles. Illinois J. Math. 52 (2008), no. 2, 493--514.

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  • D. Abramovich, T. Graber and A. Vistoli, Algebraic orbifold quantum product, Orbifolds in mathematics and physics (Madison, WI, 2001), Contemp. Math., vol. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 1--24.
  • E. Andreini, Y. Jiang and H. Tseng, The Kunneth type formula for orbifold Gromov--Witten invariants, preprint.
  • L. Borisov, L. Chen and G. Smith, The orbifold Chow ring of toric Deligne--Mumford stacks, J. Amer. Math. Soc. 18 (2005), 193--215, math.AG/0309229.
  • W. Chen and Y. Ruan, A new cohomology theory for orbifolds, Comm. Math. Phys. 248 (2004), 1--31, math.AG/0004129.
  • W. Chen and Y. Ruan, Orbifold Gromov--Witten theory, Orbifolds in mathematics and physics (Madison, WI, 2001), Contemp. Math., vol. 310, Amer. Math. Soc. Providence, RI, 2002, pp. 25--85.
  • D. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), 17--50.
  • R. Donagi and T. Pantev, Torus fibrations, gerbes and duality, math.AG /0306213.
  • D. Edidin, Notes on the construction of the moduli space of curves, Recent progress in intersection theory (Bologna, 1997), Birkhauser, Boston, 2000, pp. 85--113.
  • W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993.
  • Y. Jiang, The Chen--Ruan cohomology of weighted projective spaces, Canad. J. Math. 59 (2007), 981--1007.
  • Y. Jiang, A note on finite Abelian gerbes over toric Deligne--Mumford stacks, Proc. Amer. Math. Soc. 136 (2008), 4151--4156.
  • S. Lang, Algebra, Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002.
  • S. Keel and S. Mori, Quotients by groupoids, Ann. of Math. (2) 145 (1997), 193--213.
  • T. Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 15, Springer-Verlag, Berlin, 1988.
  • F. Perroni, Orbifold cohomology of ADE-singularities, Internat. J. Math. 18 (2007), 1009--1059.
  • M. Poddar, Orbifold Hodge numbers of Calabi--Yau hypersurfaces, Pacific J. Math. 208 (2003), 151--167.
  • Y. Ruan, Cohomology ring of crepant resolutions of orbifolds, Gromov--Witten theory of spin curves and orbifolds, Contemp. Math., vol. 403, Amer. Math. Soc., Providence, RI, 2006, pp. 117--126.
  • P. Sankaran and V. Uma, Cohomology of toric bundles, Comment. Math. Helv. 78 (2003), 540--554.%
  • A. Vistoli., Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), 613--670.