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The orbifold Chow ring of toric Deligne-Mumford stacks. (English) Zbl 1178.14057
Summary: Generalizing toric varieties, we introduce toric Deligne-Mumford stacks. The main result in this paper is an explicit calculation of the orbifold Chow ring of a toric Deligne-Mumford stack. As an application, we prove that the orbifold Chow ring of the toric Deligne-Mumford stack associated to a simplicial toric variety is a flat deformation of (but is not necessarily isomorphic to) the Chow ring of a crepant resolution.

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14C15 (Equivariant) Chow groups and rings; motives
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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References:
[1] Dan Abramovich, Alessio Corti, and Angelo Vistoli, Twisted bundles and admissible covers, Comm. Algebra 31 (2003), no. 8, 3547 – 3618. Special issue in honor of Steven L. Kleiman. · Zbl 1077.14034
[2] Dan Abramovich, Tom Graber, and Angelo Vistoli, Algebraic orbifold quantum products, Orbifolds in mathematics and physics (Madison, WI, 2001) Contemp. Math., vol. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 1 – 24. · Zbl 1067.14055
[3] Victor V. Batyrev, Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 1, 5 – 33. · Zbl 0943.14004
[4] Lev A. Borisov, String cohomology of a toroidal singularity, J. Algebraic Geom. 9 (2000), no. 2, 289 – 300. · Zbl 0949.14029
[5] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. · Zbl 0788.13005
[6] David A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), no. 1, 17 – 50. · Zbl 0846.14032
[7] W. Chen and Y. Ruan, A New Cohomology Theory for Orbifold, arXiv:math.AG/0004129.
[8] Weimin Chen and Yongbin 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. · Zbl 1091.53058
[9] P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 143 – 316. Lecture Notes in Math., Vol. 349 (French).
[10] Dan Edidin, Notes on the construction of the moduli space of curves, Recent progress in intersection theory (Bologna, 1997) Trends Math., Birkhäuser Boston, Boston, MA, 2000, pp. 85 – 113. · Zbl 0990.14008
[11] David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. · Zbl 0819.13001
[12] David Eisenbud and Joe Harris, The geometry of schemes, Graduate Texts in Mathematics, vol. 197, Springer-Verlag, New York, 2000. · Zbl 0960.14002
[13] David Eisenbud and Bernd Sturmfels, Binomial ideals, Duke Math. J. 84 (1996), no. 1, 1 – 45. · Zbl 0873.13021
[14] Barbara Fantechi and Lothar Göttsche, Orbifold cohomology for global quotients, Duke Math. J. 117 (2003), no. 2, 197 – 227. · Zbl 1086.14046
[15] William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. · Zbl 0813.14039
[16] Y. Jiang, The Chen-Ruan cohomology of weighted projective spaces, arXiv:math.AG/0304140. · Zbl 1235.14052
[17] János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. · Zbl 0877.14012
[19] Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39, Springer-Verlag, Berlin, 2000 (French). · Zbl 0945.14005
[20] Ieke Moerdijk, Orbifolds as groupoids: an introduction, Orbifolds in mathematics and physics (Madison, WI, 2001) Contemp. Math., vol. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 205 – 222. · Zbl 1041.58009
[21] Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. · Zbl 0628.52002
[22] Mainak Poddar, Orbifold cohomology group of toric varieties, Orbifolds in mathematics and physics (Madison, WI, 2001) Contemp. Math., vol. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 223 – 231. · Zbl 1027.14023
[23] Y. Ruan,Cohomology Ring of Crepant Resolutions of Orbifolds, arXiv:math.AG/0108195. · Zbl 1105.14078
[24] Richard Stanley, Generalized \?-vectors, intersection cohomology of toric varieties, and related results, Commutative algebra and combinatorics (Kyoto, 1985) Adv. Stud. Pure Math., vol. 11, North-Holland, Amsterdam, 1987, pp. 187 – 213. · Zbl 0652.52007
[25] B. Uribe, Orbifold cohomology of the symmetric product, Comm. Anal. Geom. (to appear), arXiv:math.AT/0109125. · Zbl 1087.32012
[26] Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613 – 670. · Zbl 0694.14001
[27] Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. · Zbl 0797.18001
[28] Takehiko Yasuda, Twisted jets, motivic measures and orbifold cohomology, Compos. Math. 140 (2004), no. 2, 396 – 422. · Zbl 1092.14028
[29] Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. · Zbl 0823.52002
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