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Quantum cohomology of [ℂN/μr]

  • Arend Bayer (a1) and Charles Cadman (a2)

Abstract

We give a construction of the moduli space of stable maps to the classifying stack r of a cyclic group by a sequence of rth root constructions on . We prove a closed formula for the total Chern class of μr-eigenspaces of the Hodge bundle, and thus of the obstruction bundle of the genus-zero Gromov–Witten theory of stacks of the form [ℂN/μr]. We deduce linear recursions for genus-zero Gromov–Witten invariants.

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References

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[1]Abramovich, D., Lectures on Gromov–Witten invariants of orbifolds, in Enumerative invariants in algebraic geometry and string theory, Lecture Notes in Mathematics, vol. 1947 (Springer, Berlin, 2008), 148, arXiv:math/0512372.
[2]Abramovich, D., Corti, A. and Vistoli, A., Twisted bundles and admissible covers, Commun. Algebra 31 (2003), 35473618, arXiv:math/0106211. Special issue in honor of Steven L. Kleiman.
[3]Abramovich, D. and Vistoli, A., Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (2002), 2775, arXiv:math/9908167 (electronic).
[4]Aganagic, M., Bouchard, V. and Klemm, A., Topological strings and (almost) modular forms, Commun. Math. Phys. 277 (2008), 771819, arXiv:hep-th/0607100.
[5]Alexeev, V. and Guy, G. M., Moduli of weighted stable maps and their gravitational descendants, J. Inst. Math. Jussieu 7 (2008), 425456, arXiv:math/0607683.
[6]Bayer, A. and Manin, Yu. I., Stability conditions, wall-crossing and weighted Gromov–Witten invariants, Mosc. Math. J. 9 (2009), 332, arXiv:math/0607580.
[7]Berthelot, P., Grothendieck, A., Illusie, L.et al., Théorie des intersections et théorème de Riemann–Roch, séminaire de géométrie algébrique du Bois-Marie 1966–1967 (SGA 6), Lecture Notes in Mathematics, vol. 225 (Springer, Berlin, 1971).
[8]Bryan, J. and Graber, T., The crepant resolution conjecture, in Algebraic geometry – Seattle 2005. Part 1, Proceedings of Symposia in Pure Mathematics, vol. 80 (American Mathematical Society, Providence, RI, 2009), 2342, arXiv:math/0610129.
[9]Bryan, J., Graber, T. and Pandharipande, R., The orbifold quantum cohomology of and Hurwitz–Hodge integrals, J. Algebraic Geom. 17 (2008), 128, arXiv:math/0510335.
[10]Cadman, C., Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), 405427.
[11]Cadman, C. and Cavalieri, R., Gerby localization, -Hodge integrals and the GW theory of , Amer. J. Math. 131 (2009), 10091046, arXiv:0705.2158.
[12]Cadman, C. and Chen, L., Enumeration of rational plane curves tangent to a smooth cubic, Adv. Math. 219 (2008), 316343, arXiv:math/0701406.
[13]Coates, T., Corti, A., Iritani, H. and Tseng, H.-H., The crepant resolution conjecture for type A surface singularities (2007), arXiv:0704.2034.
[14]Coates, T., Corti, A., Iritani, H. and Tseng, H.-H., Computing genus-zero twisted Gromov–Witten invariants, Duke Math. J. 147 (2009), 377438, arXiv:math/0702234.
[15]Coates, T. and Givental, A., Quantum Riemann–Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), 1553, arXiv:math/0110142.
[16]Coates, T., Iritani, H. and Tseng, H.-H., Wall-crossings in toric Gromov–Witten theory. I. Crepant examples, Geom. Topol. 13 (2009), 26752744, arXiv:math/0611550.
[17]de Concini, C. and Procesi, C., Hyperplane arrangements and holonomy equations, Selecta Math. (N.S.) 1 (1995), 495535.
[18]Faber, C. and Pandharipande, R., Logarithmic series and Hodge integrals in the tautological ring, Michigan Math. J. 48 (2000), 215252, (with an appendix by Don Zagier, dedicated to William Fulton on the occasion of his 60th birthday), arXiv:math/0002112.
[19]Givental, A. B., Gromov–Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J. 1 (2001), 551–568, 645; dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary, arXiv:math/0108100.
[20]Givental, A. B., Symplectic geometry of Frobenius structures, in Frobenius manifolds, Aspects of Mathematics, vol. E36 (Vieweg, Wiesbaden, 2004), 91112, arXiv:math/0305409.
[21]Hassett, B., Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), 316352, arXiv:math/0205009.
[22]Jarvis, T. J. and Kimura, T., Orbifold quantum cohomology of the classifying space of a finite group, in Orbifolds in mathematics and physics (Madison, WI, 2001), Contemporary Mathematics, vol. 310 (American Mathematical Society, Providence, RI, 2002), 123134, arXiv:math/0112037.
[23]Keel, S., Intersection theory of moduli space of stable n-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), 545574.
[24]Laumon, G. and Moret-Bailly, L., Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 39 [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] (Springer, Berlin, 2000).
[25]Mustaţǎ, A. M. and Mustaţǎ, A., The Chow ring of , J. Reine Angew. Math. 615 (2008), 93119, arXiv:math/0507464.
[26]Ruan, Y., Stringy geometry and topology of orbifolds, in Symposium in honor of C. H. Clemens (Salt Lake City, UT, 2000), Contemporary Mathematics, vol. 312 (American Mathematical Society, Providence, RI, 2002), 187233.
[27]Ruan, Y., The cohomology ring of crepant resolutions of orbifolds, in Gromov–Witten theory of spin curves and orbifolds, Contemporary Mathematics, vol. 403 (American Mathematical Society, Providence, RI, 2006), 117126, arXiv:math/0108195.
[28]Tseng, H.-H., Orbifold quantum Riemann–Roch, Lefschetz and Serre, Geom. Topol. 14 (2010), 181, arXiv:math/0506111.
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Quantum cohomology of [ℂN/μr]

  • Arend Bayer (a1) and Charles Cadman (a2)

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