Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-26T15:30:54.663Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantum cohomology of [ℂN/μr]

Part of: Projective and enumerative geometry Families, fibrations Curves

Published online by Cambridge University Press:  22 June 2010

Arend Bayer  and
Charles Cadman
Arend Bayer
Affiliation:
Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USA
Charles Cadman
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Rd, Vancouver, BC, Canada V6P 4M8
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give a construction of the moduli space of stable maps to the classifying stack Bμr 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.

Keywords

stable mapsroot constructionGromov–Witten invariantsstacksquantum orbifold cohomology

MSC classification

Secondary: 14D23: Stacks and moduli problems 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 14H10: Families, moduli (algebraic)
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 5 , September 2010 , pp. 1291 - 1322
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[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), 1–48, arXiv:math/0512372.CrossRefGoogle Scholar
[2]Abramovich, D., Corti, A. and Vistoli, A., Twisted bundles and admissible covers, Commun. Algebra 31 (2003), 3547–3618, arXiv:math/0106211. Special issue in honor of Steven L. Kleiman.CrossRefGoogle Scholar
[3]Abramovich, D. and Vistoli, A., Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (2002), 27–75, arXiv:math/9908167 (electronic).CrossRefGoogle Scholar
[4]Aganagic, M., Bouchard, V. and Klemm, A., Topological strings and (almost) modular forms, Commun. Math. Phys. 277 (2008), 771–819, arXiv:hep-th/0607100.CrossRefGoogle Scholar
[5]Alexeev, V. and Guy, G. M., Moduli of weighted stable maps and their gravitational descendants, J. Inst. Math. Jussieu 7 (2008), 425–456, arXiv:math/0607683.CrossRefGoogle Scholar
[6]Bayer, A. and Manin, Yu. I., Stability conditions, wall-crossing and weighted Gromov–Witten invariants, Mosc. Math. J. 9 (2009), 3–32, arXiv:math/0607580.CrossRefGoogle Scholar
[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).Google Scholar
[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), 23–42, arXiv:math/0610129.Google Scholar
[9]Bryan, J., Graber, T. and Pandharipande, R., The orbifold quantum cohomology of and Hurwitz–Hodge integrals, J. Algebraic Geom. 17 (2008), 1–28, arXiv:math/0510335.CrossRefGoogle Scholar
[10]Cadman, C., Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), 405–427.CrossRefGoogle Scholar
[11]Cadman, C. and Cavalieri, R., Gerby localization, -Hodge integrals and the GW theory of , Amer. J. Math. 131 (2009), 1009–1046, arXiv:0705.2158.CrossRefGoogle Scholar
[12]Cadman, C. and Chen, L., Enumeration of rational plane curves tangent to a smooth cubic, Adv. Math. 219 (2008), 316–343, arXiv:math/0701406.CrossRefGoogle Scholar
[13]Coates, T., Corti, A., Iritani, H. and Tseng, H.-H., The crepant resolution conjecture for type A surface singularities (2007), arXiv:0704.2034.Google Scholar
[14]Coates, T., Corti, A., Iritani, H. and Tseng, H.-H., Computing genus-zero twisted Gromov–Witten invariants, Duke Math. J. 147 (2009), 377–438, arXiv:math/0702234.CrossRefGoogle Scholar
[15]Coates, T. and Givental, A., Quantum Riemann–Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), 15–53, arXiv:math/0110142.CrossRefGoogle Scholar
[16]Coates, T., Iritani, H. and Tseng, H.-H., Wall-crossings in toric Gromov–Witten theory. I. Crepant examples, Geom. Topol. 13 (2009), 2675–2744, arXiv:math/0611550.CrossRefGoogle Scholar
[17]de Concini, C. and Procesi, C., Hyperplane arrangements and holonomy equations, Selecta Math. (N.S.) 1 (1995), 495–535.CrossRefGoogle Scholar
[18]Faber, C. and Pandharipande, R., Logarithmic series and Hodge integrals in the tautological ring, Michigan Math. J. 48 (2000), 215–252, (with an appendix by Don Zagier, dedicated to William Fulton on the occasion of his 60th birthday), arXiv:math/0002112.CrossRefGoogle Scholar
[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.Google Scholar
[20]Givental, A. B., Symplectic geometry of Frobenius structures, in Frobenius manifolds, Aspects of Mathematics, vol. E36 (Vieweg, Wiesbaden, 2004), 91–112, arXiv:math/0305409.CrossRefGoogle Scholar
[21]Hassett, B., Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), 316–352, arXiv:math/0205009.CrossRefGoogle Scholar
[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), 123–134, arXiv:math/0112037.CrossRefGoogle Scholar
[23]Keel, S., Intersection theory of moduli space of stable n-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), 545–574.Google Scholar
[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).CrossRefGoogle Scholar
[25]Mustaţǎ, A. M. and Mustaţǎ, A., The Chow ring of , J. Reine Angew. Math. 615 (2008), 93–119, arXiv:math/0507464.Google Scholar
[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), 187–233.CrossRefGoogle Scholar
[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), 117–126, arXiv:math/0108195.CrossRefGoogle Scholar
[28]Tseng, H.-H., Orbifold quantum Riemann–Roch, Lefschetz and Serre, Geom. Topol. 14 (2010), 1–81, arXiv:math/0506111.CrossRefGoogle Scholar