Skip to main content
SpringerLink
Log in

On the Crepant Resolution Conjecture in the Local Case

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

In this paper we analyze four examples of birational transformations between local Calabi–Yau 3-folds: two crepant resolutions, a crepant partial resolution, and a flop. We study the effect of these transformations on genus-zero Gromov–Witten invariants, proving the Coates–Iritani–Tseng/Ruan form of the Crepant Resolution Conjecture in each case. Our results suggest that this form of the Crepant Resolution Conjecture may also hold for more general crepant birational transformations. They also suggest that Ruan’s original Crepant Resolution Conjecture should be modified, by including appropriate “quantum corrections”, and that there is no straightforward generalization of either Ruan’s original Conjecture or the Cohomological Crepant Resolution Conjecture to the case of crepant partial resolutions. Our methods are based on mirror symmetry for toric orbifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramovich, D., Graber, T., Vistoli, A.: Algebraic orbifold quantum products. In: Orbifolds in mathematics and physics (Madison, WI, 2001), pp. 1–24. Contemp. Math. 310. Amer. Math. Soc., Providence, RI, 2002

  2. Abramovich D., Graber T., Vistoli A.: Gromov–Witten theory of Deligne–Mumford stacks. Amer. J. Math. 130, 1337–1398 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Aganagic M., Bouchard V., Klemm A.: Topological strings and (almost) modular forms. Comm. Math. Phys. 277, 771–819 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Audin, M.: Torus actions on symplectic manifolds. Progress in Mathematics 93. Basel: Birkhäuser Verlag (2004)

  5. Barannikov S.: Quantum periods. I. Semi-infinite variations of Hodge structures. Internat. Math. Res. Notices 23, 1243–1264 (2001)

    Article  MathSciNet  Google Scholar 

  6. Bertram A., Kley H.P.: New recursions for genus-zero Gromov-Witten invariants. Topology 44, 1–24 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Boissiere, S., Mann, E., Perroni, F.: Crepant resolutions of weighted projective spaces and quantum deformations. http://arxiv.org/abs/math/0610617v2 (2006)

  8. Boissiere, S., Mann, E., Perroni, F.: The cohomological crepant resolution conjecture for \({\mathbb{P}}\) (1, 3, 4, 4). http://arxiv.org/abs/0712.3248v1 (2007)

  9. Borisov L.A., Horja R.P.: Mellin–Barnes integrals as Fourier–Mukai transforms. Adv. Math. 207(2), 876–927 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bryan, J., Gholampour, A.: Hurwitz-Hodge integrals, the E6 and D4 root systems, and the crepant resolution conjecture. http://arxiv.org/abs/0708.4244v1 (2007)

  11. Bryan, J., Graber, T.: The crepant resolution conjecture. http://arxiv.org/abs/math/0610129v2 (2006)

  12. Bryan J., Graber T., Pandharipande R.: The orbifold quantum cohomology of \({\mathbb{C}^2/\mathbb{Z}_3}\) and Hurwitz–Hodge integrals. J. Algebraic Geom. 17, 1–28 (2008)

    MATH  MathSciNet  Google Scholar 

  13. Candelas P., de la Ossa X.C., Parkes L.: A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. Nuclear Phys. B 359, 21–74 (1991)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Cadman, C., Cavalieri, R.: Gerby localization, \({\mathbb{Z}_3}\) -hodge integrals and the GW theory of \({\mathbb{C}^{3}/\mathbb{Z}_{3}}\). http://arxiv.org/abs/0705.2158v3 (2007)

  15. Chen, W., Ruan, Y.: Orbifold Gromov–Witten theory. In: Orbifolds in mathematics and physics (Madison, WI, 2001) pp. 25–85. Contemp. Math. 310. Amer. Math. Soc., Providence, RI, 2002

  16. Chen W., Ruan Y.: A new cohomology theory of orbifold. Comm. Math. Phys. 248, 1–31 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. Chiang T.-M., Klemm A., Yau S.-T., Zaslow E.: Local mirror symmetry: calculations and interpretations. Adv. Theor. Math. Phys. 3, 495–565 (1999)

    MATH  MathSciNet  Google Scholar 

  18. Coates, T., Iritani, H., Tseng, H.-H.: Wall-crossings in toric Gromov–Witten theory I: crepant examples. http://arxiv.org/abs/math/0611550v4 (2006)

  19. Coates, T., Corti, A., Iritani, H., Tseng, H.-H.: Computing genus-zero twisted Gromov–Witten invariants. http://arxiv.org/abs/math/0702234v3 (2007)

  20. Coates, T., Corti, A., Iritani, H., Tseng, H.-H.: The crepant resolution conjecture for type A surface singularities. http://arxiv.org/abs/0704.2034v3 (2007)

  21. Coates, T., Corti, A., Iritani, H., Tseng, H.-H.: The small quantum orbifold cohomology of Fano toric Deligne–Mumford stacks, in preparation

  22. Coates, T., Corti, A., Lee, Y.-P., Tseng, H.-H.: The quantum orbifold cohomology of weighted projective spaces. http://arxiv.org/abs/math/0608481v6 (2006)

  23. Coates T., Givental A.: Quantum Riemann–Roch, Lefschetz and Serre. Ann. Math. (2) 165, 15–53 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Coates, T., Ruan, Y.: Quantum cohomology and crepant resolutions: a conjecture. http://arxiv.org/abs/0710.5901v3 (2007)

  25. Corti A.: What is... a flip?. Notices Amer. Math. Soc. 51, 1350–1351 (2004)

    MATH  MathSciNet  Google Scholar 

  26. Dubrovin, B.: Geometry of 2D topological field theories. In: Integrable systems and quantum groups (Montecatini Terme, 1993), pp. 120–348. Lecture Notes in Math. 1620. Springer, Berlin (1996)

  27. Elezi A.: Mirror symmetry for concavex vector bundles on projective spaces. Int. J. Math. Math. Sci. 2003(3), 159–197 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. Gillam, W.D.: The crepant resolution conjecture for 3-dimensional flags modulo an involution. http://arxiv.org/abs/0708.0842v1 (2007)

  29. Givental A.B.: Equivariant Gromov-Witten invariants. Internat. Math. Res. Notices 13, 613–663 (1996)

    Article  MathSciNet  Google Scholar 

  30. Givental, A.B.: A mirror theorem for toric complete intersections. In: Topological field theory, primitive forms and related topics (Kyoto, 1996), pp. 141–175. Progr. Math. 160. Birkhäuser Boston, Boston, MA (1998)

  31. Givental, A.B.: Elliptic Gromov-Witten invariants and the generalized mirror conjecture. In: Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), pp. 107–155. World Sci. Publ., River Edge, NJ (1998)

  32. Givental A.B.: Gromov-Witten invariants and quantization of quadratic Hamiltonians. Mosc. Math. J. 1, 551–568 (2001) 645

    MATH  MathSciNet  Google Scholar 

  33. Givental, A.B.: Symplectic geometry of Frobenius structures. In: Frobenius manifolds, pp. 91–112. Aspects Math., E36. Vieweg, Wiesbaden (2004)

  34. Graber T., Pandharipande R.: Localization of virtual classes. Invent. Math. 135, 487–518 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  35. Hori, K., Vafa, C.: Mirror symmetry. http://arxiv.org/abs/hep-th/0002222v3 (2000)

  36. Iritani H.: Quantum D-modules and generalized mirror transformations. Topology 47, 225–276 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Iritani, H.: Real and integral structures in quantum cohomology I: toric orbifolds. http://arxiv.org/abs/0712.2204v2 (2007)

  38. Iritani, H.: Wall-crossings in toric Gromov–Witten theory III, work in progress

  39. Kontsevich M., Manin Yu.: Gromov-Witten classes, quantum cohomology, and enumerative geometry. Comm. Math. Phys. 164, 525–562 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  40. Lee Y.-P., Pandharipande R.: A reconstruction theorem in quantum cohomology and quantum K-theory. Amer. J. Math. 126, 1367–1379 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  41. Lian B.H., Liu K., Yau S.-T.: Mirror principle. I. Asian J. Math. 1, 729–763 (1997)

    MATH  MathSciNet  Google Scholar 

  42. Perroni F.: Chen-Ruan cohomology of ADE singularities. Internat. J. Math. 18, 1009–1059 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  43. Reid, M.: Young person’s guide to canonical singularities. In: Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), pp. 345–414. Proc. Sympos. Pure Math. 46. Amer. Math. Soc., Providence, RI, 1987

  44. Rose M.A.: A reconstruction theorem for genus zero Gromov–Witten invariants of stacks. Amer. J. Math. 130, 1427–1443 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  45. Ruan, Y.: The cohomology ring of crepant resolutions of orbifolds. In: Gromov-Witten theory of spin curves and orbifolds, pp. 117–126. Contemp. Math. 403. Amer. Math. Soc., Providence, RI, 2006

  46. Wise, J.: The genus zero Gromov-Witten invariants of \({[Sym^{2} \mathbb{P}^{2}]}\). http://arxiv.org/abs/math/0702219v3 (2007)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Imperial College London, 180 Queen’s Gate, London, SW7 2AZ, UK

    Tom Coates

Authors
  1. Tom Coates
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tom Coates.

Additional information

Communicated by N. A. Nekrasov

Rights and permissions

Reprints and permissions

About this article

Cite this article

Coates, T. On the Crepant Resolution Conjecture in the Local Case. Commun. Math. Phys. 287, 1071–1108 (2009). https://doi.org/10.1007/s00220-008-0715-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-008-0715-y

Keywords

Search

Navigation