Skip to main content
SpringerLink
Log in

Asymptotic Curvature of Moduli Spaces for Calabi–Yau Threefolds

  • Published:
Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

Motivated by the classical statements of Mirror Symmetry, we study certain Kähler metrics on the complexified Kähler cone of a Calabi–Yau threefold, conjecturally corresponding to approximations to the Weil–Petersson metric near large complex structure limit for the mirror. In particular, the naturally defined Riemannian metric (defined via cup-product) on a level set of the Kähler cone is seen to be analogous to a slice of the Weil–Petersson metric near large complex structure limit. This enables us to give counterexamples to a conjecture of Ooguri and Vafa that the Weil–Petersson metric has non-positive scalar curvature in some neighborhood of the large complex structure limit point.

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. Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. J. Algebraic Geom. 3, 493–535 (1994)

    MathSciNet  MATH  Google Scholar 

  2. Batyrev, V.V., Materov, E.N.: Toric residues and mirror symmetry. Mosc. Math. J. 2, 435–475 (2002)

    MathSciNet  MATH  Google Scholar 

  3. Bertin, J., Peters, C.: Variations de structures de Hodge, variétés de Calabi–Yau et symétrie miroir. In: Bertin, J., Demailly, J.-P., Illusie, L., Peters, C. (eds.) Introduction à la Théorie de Hodge, pp. 169–256. Société Mathématique de France, Paris (1996)

    Google Scholar 

  4. Bryant, R., Griffiths, P.: Some observations on the infinitesimal period relations for regular threefolds with trivial canonical bundle. In: Artin, M., Tate, J. (eds.) Arithmetic and Geometry, vol. II. Progress in Math., vol. 36, pp. 77–102. Birkhäuser, Boston (1983)

    Google Scholar 

  5. Candelas, P., De la Ossa, X.C., Green, P.S., Parkes, L.: An exactly soluble superconformal theory from a mirror pair of Calabi–Yau manifolds. Phys. Lett. B 258, 118–126 (1991)

    Article  MathSciNet  Google Scholar 

  6. Cox, D., Katz, S.: Mirror Symmetry and Algebraic Geometry. Mathematical Surveys and Monographs, vol. 68. AMS, Providence (1999)

    MATH  Google Scholar 

  7. Freed, D.S.: Special Kähler manifolds. Commun. Math. Phys. 203, 31–52 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Givental, A.: A mirror theorem for toric complete intersections. In: Kashiwara, M., et al. (eds.) Topological Field Theory, Primitive Forms and Related Topics. Proceedings of the 38th Taniguchi Symposium, Kyoto, Japan, December 9–13, 1996. Prog. Math., vol. 160, pp. 141–175. Birkhäuser, Boston (1998)

    Google Scholar 

  9. Gross, M., Siebert, B.: Mirror symmetry via logarithmic degeneration data II. arXiv:0709.2290

  10. Hosono, S., Klemm, A., Theisen, S., Yau, S.-T.: Mirror symmetry, mirror map and applications to Calabi–Yau hypersurfaces. Commun. Math. Phys. 167, 301–350 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hosono, S., Klemm, A., Theisen, S., Yau, S.-T.: Mirror symmetry, mirror map and applications to complete intersection Calabi–Yau spaces. Nucl. Phys. B 433, 501–552 (1995)

    Article  MathSciNet  Google Scholar 

  12. Hosono, S., Lian, B.H., Yau, S.T.: GKZ–generalized hypergeometric systems in mirror symmetry of Calabi–Yau hypersurfaces. Commun. Math. Phys. 182, 535–577 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Karu, K.: Toric residue mirror conjecture for Calabi–Yau complete intersections. J. Algebraic Geom. 14, 741–760 (2005)

    MathSciNet  MATH  Google Scholar 

  14. Kobayashi, S.: Transformation Groups in Differential Geometry. Springer, Berlin (1985)

    Google Scholar 

  15. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. II. Wiley, New York (1969)

    MATH  Google Scholar 

  16. Loftin, J.C.: Affine spheres and Kähler-Einstein metrics. Math. Res. Lett. 9, 425–432 (2002)

    MathSciNet  MATH  Google Scholar 

  17. Lu, Z., Sun, X.: Weil–Petersson geometry on the moduli space of polarized Calabi–Yau manifolds. J. Inst. Math. Jussieu 3, 185–229 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Morrison, D., Vafa, C.: Compactifications of F-theory on Calabi–Yau threefolds—I. Nucl. Phys. B 473, 74–92 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nakayama, N.: On Weierstrass models. In: Algebraic Geometry and Commutative Algebra, vol. II, pp. 405–431. Kinokuniya, Tokyo (1988)

    Google Scholar 

  20. Ooguri, H., Vafa, C.: On the geometry of the string landscape and the swampland. Nucl. Phys. B 766, 21–33 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Schumacher, G.: The curvature of the Petersson–Weil metric on the moduli space of Kähler–Einstein manifolds. In: Ancona, V., Silva, A. (eds.) Complex Analysis and Geometry, pp. 339–354. Plenum, New York (1993)

    Google Scholar 

  22. Strominger, A.: Special geometry. Commun. Math. Phys. 133, 163–180 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tian, G.: Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Petersson–Weil metric. In: Yau, S.-T. (ed.) Mathematical Aspects of String Theory, vol. 1, pp. 629–646. World Scientific, Singapore (1987)

    Google Scholar 

  24. Todorov, A.N.: The Weil–Petersson geometry of the moduli space of SU(n≤3) (Calabi-Yau) manifolds. I. Commun. Math. Phys. 126, 325–346 (1989)

    Article  MATH  Google Scholar 

  25. Totaro, B.: The curvature of a Hessian metric. Int. J. Math. 15, 369–391 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang, C.-L.: Curvature properties of the Calabi–Yau moduli. Doc. Math. 8, 577–590 (2003)

    MathSciNet  MATH  Google Scholar 

  27. Wilson, P.M.H.: The Kähler cone on Calabi–Yau threefolds. Invent. Math. 107, 561–583 (1992). Erratum: Invent. Math. 114, 231–232 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wilson, P.M.H.: Sectional curvatures of Kähler moduli. Math. Ann. 330, 631–664 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wilson, P.M.H.: Some geometry and combinatorics for the S-invariant of ternary cubics. Exp. Math. 15, 479–490 (2006)

    MATH  Google Scholar 

  30. Zheng, F.: Complex Differential Geometry. Studies in Advanced Mathematics, vol. 18. AMS/International Press, Providence/Somerville (2000)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Pure Mathematics, University of Cambridge, 16 Wilberforce Road, Cambridge, CB3 0WB, UK

    Thomas Trenner & P. M. H. Wilson

Authors
  1. Thomas Trenner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. M. H. Wilson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. M. H. Wilson.

Additional information

Communicated by P. Gilkey.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Trenner, T., Wilson, P.M.H. Asymptotic Curvature of Moduli Spaces for Calabi–Yau Threefolds. J Geom Anal 21, 409–428 (2011). https://doi.org/10.1007/s12220-010-9152-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-010-9152-1

Keywords

Mathematics Subject Classification (2000)

Search

Navigation