Skip to main content
SpringerLink
Log in

Homological Mirror Symmetry for Toric del Pezzo Surfaces

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

Abstract

We prove the homological mirror conjecture for toric del Pezzo surfaces. In this case, the mirror object is a regular function on an algebraic torus We show that the derived Fukaya category of this mirror coincides with the derived category of coherent sheaves on the original manifold.

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. Auroux, D., Katzarkov, L., Orlov, D.: Mirror symmetry for weighted projective planes and their noncommutative deformations. http://arxiv.org/list/math.AG/0404281, 2004.

  2. Auroux, D., Katzarkov, L., Orlov, D.: Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves. http://arxiv.org/list/math.AG/0506166, 2005

  3. Batyrev, V.: Quantum cohomology rings of toric manifolds. Astérisque 218, 9, 34 (1993)

    MATH  MathSciNet  Google Scholar 

  4. Beilinson, A.: Coherent sheaves on and problems in linear algebra. Funkts. Anal. i. Prilozhen. 12, no. 3, 68–69 (1978)

  5. Bondal, A.: Representation of associative algebras and coherent sheaves. Izv. Ross. Akad. Nauk Ser. Mat. 53, no. 1, 25–44 (1989)

  6. Eguchi, T., Hori, K., Xiong, C.-S.: Gravitational quantum cohomology. Int. J. Mod. Phys. A 12, no. 9, 1743–1782 (1997)

    Google Scholar 

  7. Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory. Preprint, available at http://www.math.kyoto-u.ac.jp/~fukaya, 2000

  8. Givental, A.: Homological geometry and mirror symmetry. In: Proceedings of the International Congress of Mathematicians. Proceedings, Zurich 1994. Basel: Birkhäuser, 1994, pp. 472-480

  9. Hori, K., Iqbal, A., Vafa, C.: D-Branes and mirror symmetry. http://arxiv.org/list/hep-th/0005247, 2000

  10. Hori, K., Vafa, C.: Mirror symmetry. http:// arxiv.org/list/hep-th/0002222, 2000

  11. Kontsevich, M.: Homological algebra of mirror symmetry. In: Proceedings of the International Congress of Mathematicians. Proceedings, Zurich 1994. Basel: Birkhäuser, 1994, pp. 120–139

  12. Kouchnirenko, A.: Polyédres de Newton et nombres de Milnor. Invent. Math. 32, no. 1, 1–31 (1976)

  13. Orlov, D.: Projective bundles, monoidal transformations and derived categories of coherent sheaves. Izv. Ross. Akad. Nauk Ser. Math. 56, no. 4, 852–862 (1992)

  14. Robbin, J., Salamon D.: Maslov index for path. Topology 32, no. 4, 827–844 (1993)

  15. Seidel, P.: Graded Lagrangian submanifolds. Bull. Soc. Math. France 128, no. 1, 103–149 (2000)

  16. Seidel, P.: Vanishing cycles and mutations. In: European Congress of Mathematics, Vol. II. Proceedings, Barcelona 2000. Basel: Birkhäuser, 2001, pp. 65–85

  17. Seidel, P.: More about vanishing cycles and mutations. In: Symplectic geometry and mirror symmetry. Proceedings, Seoul 2000. Singapore:World Sci. Publishing, 2001, pp. 429–465

  18. Seidel, P.: Homological mirror symmetry for the quartic surface. http://arxiv.org/list/math.AG/0310414 (2003)

  19. Witten, E.: Phases of N=2 theories in two dimensions. Nucl. Phys. B 403, 159–222 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Kazushi Ueda

Authors
  1. Kazushi Ueda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kazushi Ueda.

Additional information

Communicated by N.A. Nekrasov

Supported by JSPS Fellowships for Young Scientists No.15-5561.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ueda, K. Homological Mirror Symmetry for Toric del Pezzo Surfaces. Commun. Math. Phys. 264, 71–85 (2006). https://doi.org/10.1007/s00220-005-1509-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-005-1509-0

Keywords

Search

Navigation