Skip to main content
Log in

Homological Mirror Symmetry for manifolds of general type

  • Research Article
  • Published:
Central European Journal of Mathematics

Abstract

In this paper we outline the foundations of Homological Mirror Symmetry for manifolds of general type. Both Physics and Categorical prospectives are considered.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Abouzaid M., On the Fukaya categories of higher genus surfaces, Adv. Math., 2008, 217(3), 1192–1235

    Article  MATH  MathSciNet  Google Scholar 

  2. Auroux D., Katzarkov L., Orlov D., Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves, Invent. Math., 2006, 166(3), 537–582

    Article  MATH  MathSciNet  Google Scholar 

  3. Auroux D., Katzarkov L., Orlov D., Mirror symmetry for weighted projective planes and their noncommutative deformations, preprint available at http://arxiv.org/abs/math/0404281

  4. Bondal A., Kapranov M., Framed triangulated categories, Mat. Sb., 1990, 181(5), 669–683 (in Russian), English translation: Math. USSR-Sb., 1991, 70(1), 93–107

    MATH  Google Scholar 

  5. Bondal A., Orlov D., Semiorthogonal decomposition for algebraic varieties, preprint available at http://arxiv.org/abs/alg-geom/9506012

  6. Bridgeland T., King A., Reid M., The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc., 2001, 14(3), 535–554

    Article  MATH  MathSciNet  Google Scholar 

  7. Candelas P., de la Ossa X., Green P., Parkes L., A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B, 1991, 359(1), 21–74

    Article  MATH  MathSciNet  Google Scholar 

  8. Cox D., Katz S., Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, 68, American Mathematical Society, Providence, RI, 1999

    MATH  Google Scholar 

  9. Efimov A., Homological mirror symmetry for curves of higher genus, preprint available at http://arxiv.org/abs/0907.3903

  10. Fukaya K., Mirror symmetry of abelian varieties and multi-theta functions, J. Algebraic Geom., 2002, 11(3), 393–512

    MATH  MathSciNet  Google Scholar 

  11. Fukaya K., Oh Y.-G., Ohta H., Ono K., Lagrangian intersection Floer theory — anomaly and obstruction, preprint available at http://www.math.kyoto-u.ac.jp/fukaya/fukaya.html

  12. Hori K., Katz S., Klemm A., Pandharipande R., Thomas R., Vafa C., Vakil R., Zaslow E., Mirror symmetry, Volume 1, Clay Mathematics Monographs, American Mathematical Society, Providence, RI, 2003

    MATH  Google Scholar 

  13. Hori K., Vafa C., Mirror symmetry, preprint available at http://arxiv.org/abs/hep-th/0002222

  14. Kapustin A., Orlov D., Remarks on A-branes, mirror symmetry, and the Fukaya category, J. Geom. Phys., 2003, 48(1), 84–99

    Article  MATH  MathSciNet  Google Scholar 

  15. Kapustin A., Orlov D., Lectures on mirror symmetry, derived categories, and D-branes, Russian Math. Surveys, 2004, 59(5), 907–940

    Article  MATH  MathSciNet  Google Scholar 

  16. Kawamata Y., D-equivalence and K-equivalence, J. Differential Geom., 2002, 61(1), 147–171

    MATH  MathSciNet  Google Scholar 

  17. Kuznetsov A., Derived category of V 12 Fano threefolds, preprint available at http://arxiv.org/abs/math/0310008

  18. Mukai S., Non-Abelian Brill Noether theory and Fano 3 folds, preprint available at http://arxiv.org/abs/alg-geom/9704015

  19. Narasimhan M.S., Ramanan S., Moduli of vector bundles on a compact Riemann surface, Ann. of Math. (2), 1969, 89, 14–51

    Article  MathSciNet  Google Scholar 

  20. Orlov D., Equivalences of derived categories and K3 surfaces, J. Math. Sci. (New York), 1997, 84(5), 1361–1381

    Article  MATH  MathSciNet  Google Scholar 

  21. Orlov D., Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Tr. Mat. Inst. Steklova, 2004, 246, Algebr. Geom. Metody, Svyazi i Prilozh., 240–262, English translation: Proc. Steklov Inst. Math., 2004, 3, 227–248

  22. Orlov D., Mirror symmetry for higher genus curves, Lectures at University of Miami, January 2008, IAS, March 2008

  23. Orlov D., Formal completions and idempotent completions of triangulated categories of singularities, preprint available at http://arxiv.org/abs/0901.1859

  24. Polishchuk A., Zaslow E., Categorical mirror symmetry: the elliptic curve, Adv. Theor. Math. Phys. 2, 1998, 2, 443–470

    MATH  MathSciNet  Google Scholar 

  25. Seidel P., More about vanishing cycles and mutation, Symplectic geometry and mirror symmetry (Seoul, 2000), 429–465, World Sci. Publ., River Edge, NJ, 2001

    Google Scholar 

  26. Seidel P., Fukaya categories and deformations, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 351–360, Higher Ed. Press, Beijing, 2002

    Google Scholar 

  27. Seidel P., Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, 2008

  28. Seidel P., Homological mirror symmetry for the quartic surface, preprint available at http://arxiv.org/abs/math/0310414

  29. Seidel P., Homological mirror symmetry for the genus two curve, preprint available at http://arxiv.org/abs/0812.1171.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Anton Kapustin.

Additional information

To Fedya, our teacher and friend, with admiration

About this article

Cite this article

Kapustin, A., Katzarkov, L., Orlov, D. et al. Homological Mirror Symmetry for manifolds of general type. centr.eur.j.math. 7, 571–605 (2009). https://doi.org/10.2478/s11533-009-0056-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s11533-009-0056-x

MSC

Keywords

Navigation