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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 7, Issue 4


Volume 13 (2015)

Homological Mirror Symmetry for manifolds of general type

Anton Kapustin / Ludmil Katzarkov / Dmitri Orlov / Mirroslav Yotov
Published Online: 2009-10-31 | DOI: https://doi.org/10.2478/s11533-009-0056-x


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

MSC: 57D37; 57R17; 14J33

Keywords: Homological mirror Symmetry; K theory; Categories

  • [1] Abouzaid M., On the Fukaya categories of higher genus surfaces, Adv. Math., 2008, 217(3), 1192–1235 http://dx.doi.org/10.1016/j.aim.2007.08.011Web of ScienceCrossrefGoogle 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 http://dx.doi.org/10.1007/s00222-006-0003-4CrossrefGoogle 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 Google Scholar

  • [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 Google Scholar

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

  • [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 http://dx.doi.org/10.1090/S0894-0347-01-00368-XCrossrefGoogle 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 http://dx.doi.org/10.1016/0550-3213(91)90292-6CrossrefGoogle Scholar

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

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

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

  • [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 Google Scholar

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

  • [14] Kapustin A., Orlov D., Remarks on A-branes, mirror symmetry, and the Fukaya category, J. Geom. Phys., 2003, 48(1), 84–99 http://dx.doi.org/10.1016/S0393-0440(03)00026-3CrossrefGoogle Scholar

  • [15] Kapustin A., Orlov D., Lectures on mirror symmetry, derived categories, and D-branes, Russian Math. Surveys, 2004, 59(5), 907–940 http://dx.doi.org/10.1070/RM2004v059n05ABEH000772CrossrefGoogle Scholar

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

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

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

  • [19] Narasimhan M.S., Ramanan S., Moduli of vector bundles on a compact Riemann surface, Ann. of Math. (2), 1969, 89, 14–51 http://dx.doi.org/10.2307/1970807CrossrefGoogle Scholar

  • [20] Orlov D., Equivalences of derived categories and K3 surfaces, J. Math. Sci. (New York), 1997, 84(5), 1361–1381 http://dx.doi.org/10.1007/BF02399195CrossrefGoogle 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 Google Scholar

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

  • [23] Orlov D., Formal completions and idempotent completions of triangulated categories of singularities, preprint available at http://arxiv.org/abs/0901.1859 Web of ScienceGoogle Scholar

  • [24] Polishchuk A., Zaslow E., Categorical mirror symmetry: the elliptic curve, Adv. Theor. Math. Phys. 2, 1998, 2, 443–470 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 Web of ScienceGoogle Scholar

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

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

About the article

Published Online: 2009-10-31

Published in Print: 2009-12-01

Citation Information: Open Mathematics, Volume 7, Issue 4, Pages 571–605, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-009-0056-x.

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© 2009 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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