## Abstract

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

Show Summary Details# Open Mathematics

### formerly Central European Journal of Mathematics

#### Open Access

# Homological Mirror Symmetry for manifolds of general type

### Publication History

## Abstract

## Citing Articles

*International Mathematics Research Notices*, 2015*Journal de Mathématiques Pures et Appliquées*, 2014, Volume 102, Number 4, Page 702*Compositio Mathematica*, 2014, Volume 150, Number 04, Page 621*Успехи математических наук*, 2010, Volume 65, Number 5, Page 191*Manuscripta Mathematica*, 2013, Volume 141, Number 3-4, Page 391*Advances in Mathematics*, 2012, Volume 230, Number 2, Page 493*Inventiones mathematicae*, 2012, Volume 189, Number 1, Page 149*Communications in Mathematical Physics*, 2011, Volume 304, Number 2, Page 411*Russian Mathematical Surveys*, 2011, Volume 65, Number 5, Page 987

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

IMPACT FACTOR 2015: 0.512

SCImago Journal Rank (SJR) 2015: 0.521

Source Normalized Impact per Paper (SNIP) 2015: 1.233

Impact per Publication (IPP) 2015: 0.546

Mathematical Citation Quotient (MCQ) 2015: 0.39

^{1}Department of Physics, California Institute of Technology, Pasadena, USA

^{2}Department of Mathematics, Universität Wien, Wien, Austria

^{3}Algebra Section, Steklov Mathematical Institute RAS, Moscow, Russia

^{4}Department of Mathematics and Statistics, Florida International University, Miami, USA

Citation Information: Open Mathematics. Volume 7, Issue 4, Pages 571–605, ISSN (Online) 2391-5455, DOI: 10.2478/s11533-009-0056-x, October 2009

- Published Online:
- 2009-10-31

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

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.2007.08.011 [Web of Science] [CrossRef]

[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/s00222-006-0003-4 [CrossRef]

[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

[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 http://dx.doi.org/10.1090/S0894-0347-01-00368-X [CrossRef]

[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)90292-6 [CrossRef]

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

[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

[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

[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 http://dx.doi.org/10.1016/S0393-0440(03)00026-3 [CrossRef]

[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/RM2004v059n05ABEH000772 [CrossRef]

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

[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 http://dx.doi.org/10.2307/1970807 [CrossRef]

[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/BF02399195 [CrossRef]

[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 [Web of Science]

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

[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

[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

[27] Seidel P., Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, 2008 [Web of Science]

[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.

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]

V. Przyjalkowski and C. Shramov

DOI: 10.1093/imrn/rnv024

[2]

Matthew Ballard, David Favero, and Ludmil Katzarkov

[3]

Robert Fisette and Alexander Polishchuk

[4]

Александр Иванович Ефимов and Aleksander Ivanovich Efimov

DOI: 10.4213/rm9389

[5]

Heinrich Hartmann

[6]

Alexander I. Efimov

[7]

Ivan Smith

[8]

Ed Segal

[9]

Aleksander I Efimov

## Comments (0)