Geodesic
Homological mirror symmetry for punctured spheres
Abouzaid, Mohammed1 ; Auroux, Denis2 ; Efimov, Alexander3 ; Katzarkov, Ludmil4 ; Orlov, Dmitri3
1 Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027
2 Department of Mathematics, University of California, Berkeley, Berkeley, California 94720-3840
3 Algebraic Geometry Section, Steklov Mathematical Institute, Russian Academy of Sciences, 8 Gubkin Street, Moscow 119991, Russia
4 Department of Mathematics, Universität Wien, Garnisongasse 3, Vienna A-1090, Austria, and University of Miami, P.O. Box 249085, Coral Gables, Florida 33124-4250
Journal of the American Mathematical Society, Tome 26 (2013) no. 4, pp. 1051-1083

Voir la notice de l'article provenant de la source American Mathematical Society

We prove that the wrapped Fukaya category of a punctured sphere ($S^{2}$ with an arbitrary number of points removed) is equivalent to the triangulated category of singularities of a mirror Landau-Ginzburg model, proving one side of the homological mirror symmetry conjecture in this case. By investigating fractional gradings on these categories, we conclude that cyclic covers on the symplectic side are mirror to orbifold quotients of the Landau-Ginzburg model.
DOI : 10.1090/S0894-0347-2013-00770-5

Abouzaid, Mohammed 1 ; Auroux, Denis 2 ; Efimov, Alexander 3 ; Katzarkov, Ludmil 4 ; Orlov, Dmitri 3

1 Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027
2 Department of Mathematics, University of California, Berkeley, Berkeley, California 94720-3840
3 Algebraic Geometry Section, Steklov Mathematical Institute, Russian Academy of Sciences, 8 Gubkin Street, Moscow 119991, Russia
4 Department of Mathematics, Universität Wien, Garnisongasse 3, Vienna A-1090, Austria, and University of Miami, P.O. Box 249085, Coral Gables, Florida 33124-4250 
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Abouzaid, Mohammed; Auroux, Denis; Efimov, Alexander; Katzarkov, Ludmil; Orlov, Dmitri. Homological mirror symmetry for punctured spheres. Journal of the American Mathematical Society, Tome 26 (2013) no. 4, pp. 1051-1083. doi: 10.1090/S0894-0347-2013-00770-5

[1] Abouzaid, Mohammed Morse homology, tropical geometry, and homological mirror symmetry for toric varieties Selecta Math. (N.S.) 2009 189 270

[2] Abouzaid, Mohammed A geometric criterion for generating the Fukaya category Publ. Math. Inst. Hautes Études Sci. 2010 191 240

[3] Abouzaid, Mohammed A cotangent fibre generates the Fukaya category Adv. Math. 2011 894 939

[4] Abouzaid, Mohammed, Seidel, Paul An open string analogue of Viterbo functoriality Geom. Topol. 2010 627 718

[5] Abouzaid, Mohammed, Smith, Ivan Homological mirror symmetry for the 4-torus Duke Math. J. 2010 373 440

[6] Auroux, Denis, Katzarkov, Ludmil, Orlov, Dmitri Mirror symmetry for weighted projective planes and their noncommutative deformations Ann. of Math. (2) 2008 867 943

[7] Auroux, Denis, Katzarkov, Ludmil, Orlov, Dmitri Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves Invent. Math. 2006 537 582

[8] Efimov, Alexander I. Homological mirror symmetry for curves of higher genus Adv. Math. 2012 493 530

[9] Chan, Kwokwai, Leung, Naichung Conan Mirror symmetry for toric Fano manifolds via SYZ transformations Adv. Math. 2010 797 839

[10] Cho, Cheol-Hyun Products of Floer cohomology of torus fibers in toric Fano manifolds Comm. Math. Phys. 2005 613 640

[11] Fang, Bohan, Liu, Chiu-Chu Melissa, Treumann, David, Zaslow, Eric T-duality and homological mirror symmetry for toric varieties Adv. Math. 2012 1875 1911

[12] Fukaya, Kenji Mirror symmetry of abelian varieties and multi-theta functions J. Algebraic Geom. 2002 393 512

[13] Fukaya, K., Seidel, P., Smith, I. The symplectic geometry of cotangent bundles from a categorical viewpoint 2009 1 26

[14] Kapustin, Anton, Katzarkov, Ludmil, Orlov, Dmitri, Yotov, Mirroslav Homological mirror symmetry for manifolds of general type Cent. Eur. J. Math. 2009 571 605

[15] Kapustin, Anton, Orlov, Dmitri Vertex algebras, mirror symmetry, and D-branes: the case of complex tori Comm. Math. Phys. 2003 79 136

[16] Keller, Bernhard Introduction to 𝐴-infinity algebras and modules Homology Homotopy Appl. 2001 1 35

[17] Keller, Bernhard On differential graded categories 2006 151 190

[18] Kontsevich, Maxim Homological algebra of mirror symmetry 1995 120 139

[19] Kontsevich, Maxim, Soibelman, Yan Homological mirror symmetry and torus fibrations 2001 203 263

[20] Kontsevich, M., Soibelman, Y. Notes on 𝐴_{∞}-algebras, 𝐴_{∞}-categories and non-commutative geometry 2009 153 219

[21] Lunts, Valery A. Categorical resolution of singularities J. Algebra 2010 2977 3003

[22] Orlov, D. O. Triangulated categories of singularities and D-branes in Landau-Ginzburg models Tr. Mat. Inst. Steklova 2004 240 262

[23] Orlov, Dmitri Matrix factorizations for nonaffine LG-models Math. Ann. 2012 95 108

[24] Polishchuk, Alexander, Zaslow, Eric Categorical mirror symmetry: the elliptic curve Adv. Theor. Math. Phys. 1998 443 470

[25] Schlichting, Marco Negative 𝐾-theory of derived categories Math. Z. 2006 97 134

[26] Seidel, Paul Fukaya categories and deformations 2002 351 360

[27] Seidel, Paul Fukaya categories and Picard-Lefschetz theory 2008

[28] Seidel, Paul Homological mirror symmetry for the genus two curve J. Algebraic Geom. 2011 727 769

[29] Sheridan, Nick On the homological mirror symmetry conjecture for pairs of pants J. Differential Geom. 2011 271 367

[30] Strominger, Andrew, Yau, Shing-Tung, Zaslow, Eric Mirror symmetry is 𝑇-duality Nuclear Phys. B 1996 243 259

[31] Thomason, R. W. The classification of triangulated subcategories Compositio Math. 1997 1 27

[32] Ueda, Kazushi Homological mirror symmetry for toric del Pezzo surfaces Comm. Math. Phys. 2006 71 85

[33] Weibel, Charles The negative 𝐾-theory of normal surfaces Duke Math. J. 2001 1 35

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