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Homological mirror symmetry for punctured spheres


Authors: Mohammed Abouzaid, Denis Auroux, Alexander I. Efimov, Ludmil Katzarkov and Dmitri Orlov
Journal: J. Amer. Math. Soc. 26 (2013), 1051-1083
MSC (2010): Primary 53D37, 14J33; Secondary 53D40, 53D12, 18E30, 14F05
DOI: https://doi.org/10.1090/S0894-0347-2013-00770-5
Published electronically: April 4, 2013
MathSciNet review: 3073884
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Abstract: 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.


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Additional Information

Mohammed Abouzaid
Affiliation: Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027
Email: abouzaid@math.columbia.edu

Denis Auroux
Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720-3840
Email: auroux@math.berkeley.edu

Alexander I. Efimov
Affiliation: Algebraic Geometry Section, Steklov Mathematical Institute, Russian Academy of Sciences, 8 Gubkin Street, Moscow 119991, Russia
Email: efimov13@yandex.ru

Ludmil Katzarkov
Affiliation: Department of Mathematics, Universität Wien, Garnisongasse 3, Vienna A-1090, Austria, and University of Miami, P.O. Box 249085, Coral Gables, Florida 33124-4250
Email: lkatzark@math.uci.edu

Dmitri Orlov
Affiliation: Algebraic Geometry Section, Steklov Mathematical Institute, Russian Academy of Sciences, 8 Gubkin Street, Moscow 119991, Russia
Email: orlov@mi.ras.ru

DOI: https://doi.org/10.1090/S0894-0347-2013-00770-5
Received by editor(s): March 22, 2011
Received by editor(s) in revised form: March 2, 2013
Published electronically: April 4, 2013
Additional Notes: The first author was supported by a Clay Research Fellowship
The second author was partially supported by NSF grants DMS-0652630 and DMS-1007177
The third author was partially supported by the Dynasty Foundation, NSh grant 4713.2010.1, and by AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023
The fourth author was funded by NSF grant DMS-0600800, NSF FRG grant DMS-0652633, FWF grant P20778, and an ERC grant – GEMIS
The last author was partially supported by RFBR grants 10-01-93113, 11-01-00336, 11-01-00568, NSh grant 4713.2010.1, and by AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023
Article copyright: © Copyright 2013 American Mathematical Society
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