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Homological mirror symmetry without correction


Author: Mohammed Abouzaid
Journal: J. Amer. Math. Soc. 34 (2021), 1059-1173
MSC (2020): Primary 53D37; Secondary 14G22
DOI: https://doi.org/10.1090/jams/973
Published electronically: May 24, 2021
MathSciNet review: 4301560
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Abstract: Let $X$ be a closed symplectic manifold equipped with a Lagrangian torus fibration over a base $Q$. A construction first considered by Kontsevich and Soibelman produces from this data a rigid analytic space $Y$, which can be considered as a variant of the $T$-dual introduced by Strominger, Yau, and Zaslow. We prove that the Fukaya category of tautologically unobstructed graded Lagrangians in $X$ embeds fully faithfully in the derived category of (twisted) coherent sheaves on $Y$, under the technical assumption that $\pi _2(Q)$ vanishes (all known examples satisfy this assumption). The main new tool is the construction and computation of Floer cohomology groups of Lagrangian fibres equipped with topological infinite rank local systems that correspond, under mirror symmetry, to the affinoid rings introduced by Tate, equipped with their natural topologies as Banach algebras.


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Additional Information

Mohammed Abouzaid
Affiliation: Department of Mathematics, Columbia University, 2990 Broadway Ave., New York, New York 10027
MR Author ID: 734175
Email: abouzaid@math.columbia.edu

Received by editor(s): March 22, 2017
Received by editor(s) in revised form: January 6, 2020, and December 14, 2020
Published electronically: May 24, 2021
Additional Notes: The author was supported by the Simons Foundation through its “Homological Mirror Symmetry” Collaboration grant SIMONS 385571, and by NSF grants DMS-1308179, DMS-1609148, and DMS-1564172. He was also partially supported by the Erik Ellentuck Fellowship and the IAS Fund of Math during the “Homological Mirror Symmetry” program at the Institute for Advanced Study.
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