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Homological mirror symmetry at large volume

Benjamin Gammage and Vivek Shende

Vol. 5 (2023), No. 1, 31–71
Abstract

A typical large complex-structure limit for mirror symmetry consists of toric varieties glued to each other along their toric boundaries. Here we construct the mirror large volume limit space as a Weinstein symplectic manifold. We prove homological mirror symmetry: the category of coherent sheaves on the first space is equivalent to the Fukaya category of the second.

Our equivalence intertwines the Viterbo restriction maps for a generalized pair-of-pants cover of the symplectic manifold with the restriction of coherent sheaves for a certain affine cover of the algebraic variety. We deduce a posteriori a local-to-global principle conjectured by Seidel — certain diagrams of Viterbo restrictions are cartesian — by passing Zariski descent through our mirror symmetry result.

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Keywords
mirror symmetry, Weinstein manifolds, microlocal sheaves
Mathematical Subject Classification
Primary: 14J33, 53D37
Milestones
Received: 23 November 2021
Revised: 25 July 2022
Accepted: 9 August 2022
Published: 20 April 2023
Authors
Benjamin Gammage
Department of Mathematics
Harvard University
Cambridge, MA
United States
Vivek Shende
Centre for Quantum Mathematics
Universitet Syddansk
Odense
Denmark