Homological mirror symmetry at large volume

Benjamin Gammage, Vivek Shende

Research output: Contribution to journalJournal articleResearchpeer-review

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.

Original languageEnglish
JournalTunisian Journal of Mathematics
Volume5
Issue number1
Pages (from-to)31-71
Number of pages41
ISSN2576-7658
DOIs
Publication statusPublished - 1. Apr 2023

Keywords

  • microlocal sheaves
  • mirror symmetry
  • Weinstein manifolds

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