The bijectivity of mirror functors on tori
type: Item
instance of: scholarly article
title: The bijectivity of mirror functors on tori
zbMATH Open document ID: 1502.14042
DOI: 10.1215/21562261-2022-0021
author: Kazushi Kobayashi
published in: Kyoto Journal of Mathematics
publication date: 2022-10-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1905.00692
review text: Given an \(n\)-dimensional complex torus \(X\) and its mirror \(\check{X}\), with trivial SYZ fibrations over base \(B\), the paper studies SYZ transformations between objects in the full subcategory \(\mathrm{Fuk}_{\mathrm{aff}}(\check{X})\) of the Fukaya category \(\mathrm{Fuk}(\check{X})\) consisting of affine Lagrangian multisections of \(\check{X}\to B\) with unitary local systems on them, and objects in the DG category \(DG_X\) of simple projectively flat bundles over \(X\). The main result of the paper is that for a fixed transition function, the SYZ transform \[ \iota:\mathrm{Ob}(\mathrm{Fuk}_{\mathrm{aff}}(\check{X}))\rightarrow\mathrm{Ob}(DG_X) \] admits an inverse up to isomorphism of objects. The inverse map is constructed explicitly using natural parameters. It follows that the induced map \[ \iota^{\mathrm{isom}}: \mathrm{Ob}^{\mathrm{isom}}(\mathrm{Fuk}_{\mathrm{aff}}(\check{X}))\rightarrow \mathrm{Ob}^{\mathrm{isom}}(DG_X), \] which is known to be injective, is indeed a bijection.
reviewed by: Jesse Huang
Mathematics Subject Classification ID: 14F08
Mathematics Subject Classification ID: 14J33
Mathematics Subject Classification ID: 53D37
zbMATH DE Number: 7608277
zbMATH Keywords: homological mirror symmetry
zbMATH Keywords: SYZ transform
Wikidata QID: Q121850321
MaRDI profile type: Publication
OpenAlex ID: W2943734510
arXiv ID: 1905.00692
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description: scientific article