Aspects of functoriality in homological mirror symmetry for toric varieties
type: Item
instance of: scholarly article
title: Aspects of functoriality in homological mirror symmetry for toric varieties
zbMATH Open document ID: 1493.14064
author: Yanyan Li
published in: Advances in Mathematics
publication date: 2022-04-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/2010.08817
review text: This paper is about homological mirror symmetry for toric varieties, which suggests that the bounded derived category of coherent sheaves on a toric variety is equivalent to the Fukaya-Seidel category of its mirror. For a toric variety \(X\), the mirror to it is given by the algebraic torus \(({\mathbb{C}^*})^n\) together with a potential function \(W: ({\mathbb{C}^*})^n \to \mathbb{C}\). The authors prove that the derived category of coherent sheaves on \(X\) is equivalent to the partially wrapped Fukaya category of \(({\mathbb{C}^*})^n\) relative to a closed subset of its contact boundary. When \(X\) is Fano, this closed subset is \(W^{-1}(\infty)\) and the partially wrapped Fukaya category is equivalent to the Fukaya category of \((({\mathbb{C}^*})^n,W)\). This main result also follows from the work of [\textit{B. Fang} et al., Invent. Math. 186, No. 1, 79--114 (2011; Zbl 1250.14011)], who showed that the equivalence between the derived category of coherent sheaves on \(X\) with a category of constructible sheaves, and then applied results of [\textit{S. Ganatra} et al., ``Microlocal Morse theory of wrapped Fukaya categories'', Preprint, \url{arXiv:1809.08807}] relating the category of constructible sheaves to the Fukaya category. The authors of this paper follow a direct approach, which does not involve constructible sheaves.
Mathematics Subject Classification ID: 14J33
Mathematics Subject Classification ID: 53D37
zbMATH DE Number: 7514702
zbMATH Keywords: homological mirror symmetry
zbMATH Keywords: toric mirror symmetry
zbMATH Keywords: Lagrangian cobordism
zbMATH Keywords: tropical Lagrangian section
zbMATH Keywords: linking disk
reviewed by: Hülya Argüz
MaRDI profile type: Publication
OpenAlex ID: W4220732293
arXiv ID: 2010.08817
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DOI: 10.1016/J.AIM.2022.108317
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description: scientific article