T-duality and homological mirror symmetry for toric varieties
type: Item
instance of: scholarly article
title: T-duality and homological mirror symmetry for toric varieties
zbMATH Open document ID: 1260.14049
author: Chiu-Chu Melissa Liu
author: David Treumann
author: Eric Zaslow
author: Bohan Fang
published in: Advances in Mathematics
publication date: 2012-03-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/0811.1228
review text: In the paper under review, the authors prove one direction of the homological mirror symmetry conjecture for toric varieties. Let \(X_{\Sigma}\) be a complex \(n-\)dimensional complete toric variety defined by a finite complete fan \(\Sigma\subseteq N_{\mathbb{R}}\simeq\mathbb{R}^{n}\). Write \(M\) for the dual of \(N\) and \(M_{\mathbb{R}}\) for the dual of \(N_{\mathbb{R}}\). Moreover, let \(T_{\mathbb{R}}\) be the maximal compact subgroup of the algebraic torus \(T=N\times\mathbb{C}^{*}\), and \(T^{\vee}_{\mathbb{R}}\) be its dual torus. As shown by the authors [Invent. Math. 186, No. 1, 79--114 (2011; Zbl 1250.14011)], there is a quasi-equivalence of monoidal dg-categories \(\kappa: \mathrm{Per} f_{T}(X_{\Sigma})\longrightarrow Sh_{cc}(M_{\mathbb{R}},\Lambda_{\Sigma})\), where \(\mathrm{Per} f_{T}(X_{\Sigma})\) is the dg-category of perfect complexes of \(T-\)equivariant coherent sheaves (localized with respect to acyclic complexes); \(\Lambda_{\Sigma}\) is a Lagrangian subvariety of \(T^{*}M_{\mathbb{R}}\) associated to the toric fan; \(\mathrm{Sh}_{cc}(M_{\mathbb{R}},\Lambda_{\Sigma})\) is the dg-category of bounded chain complexes of sheaves of \(\mathbb{C}-\)vector spaces on \(M_{\mathbb{R}}\) (localized with respect to acyclic complexes), whose cohomology sheaves are constructible with respect to the Whitney stratification, have compact support, and the singular support is in \(\Lambda_{\Sigma}\). \textit{D. Nadler} and \textit{E. Zaslow} [J. Am. Math. Soc. 22, No. 1, 233--286 (2009; Zbl 1227.32019)], obtained a quasi-equivalence of \(A_{\infty}-\)categories \(\mu:\mathrm{Sh}_{cc}(M_{\mathbb{R}},\Lambda_{\Sigma})\longrightarrow \mathrm{Fuk}(T^{*}M_{\mathbb{R}},\Lambda_{\Sigma}),\) where the last is the unwrapped Fuakaya category of Lagrangian branes. The authors show that on \(\mathrm{Fuk}(T^{*}M_{\mathbb{R}},\Lambda_{\Sigma})\) there is a product \(\diamond\), and that the functor \(\mu\) intertwins the convolution product and \(\diamond\). Combining all t…
reviewed by: Arvid Perego
Mathematics Subject Classification ID: 14J33
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 6011375
zbMATH Keywords: T-duality
zbMATH Keywords: homological mirror symmetry
zbMATH Keywords: toric varieties
MaRDI profile type: Publication
OpenAlex ID: W2963446977
arXiv ID: 0811.1228
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DOI: 10.1016/J.AIM.2011.10.022
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description: scientific article