The coherent-constructible correspondence and Fourier-Mukai transforms
type: Item
instance of: scholarly article
title: The coherent-constructible correspondence and Fourier-Mukai transforms
zbMATH Open document ID: 1226.14026
author: Chiu-Chu Melissa Liu
author: David Treumann
author: Bohan Fang
author: Eric Zaslow
publication date: 2011-04-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1009.3506
review text: The purpose of the paper under review is to investigate some of Kawamata's results concerning derived equivalences between toric orbifolds using the so-called coherent constructible correspondence (CCC). Given a toric \(n\)-fold \(X\), the coherent-constructible correspondence (CCC) is an equivalence between a category of coherent sheaves on \(X\) and a category of constructible sheaves on a compact \(n\)-torus. There also exists an equivariant version. It is furthermore possible to extend this correspondence to toric Deligne-Mumford stacks, in particular to toric orbifolds, that is, toric DM-stacks with generically trivial stabilizers. Let \(\mathcal{X}_{\pmb\Sigma}\) be a toric orbifold defined by a so-called stacky fan \({\pmb\Sigma}=(N,\Sigma,\beta)\). This contains the torus \(T\cong (\mathbb{C}^*)^n\) as a dense open subset. The CCC for \(\mathcal{X}_{\pmb\Sigma}\) is a quasi-equivalence of the following triangulated dg-categories: On the one hand, one has the category of \(T\)-equivariant perfect complexes on the orbifold and on the other hand a category of constructible sheaves on \(M_\mathbb{R}\) (the dual space of \(N_\mathbb{R}\)) characterized by a conical Lagrangian \(\Lambda_{\pmb{\Sigma}} \in T^*M_\mathbb{R}\) determined by the above stacky fan. The toric orbifold \(\mathcal{X}_{\pmb\Sigma}\) is the DM-stack associated with a log pair \((X_\Sigma,B)\) in the sense of \textit{Y. Kawamata} [J.\ Math.\ Sci., Tokyo 12, No.\ 2, 211--231 (2005; Zbl 1095.14014)]. Given two toric orbifolds \(\mathcal{X}_1\) and \(\mathcal{X}_2\) associated with projective toric log pairs, one says that the orbifolds are \(K\)-equivalent if there exists an orbifold \(\mathcal{W}\) and proper birational morphisms \(\mu_i: \mathcal{W}\rightarrow \mathcal{X}_i\) of orbifolds such that \(\mu_1^*K_{\mathcal{X}_1}= \mu_2^*K_{\mathcal{X}_2}\). Kawamata proved that the Fourier--Mukai functor \((\mu_1)_*\circ \mu_2^*\) is an equivalence of triangulated categories. To be more precise, h…
reviewed by: Pawel Sosna
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 5876781
zbMATH Keywords: toric orbifolds
zbMATH Keywords: coherent sheaves
zbMATH Keywords: constructible sheaves
zbMATH Keywords: Fourier-Mukai transforms
MaRDI profile type: Publication
OpenAlex ID: W2023010197
arXiv ID: 1009.3506
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DOI: 10.1007/S10114-011-0462-4
published in: Acta Mathematica Sinica, English Series
description: scientific article