A categorification of Morelli's theorem
type: Item
instance of: scholarly article
title: A categorification of Morelli's theorem
zbMATH Open document ID: 1250.14011
author: Chiu-Chu Melissa Liu
author: David Treumann
author: Eric Zaslow
author: Bohan Fang
published in: Inventiones Mathematicae
publication date: 2011-11-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1007.0053
review text: Let \(X\) be a proper toric variety of dimension \(n\) and consider \(\text{Perf}_T(X)\), the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves, that is, the category of complexes that locally admit a bounded resolution by torus-equivariant vector bundles. The main result of the paper under review is the existence of a conical Lagrangian \(\Lambda\subset T^*\mathbb{R}^n\) and an equivalence of triangulated dg categories \(\text{Perf}_T(X)\cong Sh_{cc}(\mathbb{R}^n;\Lambda)\), where the latter is the triangulated dg category of sheaves on \(\mathbb{R}^n\) which are compactly supported, constructible (``cc'') and whose singular support lies in \(\Lambda\). Moreover, this equivalence is monoidal, meaning that it intertwines the tensor product on \(X\) with the convolution product on \(\mathbb{R}^n\). This statement can, in particular, be used to prove a version of the homological mirror symmetry conjecture for toric varieties. A special case of the above result is the existence of a monoidal full embedding of the bounded derived category of torus-equivariant coherent sheaves on an \(n\)-dimensional smooth projective toric variety into the bounded derived category of sheaves on \(\mathbb{R}^n\) which are compactly supported and constructible with respect to the stratification defined by the union of all affine hyperplanes in \(\mathbb{R}^n\) that are spanned by lattice points and that are parallel to the faces of the moment polytope of \(X\). This result, in particular, gives \textit{R. Morelli}'s theorem on the \(K\)-theory of a smooth projective toric variety in [Adv.\ Math.\ 100, No.\ 2, 154--182 (1993, Zbl 0805.14025)].
reviewed by: Pawel Sosna
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 5968701
zbMATH Keywords: toric varieties
zbMATH Keywords: torus-equivariant sheaves
zbMATH Keywords: constructible sheaves
zbMATH Keywords: triangulated dg-categories
MaRDI profile type: Publication
OpenAlex ID: W3103424700
arXiv ID: 1007.0053
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DOI: 10.1007/S00222-011-0315-X
description: scientific article