A categorical invariant for cubic threefolds
type: Item
instance of: scholarly article
title: A categorical invariant for cubic threefolds
zbMATH Open document ID: 1242.14012
author: Marcello Bernardara
author: Emanuele Macrì
author: Sukhendu Mehrotra
author: Paolo Stellari
published in: Advances in Mathematics
publication date: 2012-01-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/0903.4414
review text: The authors study categorical invariants of cubic threefolds, proving a categorical version of the Clemens-Griffiths-Tyurin Torelli Theorem for cubic threefolds. If \(Y\) is a cubic threefold, the bounded derived category \(D^{b}(Y)\) of coherent sheaves on \(Y\) has a semi-orthogonal decompositions \(D^{b}(Y)=\langle\mathbb{T}_{Y},\mathcal{O}_{Y},\mathcal{O}_{Y}(1)\rangle\), where \(\mathcal{O}_{Y}(1)\) is the restriction to \(Y\) of \(\mathcal{O}_{\mathbb{P}^{4}}(1)\), and \(\mathbb{T}_{Y}\) is the orthogonal of the triangulated subcategory \(\langle\mathcal{O}_{Y},\mathcal{O}_{Y}(1)\rangle\). \textit{A. Kuznetsov} [Proc. Steklov Inst. Math. 246, 171--194 (2004); translation from Tr. Mat. Inst. Steklova 246, 183--207 (2004; Zbl 1107.14028)] conjectured that two cubic threefolds \(Y_{1}\) and \(Y_{2}\) are isomorphic if and only if \(\mathbb{T}_{Y_{1}}\) is equivalent (as triangulated category) to \(\mathbb{T}_{Y_{2}}\). The aim of the paper is to prove this conjecture, i.e., that an equivalence \(U:\mathbb{T}_{Y_{1}}\longrightarrow\mathbb{T}_{Y_{2}}\) induces an isomorphism between \(Y_{1}\) and \(Y_{2}\). By the Torelli Theorem for cubic threefolds, this is true if one is able to produce an isomorphism of principally polarized abelian varieties between the intermediate Jacobians \(J(Y_{1})\) and \(J(Y_{2})\). As these are the Albanese varieties of the Fano varieties of lines \(F(Y_{1})\) and \(F(Y_{2})\), the aim is to produce an isomorphism between \(F(Y_{1})\) and \(F(Y_{2})\) out of the equivalence \(U\). To do so, as the ideal sheaves of lines on \(Y_{i}\) are contained in \(\mathbb{T}_{Y_{i}}\), the authors produce an equivalence \(U'\) out of \(U\) giving a bijective correspondence between the set of ideal sheaves, and then verify that such a bijection is indeed an isomorphism of varieties. To produce the equivalence \(U'\), the authors realize \(\mathbb{T}_{Y}\) as a subcategory of the derived category \(D^{b}(\mathbb{P}^{2},\mathcal{B}_{0})\) of coherent …
reviewed by: Arvid Perego
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 14E08
zbMATH DE Number: 5994256
zbMATH Keywords: derived categories
zbMATH Keywords: cubic threefolds
MaRDI profile type: Publication
OpenAlex ID: W2964313687
arXiv ID: 0903.4414
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DOI: 10.1016/J.AIM.2011.10.007
description: scientific article