Derived equivalent Calabi-Yau threefolds from cubic fourfolds
type: Item
instance of: scholarly article
title: Derived equivalent Calabi-Yau threefolds from cubic fourfolds
zbMATH Open document ID: 1342.14036
author: John Calabrese
author: Richard P. Thomas
published in: Mathematische Annalen
publication date: 2016-05-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1408.4063
review text: The bounded derived category \(D(X)\) of coherent sheaves on a smooth variety \(X\) is a powerful tool for revealing ``hidden'' relationships between algebraic varieties. Although these derived equivalences are often constructed in a case-by-case basis, a powerful tool has been developed by \textit{A. Kuznetsov} [Publ. Math., Inst. Hautes Étud. Sci. 105, 157--220 (2007; Zbl 1131.14017)] which establishes derived equivalences in a very general setup. Generalizing the classical projective duality, this Homological Projective Duality (HPD) relates the derived categories of universal hypersurfaces over linear systems with the derived categories of their base loci. The authors of this article investigate the most elementary case of HPD, that of pencils of hypersurfaces, in the much-studied case of cubic fourfolds. In two beautiful examples of very different flavor, the authors establish derived equivalences between two different pairs \((X,Y)\) of Calabi-Yau threefolds. In the first example, the authors consider a generic pencil of cubic fourfolds containing a fixed plane \(P\subset\mathbb P^5\) with baselocus a (3,3) complete intersection \(X_0\) with 12 ordinary double points (ODPs) along \(P\). They let \(X:=\mathrm{Bl}_P X_0\) be the blowup along \(P\), which gives a projective Calabi-Yau small resolution of \(X_0\). To get \(Y\), they choose a plane \(\mathbb P^2\subset\mathbb P^5\) disjoint from \(P\) and note that the projection from \(P\) to \(\mathbb P^2\) gives a quadric fibration \(\mathcal Y\to\mathbb P^2\times \mathbb P^1\), which is a relative version of the corresponding quadric fibration induced by the projection on each cubic fourfold in the family. Taking \(Y_0\) to be the double cover of \(\mathbb P^2\times\mathbb P^1\) branched along the degeneracy locus of \(\mathcal Y\to\mathbb P^2\times\mathbb P^1\), which has 66 ODPs, they finally let \(Y\) be any small resolution of \(Y_0\) (and show that all of these are non-Kähler). They show that \(D(X)\cong D…
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14J32
zbMATH DE Number: 6587937
zbMATH Keywords: homological projective duality
zbMATH Keywords: Calabi-Yau threefolds
zbMATH Keywords: cubic fourfolds
zbMATH Keywords: K3 surfaces
reviewed by: Howard Nuer
MaRDI profile type: Publication
OpenAlex ID: W3104150500
arXiv ID: 1408.4063
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DOI: 10.1007/S00208-015-1260-6
description: scientific article; zbMATH DE number 6587937