Derived equivalent Calabi-Yau threefolds from cubic fourfolds
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 \(D(X)\cong D(Y,\alpha)\) where \(\alpha\in\mathrm H^2_{et}(Y,\mathcal O_Y^{\times}) \) is the Brauer class corresponding to the quadric fibration. The second example comes instead from a pencil of cubic fourfolds all having an ODP at a fixed point \(0\in\mathbb P^5\). Similar to above, they let \(X=\mathrm{Bl}_0 X_0\), where \(X_0\) is the base locus of the pencil. The authors then project the pencil's universal hypersurface \(\mathcal H\subset\mathbb P^5\times\mathbb P^1\) away from \(\{0\}\times\mathbb P^1\) to a disjoint \(\mathbb P^4\times\mathbb P^1\) to obtain a birational map \(\mathrm{Bl}_{\{0\}\times\mathbb P^1} \mathcal H\to\mathbb P^4\times\mathbb P^1\). It follows that \(\mathrm{Bl}_{\{0\}\times\mathbb P^1} \mathcal H\cong \mathrm{Bl}_Y(\mathbb P^4\times\mathbb P^1)\) where \(Y\) is the smooth intersection of a (2,1) divisor and a (3,1) divisor. The authors show that \(X\) and \(Y\) are derived equivalent smooth projective Calabi-Yau threefolds. It turns out that \(X\) and \(Y\) are in fact birational, and thus already derived equivalent though in a different way, so the authors obtain an exotic autoequivalence of \(D(X)\).
scientific article; zbMATH DE number 6587937
scientific article