A categorical invariant for cubic threefolds
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 sheaves with an action of an algebra \(\mathcal{B}_{0}\). Then they produce a Bridgeland stability condition on \(\mathbb{T}_{Y}\) such that the ideal sheaves of lines on \(Y\) are all stable, and they are the only stable objects in their numerical class. Composing with shifts and powers of the Serre functor of \(\mathbb{T}_{Y}\), we get the equivalence \(U'\). More precisely: let \(l_{0}\subseteq Y\) be a line, and \(\widetilde{Y}\) the strict transform of \(Y\) via the blow-up \(\widetilde{\mathbb{P}}^{4}\) of \(\mathbb{P}^{4}\) along \(l_{0}\), and let \(\sigma:\widetilde{Y}\longrightarrow Y\) be the natural morphism. The \(\mathbb{P}^{2}\)-bundle \(q:\widetilde{\mathbb{P}}^{4}\longrightarrow\mathbb{P}^{2}\) induced by the projection from \(l_{0}\) to a plane gives a conic fibration \(\pi:\widetilde{Y}\longrightarrow\mathbb{P}^{2}\). Let \(\mathcal{B}_{0}\) be the even part of the Clifford algebra corresponding to \(\pi\). The category \(D^{b}(\mathbb{P}^{2},\mathcal{B}_{0})\) is the bounded derived category of the category \(\mathrm{Coh}(\mathbb{P}^{2},\mathcal{B}_{0})\) of coherent sheaves on \(\mathbb{P}^{2}\) having the structure of right \(\mathcal{B}_{0}\)-module. We have a fully faithful functor \(\Phi:D^{b}(\mathbb{P}^{2},\mathcal{B}_{0})\longrightarrow D^{b}(\widetilde{Y})\), using which the authors produce a fully faithful functor \(\Xi:\mathbb{T}_{Y}\longrightarrow D^{b}(\mathbb{P}^{2},\mathcal{B}_{0})\), identifying \(\mathbb{T}_{Y}\) with a full triangulated subcategory of \(D^{b}(\mathbb{P}^{2},\mathcal{B}_{0})\). The stability condition on \(\mathbb{T}_{Y}\) is constructed in section 3. After studying the numerical Grothendieck group of \(\mathbb{T}_{Y}\) and of \(D^{b}(\mathbb{P}^{2},\mathcal{B}_{0})\), the authors introduce a slope for the objects in \(\mathrm{Coh}(\mathbb{P}^{2},\mathcal{B}_{0})\), which is used to define a torsion pair on \(\mathrm{Coh}(\mathbb{P}^{2},\mathcal{B}_{0})\), and hence a bounded \(t\)-structure on \(D^{b}(\mathbb{P}^{2},\mathcal{B}_{0})\). Restricting it to \(\mathbb{T}_{Y}\), the authors obtain a bounded \(t\)-structure on \(\mathbb{T}_{Y}\), whose heart is denoted \(\mathbb{B}\). Finally, they define a Bridgeland stability condition \(\sigma=(Z,\mathbb{B})\) on \(\mathbb{T}_{Y}\) by letting \(Z([A])=rk(A)+i(\deg(A)-\mu(\mathcal{B}_{0})\mathrm{rk}(A))\) for \(A\in\mathbb{T}_{Y}\). The third step in the proof is to show that the ideal sheaves of lines on \(Y\) are \(\sigma\)-stable objects of \(\mathbb{T}_{Y}\), and they are the only \(\sigma\)-stable objects in their numerical class: this is the content of section 4 of the paper. In section 5 the authors show (using convolutions) that the bijective correspondence between ideal sheaves of lines induced by the equivalence \(U:\mathbb{T}_{Y_{1}}\longrightarrow\mathbb{T}_{Y_{2}}\) is an isomorphism between \(F(Y_{1})\) and \(F(Y_{2})\). Section 6 is devoted to some results for cubic fourfolds containing a plane, for which there is an analogous semi-orthogonal decomposition of the bounded derived category. The only non-trivial part is \(\mathbb{T}_{Y}\), which is equivalent to the bounded derived category \(D^{b}(S,\alpha)\) of \(\alpha\)-twisted sheaves on a \(K3\) surface \(S\). Under some additional hypothesis, the authors show that if \(Y\) is a generic cubic fourfold containing a plane, and if there is another cubic fourfold \(Y'\) such that \(\mathbb{T}_{Y}\) is equivalent to \(\mathbb{T}_{Y'}\), then \(Y\simeq Y'\).
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