A refined derived Torelli theorem for Enriques surfaces. II: The non-generic case (Q2118192)

A famous result by Bondal and Orlov states that a smooth projective variety with either ample or antiample canonical bundle can be reconstructed from its bounded derived category, up to isomorphism. This is no longer true in general without the assumption on the canonical bundle: counterexamples are known in the case of surfaces with trivial canonical bundle [\textit{S. Mukai}, Stud. Math., Tata Inst. Fundam. Res. 11, 341--413 (1987; Zbl 0674.14023)]. In a positive direction, the isomorphism class of an Enriques surface defined over an algebraically closed field of characteristic different from \(2\), which is a smooth projective surface \(X\) such that \(H^1(X, \mathcal{O}_X)=0\) and whose dualizing sheaf is \(2\)-torsion, is determined by its bounded derived category. This result, known as Derived Torelli Theorem for Enriques surfaces, has been proved over the complex numbers in [\textit{T. Bridgeland} and \textit{A. Maciocia}, Math. Z. 236, 677--697 (2001; Zbl 1081.14023)], and then generalized in [\textit{K. Honigs} et al., Math. Res. Lett. 28, No. 1, 65--91 (2021; Zbl 1471.14040)]. In the paper under review, the authors prove a refined version of the Derived Torelli Theorem for Enriques surfaces defined over an algebraically closed field \(\mathbb{K}\) of characteristic different from \(2\). More precisely, let \(X\) be an Enriques surface defined over \(\mathbb{K}\). Its bounded derived category admits a semiorthogonal decomposition consisting of a component \(\mathcal{L}\), generated by \(10\) line bundles on \(X\), and its right orthogonal, denoted by \(\mathcal{K}u(X, \mathcal{L})\) and called Kuznetsov component. The main result is Theorem A, which states that if two Enriques surfaces \(X_1\) and \(X_2\) with semiorthogonal decompositions of the form \(\text{D}^b(X_i)=\langle \mathcal{K}u(X_i, \mathcal{L}_i), \mathcal{L}_i \rangle\) admit an exact equivalence \(\mathcal{K}u(X_1, \mathcal{L}_1) \to \mathcal{K}u(X_2, \mathcal{L}_2)\) of Fourier--Mukai type, then \(X_1 \cong X_2\). The refined Derived Torelli theorem has been previously shown by the same authors for generic Enriques surfaces in [\textit{C. Li} et al., Math. Ann. 379, 1475--1505 (2020; Zbl 1484.14035)]. In both papers, the strategy involves techniques to extend equivalences among the Kuznetsov components to the whole derived category, in order to apply the Derived Torelli Theorem, together with classification results of special objects in the Kuznetsov component. This second ingredient is much harder removing the genericity assumption on the surface, because of the presence not only of \(3\)-spherical objects, but also of \(3\)-pseudoprojective objects. The paper is organized as follows. In Section 1, the authors review some material about semiorthogonal decompositions, the special case of Enriques surfaces and some classification results for special objects in the Kuznetsov component. Section 2 is devoted to the classification of \(3\)-spherical and \(3\)-pseudoprojective objects in the Kuznetsov component of an Enriques surface, while Section 3 contains the proof of the main theorem.