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1. A refined derived Torelli theorem for enriques surfaces, II: the non-generic case

   https://doi.org/10.1007/s00209-021-02930-4  

   Li, Chunyi; Stellari, Paolo; Zhao, Xiaolei (April 2022, Mathematische Zeitschrift)

   Abstract We prove that two Enriques surfaces defined over an algebraically closed field of characteristic different from 2 are isomorphic if their Kuznetsov components are equivalent. This improves and completes our previous result joint with Nuer where the same statement is proved for generic Enriques surfaces. 

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2. Derived categories of hearts on Kuznetsov components

   https://doi.org/10.1112/jlms.12804  

   Li, Chunyi; Pertusi, Laura; Zhao, Xiaolei (August 2023, Journal of the London Mathematical Society)

   Abstract We prove a general criterion that guarantees that an admissible subcategory of the derived category of an abelian category is equivalent to the bounded derived category of the heart of a bounded t‐structure. As a consequence, we show that has a strongly unique dg enhancement, applying the recent results of Canonaco, Neeman, and Stellari. We apply this criterion to the Kuznetsov component when is a cubic fourfold, a GM variety, or a quartic double solid. In particular, we obtain that these Kuznetsov components have strongly unique dg enhancement and that exact equivalences of the form are of Fourier–Mukai type when , belong to these classes of varieties, as predicted by a conjecture of Kuznetsov. 

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3. Stability conditions in families

   https://doi.org/10.1007/s10240-021-00124-6  

   Bayer, Arend; Lahoz, Martí; Macrì, Emanuele; Nuer, Howard; Perry, Alexander; Stellari, Paolo (June 2021, Publications mathématiques de l'IHÉS)

   Abstract We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers. Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type. Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration. 

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4. Some remarks about deformation theory and formality conjecture

   https://doi.org/10.1007/s11565-024-00500-0  

   Chen, Huachen; Pertusi, Laura; Zhao, Xiaolei (February 2024, ANNALI DELL'UNIVERSITA' DI FERRARA)

   Abstract Using the algebraic criterion proved by Bandiera, Manetti and Meazzini, we show the formality conjecture for universally gluable objects with linearly reductive automorphism groups in the bounded derived category of a K3 surface. As an application, we prove the formality conjecture for polystable objects in the Kuznetsov components of Gushel–Mukai threefolds and quartic double solids. 

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5. Kleene Algebra with Commutativity Conditions Is Undecidable

   https://doi.org/10.4230/LIPICS.CSL.2025.36  

   Azevedo_de_Amorim, Arthur; Zhang, Cheng; Gaboardi, Marco (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Endrullis, Jörg; Schmitz, Sylvain (Ed.)

   We prove that the equational theory of Kleene algebra with commutativity conditions on primitives (or atomic terms) is undecidable, thereby settling a longstanding open question in the theory of Kleene algebra. While this question has also been recently solved independently by Kuznetsov, our results hold even for weaker theories that do not support the induction axioms of Kleene algebra. 

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