Stability conditions in families

Arend Bayer, Martí Lahoz, Emanuele Macrì, Howard Nuer, Alexander Perry, Paolo Stellari

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)157-325
Number of pages169
JournalPublications Mathematiques de l'Institut des Hautes Etudes Scientifiques
Issue number1
StatePublished - 17 May 2021

All Science Journal Classification (ASJC) codes

  • General Mathematics


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