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 hyperkahler 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.

Stability conditions in families / A. Bayer, M. Lahoz, E. Macrì, H. Nuer, A. Perry, P. Stellari. - In: PUBLICATIONS MATHEMATIQUES. - ISSN 0073-8301. - 133:1(2021 Jun), pp. 157-325. [10.1007/s10240-021-00124-6]

Stability conditions in families

P. Stellari
2021-06

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 hyperkahler 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.
Settore MAT/03 - Geometria
Moduli and Lie Theory
FARE20PSTEL_01 - Higher categorical and stability structures in algebraic geometry (HighCaSt) - STELLARI, PAOLO - FARE -FARE Ricerca in Italia - 2020
H2020_ERC18PSTEL_01 - Stability Conditions, Moduli Spaces and Enhencements (StabCondEn) - STELLARI, PAOLO - H2020_ERC - Horizon 2020_Europern Research Council - 2018
mag-2021
PUBLICATIONS MATHEMATIQUES
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/860033
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