Stability conditions in families

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