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.

我们开发了一个布里奇兰稳定性条件和半稳定对象的模空间的理论。我们的方法基于并概括了Abramovich–Polishchuk，Kuznetsov，Lieblich和Piyaratne–Toda的先前工作。我们的概念包括稳定性的开放性，半稳定的减少，整个族群的均匀支撑特性以及半稳定对象的有界性。我们表明，只要已知在纤维上存在稳定条件，就存在这种结构。

我们的主要应用是将Mukai理论在K3曲面上的半稳定绳轮的模空间推广到与三次三次方相关的Kuznetsov分量中Bridgeland半稳定物体的模空间。这导致Addington-Thomas和Huybrechts扩展了特殊三次三次折叠的定理的定理，为新的Hodge猜想提供了新的证明，并构造了无穷的无穷级的极化超kähler流形的局部完全族。 K3型。

其他应用包括唐纳森-托马斯不变量的变形不变性，其对卡拉比–丘上的布里奇兰稳定物体进行三倍计数，以及通过退化构造三倍稳定性条件的方法。