Stability conditions in families
Stability conditions in families
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DOI：
10.1007/s10240-021-00124-6
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发表时间：
2019-02
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影响因子：
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通讯作者：
Arend Bayer;Mart'i Lahoz;Emanuele Macrì;H. Nuer;Alexander Perry;P. Stellari
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作者：
Arend Bayer;Mart'i Lahoz;Emanuele Macrì;H. Nuer;Alexander Perry;P. Stellari
我们发展了一个理论的Bridgeland稳定性条件和模空间的半稳定对象的一个家庭的品种。我们的方法是基于和推广以前的工作由阿布拉莫维奇Polishchuk，库兹涅佐夫，Lieblich和Piyaratne-Toda。我们的概念包括开放性的稳定性，半稳定的减少，支持属性均匀的家庭，半稳定对象的有界性。我们表明，这样的结构存在时，稳定性条件是已知存在的纤维。我们的主要应用是Mukai的理论的模空间的K3表面上的半稳定层的Bridgeland半稳定对象的模空间的Kuznetsov组件相关联的立方四倍。推广了Addington-Thomas和Huybrechts关于特殊三次四重流形的导出范畴的定理，给出了积分Hodge猜想的新证明，构造了K3型极化超kähler流形的单有理局部完备族的无穷级数.其他应用包括计算Calabi-Yau三重流形上Bridgeland稳定对象的Donaldson-Thomas不变量的形变不变性，以及通过退化构造三重稳定性条件的方法。
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.