论文标题
Kuznetsov通过K3类别和增强的小组动作的Fano三重猜想
Kuznetsov's Fano threefold conjecture via K3 categories and enhanced group actions
论文作者
论文摘要
我们解决了Kuznetsov的最后一个开放式猜想，这是Fano三倍的派生类别。与原始的猜想相反，我们证明了四分之一固体和Gushel-Mukai三倍的Kuznetsov组件永远不会等效，正如张最近独立显示的那样。另一方面，我们证明了修改后的猜想，主张其变形等效性。我们的不重要证明将一个分类的Enriques-K3对应与霍奇类别理论结合在一起。在此过程中，我们获得了Gushel-Mukai品种时期的分类描述，我们用来解决Kuznetsov的猜想，以及对Birational类别Torelli问题的第二作者，并简单地证明了Deparre和Kuznetsov在该周期映射的纤维上的简单证明。我们的变形等效证明取决于对障碍的独立兴趣结果，以增强类别的群体行动。
We settle the last open case of Kuznetsov's conjecture on the derived categories of Fano threefolds. Contrary to the original conjecture, we prove the Kuznetsov components of quartic double solids and Gushel-Mukai threefolds are never equivalent, as recently shown independently by Zhang. On the other hand, we prove the modified conjecture asserting their deformation equivalence. Our proof of nonequivalence combines a categorical Enriques-K3 correspondence with the Hodge theory of categories. Along the way, we obtain a categorical description of the periods of Gushel-Mukai varieties, which we use to resolve a conjecture of Kuznetsov and the second author on the birational categorical Torelli problem, as well as to give a simple proof of a theorem of Debarre and Kuznetsov on the fibers of the period map. Our proof of deformation equivalence relies on results of independent interest about obstructions to enhancing group actions on categories.
