论文标题
Hyper-Kähler品种的未脱脂Shafarevich猜想
Unpolarized Shafarevich conjectures for hyper-Kähler varieties
论文作者
论文摘要
SHAFAREVICH的猜想/问题是关于在数字领域定义的同构类别的有限类别的有限型，在有限的地方集合以外降低了良好的降低。对于K3表面，Y。她证明了这样的有限结果。对于K3表面的高维类似物的Hyper-Kähler品种，André证明了Shafarevich的Hyper-Kähler品种的Shafarevich猜想，具有给定维度的高度，并承认有界限的极化。在本文中，我们通过证明（非极化的）Shafarevich猜想在给定的变形类型中为Hyper-Kähler品种提供了两种结果的统一。我们还通过通过共同体的未来替换良好的还原条件来讨论Shafarevich猜想的共同体学概括，在这种情况下，我们的结果受到自动形态群体对同胞的行动的忠诚度的必要假设。以类似的方式，将ORR和Skorobogatov的结果推广到K3表面上，我们证明了CM类型的Hyper-Kähler品种的几何学同构类别的有限性在给定的变形类型中定义为具有有限程度的数字字段定义。我们采用这些结果方法的关键是统一的kuga-satake图，灵感来自她的作品，我们研究了其算术特性，它们具有独立的兴趣。
The Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was proved by Y. She. For hyper-Kähler varieties, which are higher-dimensional analogs of K3 surfaces, Y. André proved the Shafarevich conjecture for hyper-Kähler varieties of a given dimension and admitting a very ample polarization of bounded degree. In this paper, we provide a unification of both results by proving the (unpolarized) Shafarevich conjecture for hyper-Kähler varieties in a given deformation type. We also discuss the cohomological generalization of the Shafarevich conjecture by replacing the good reduction condition by the unramifiedness of the cohomology, where our results are subject to a certain necessary assumption on the faithfulness of the action of the automorphism group on cohomology. In a similar fashion, generalizing a result of Orr and Skorobogatov on K3 surfaces, we prove the finiteness of geometric isomorphism classes of hyper-Kähler varieties of CM type in a given deformation type defined over a number field with bounded degree. A key to our approach to these results is a uniform Kuga--Satake map, inspired by She's work, and we study its arithmetic properties, which are of independent interest.
