Unpolarized Shafarevich conjectures for hyper-Kähler varieties

Abstract: 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-K\"ahler varieties, which are higher-dimensional analogs of K3 surfaces, Y. Andr\'e has verified the Shafarevich conjecture for hyper-K\"ahler 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-K\"ahler varieties in a given deformation type. In a similar fashion, generalizing a result of Orr and Skorobogatov on K3 surfaces, we prove the finiteness of geometric isomorphism classes of hyper-K\"ahler varieties of CM type in a given deformation type defined over a number field with bounded degree. A key to our approach is a uniform Kuga--Satake map, inspired by She's work, and we study its arithmetic properties, which are of independent interest.