Unpolarized Shafarevich conjectures for hyper-K\"ahler varieties

TL;DR

This paper proves the unpolarized Shafarevich conjecture for hyper-K"ahler varieties of a fixed deformation type, extending known results for K3 surfaces and polarized cases, and explores related cohomological and CM type finiteness results.

Contribution

It unifies and extends the Shafarevich conjecture to unpolarized hyper-K"ahler varieties and studies the arithmetic properties of a uniform Kuga--Satake map.

Findings

Proves the unpolarized Shafarevich conjecture for hyper-K"ahler varieties in a fixed deformation type.

Establishes finiteness of hyper-K"ahler varieties of CM type over number fields with bounded degree.

Studies the arithmetic properties of a uniform Kuga--Satake map.

Abstract

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-K\"ahler varieties, which are higher-dimensional analogs of K3 surfaces, Y. Andr\'e proved 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. 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-K\"ahler 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.