Unpolarized Shafarevich conjectures for hyper-Kähler varieties

Abstract

We prove the unpolarized Shafarevich conjecture for hyper-Kähler varieties of a fixed deformation type, unifying earlier results for K3 surfaces and polarized hyper-Kähler varieties.We also study a cohomological variant in which good reduction is replaced by unramifiedness of cohomology; the resulting finiteness statements depend on the faithfulness of the automorphism action on cohomology.For hyper-Kähler varieties of CM type, we prove finiteness of geometric isomorphism classes in a fixed deformation type over number fields of bounded degree, extending a theorem of Orr and Skorobogatov for K3 surfaces.Our main tool is a uniform Kuga-Satake map, whose arithmetic properties are developed along the way.