Finiteness theorems for K3 surfaces and abelian varieties of CM type

Abstract: We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational numbers. As an application we confirm finiteness conjectures of Shafarevich and Coleman in the CM case. In addition we prove the uniform boundedness of the Galois invariant subgroup of the geometric Brauer group for forms of a smooth projective variety satisfying the integral Mumford--Tate conjecture. When applied to K3 surfaces, this affirms a conjecture of V\'arilly-Alvarado in the CM case.