In his proof of the Mordell conjecture, Faltings deduces it from the finiteness, up to isomorphism, of curves and abelian varieties with good reduction, a statement conjectured by Shafarevich. Since then, analogous finiteness statements, often referred to as “Shafarevich conjectures”, have been proved in a number of specific cases, relying on explicit classification and reduction to Faltings’ theorem. Using a recent technique introduced by Lawrence, Venkatesh, and Sawin, together with T. Krämer we prove the conjecture for a broad class of canonically polarized varieties, which includes all complete intersections in abelian varieties (and therefore eludes any explicit classification). The key geometric ingredient is a big monodromy theorem obtained in joint work with Krämer, Javanpeykar, and Lehn.