Uniformly effective boundedness of Shafarevich Conjecture-type for families of canonically polarized manifolds

Abstract

The main result of this note is an effective uniform bound for the number of deformation types of certain nonisotrivial families of canonically polarized manifolds. It extends the author's earlier such bound for the classical Shafarevich Conjecture over function fields to the case of higher dimensional fibers, but without the disadvantageous iterated use of Chow or Hilbert varieties that was the core of the proofs in the earlier approaches.