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

TL;DR

This paper establishes an effective uniform bound on the number of deformation types for certain nonisotrivial families of canonically polarized manifolds, extending previous results to higher dimensions without complex auxiliary constructions.

Contribution

It provides a new, more efficient proof of uniform boundedness for families of higher-dimensional canonically polarized manifolds, avoiding iterative Chow or Hilbert variety methods.

Findings

Effective bound for deformation types established

Extension from classical to higher-dimensional fibers

Avoids iterative Chow or Hilbert variety techniques

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.