Boundedness of families of canonically polarized manifolds: A higher dimensional analogue of Shafarevich’s conjecture (PLEASE NOTE: The version of record for this article is published as an Erratum in Volume 173, no. 1, pp. 585–617.)

Abstract

We show that the number of deformation types of canonically polarized manifolds over an arbitrary variety with proper singular locus is finite, and that this number is uniformly bounded in any finite type family of base varieties. As a corollary we show that a direct generalization of the geometric version of Shafarevich’s original conjecture holds for infinitesimally rigid families of canonically polarized varieties.