Boundedness of families of canonically polarized manifolds: A higher dimensional analogue of Shafarevich's conjecture

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.

19 pages, v2: minor changes; one reference added to make a step cleaner and one name added to acknowledgements, v3: final version to appear in Annals of Mathematics

References in corpus (1)

• Twisted sheaves and the period-index problem

Cited by in corpus (6)

• Demailly's notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps
• Period and index in the Brauer group of an arithmetic surface (with an appendix by Daniel Krashen)
• Compactifications of smooth families and of moduli spaces of polarized manifolds
• Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures
• Arakelov (in)equalities
• Arakelov-Parshin rigidity of towers of curve fibrations