The Shafarevich conjecture revisited: Finiteness of pointed families of polarized varieties

Abstract

Motivated by Lang-Vojta's conjectures on hyperbolic varieties, we prove a new version of the Shafarevich conjecture in which we establish the finiteness of pointed families of polarized varieties. We then give an arithmetic application to the finiteness of integral points on moduli spaces of polarized varieties.

Non-archimedean hyperbolicity of the moduli space of curves

“…Motivated by conjectures of Green-Griffiths-Lang and higher dimensional generalizations of the Shafarevich problem, there has recently been much work on different notions of hyperbolicity [Lan86,Jav20] and the verification of them on the moduli spaces of smooth projective varieties [KL11,CP15,JSZ]. In this paper, we study the non-archimedean analogue of Borel hyperbolicity introduced in [JK20], and verify it for M…”

Hodge Theory and Higgs Bundle on Moduli Spaces of Smooth Varieties with Semi-Ample Canonical Line Bundles, Negativity and Hyperbolicity

“…(6) Very recently Javanpeykar-Sun-Zuo [22] proved a new version of the Shafarevich conjecture, the socalled finiteness of pointed families f : X → Y of h-dimensional polarized varieties. A log map from a log curve φ : (C, S C ) → (Y, S) has only the trivial deformation φ t if φ t fixes no less than…”

Unpolarized Shafarevich conjectures for hyper-Kähler varieties

“…intermediate Jacobian): complete intersections of Hodge niveau ≤ 1, prime Fano threefolds of index 2, sextic surfaces etc. These results can often be reinterpreted as the finiteness of O F,S -points in certain moduli spaces; see works of Javanpeykar and his coauthors [29], [27], [30], [31] for related studies from this point of view of arithmetic hyperbolicity. More recently, Lawrence and Sawin [37] proved some analogous finiteness result for hypersurfaces in abelian varieties based on the techniques in [38].…”