Finiteness of pointed maps to moduli spaces of polarized varieties

TL;DR

This paper proves a finiteness property for pointed maps to moduli spaces of polarized varieties, leveraging hyperbolicity and Viehweg's conjecture, with implications for the structure of such maps.

Contribution

It establishes a finiteness theorem for pointed maps to moduli spaces using hyperbolicity and Viehweg's conjecture, providing new bounds on the Hom scheme.

Findings

Finiteness of pointed maps to moduli spaces of polarized varieties.

Rigidity of pointed maps to hyperbolic varieties.

Optimal dimension bounds on the Hom scheme from curves.

Abstract

We establish a finiteness result for pointed maps to the base space $U$ of a smooth projective family of varieties with maximal variation in moduli. For its proof, we establish the rigidity of pointed maps to a (not necessarily compact) variety which is hyperbolic modulo a proper closed subset. Together with Viehweg's hyperbolicity conjecture on the bigness of log-canonical bundles of moduli spaces, resolved by Campana-Paun, we derive an optimal dimension bound on the Hom scheme from a curve to $U$ among other applications.