Finiteness of pointed maps to moduli spaces of polarized varieties

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.

15 pages. Substantially revised and improved. The main addition is a new rigidity result for pseudohyperbolic varieties (Theorem 4.3), which significantly simplifies several earlier arguments