Finiteness of pointed maps to moduli spaces of polarized varieties

Ariyan Javanpeykar; Steven Lu; Ruiran Sun; Kang Zuo

Finiteness of pointed maps to moduli spaces of polarized varieties

Algebraic Geometry 2025-06-19 v2 Complex Variables

Authors: Ariyan Javanpeykar , Steven Lu , Ruiran Sun , Kang Zuo

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.

Keywords

pointed maps moduli spaces polarized varieties hyperbolicity rigidity maximal variation hom scheme

Cite

@article{arxiv.2310.06784,
  title  = {Finiteness of pointed maps to moduli spaces of polarized varieties},
  author = {Ariyan Javanpeykar and Steven Lu and Ruiran Sun and Kang Zuo},
  journal= {arXiv preprint arXiv:2310.06784},
  year   = {2025}
}

Comments

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

Finiteness of pointed maps to moduli spaces of polarized varieties

Abstract

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