Isotriviality of families of curves parametrized by $\mathcal{A}_g(n)$

TL;DR

The paper proves that for integers $g,h\ge 2$ and $n\ge 3$, all but finitely many primes $p$ yield isotrivial separable families of genus $h$ curves over $\mathcal{A}_g(n)\otimes k$ in characteristic $0$ or $p$. The core method analyzes absolutely logarithmic symmetric forms on log compactifications and uses base-locus theory, augmented base loci, and the growth properties of differential forms to force vanishing constraints that imply isotriviality. It extends Mok's base-locus results to noncompact and arithmetic settings via real polarized variations of Hodge structures and PVHS techniques, including a generalization to integral Shimura-type models. The work connects the geometry of bounded symmetric domains, Hodge-theoretic metrics, and log-geometry to derive global rigidity results for families over moduli spaces like $\mathcal{A}_g(n)$. This yields new isotriviality criteria in the noncompact and arithmetic context and provides a framework for applying Hodge-theoretic growth estimates to moduli problems across characteristics.

Abstract

We prove that for every integers $g, h\geq 2, n \geq 3$, for all but finitely many prime numbers $p$, for every field $k$ of characteristic $0$ or $p$, every separable family of smooth projective curves of genus $h$ over $\mathcal{A}_g(n) \otimes k$ is isotrivial. To prove this, we compute the common vanishing locus of the absolutely logarithmic symmetric forms on a smooth complex algebraic variety whose universal covering is biholomorphic to an irreducible bounded symmetric domain of rank at least $2$.

Theorems & Definitions (80)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• Lemma 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• proof