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

TL;DR

The paper proves that most families of smooth projective curves over certain moduli spaces are isotrivial, using complex geometric methods involving symmetric forms and bounded symmetric domains.

Contribution

It establishes the isotriviality of families of curves over $\

Findings

Most such families are isotrivial over the specified moduli spaces.

The proof involves computing the vanishing locus of symmetric forms on complex algebraic varieties.

The result holds for all but finitely many primes and over fields of characteristic zero or p.

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 $2$.