Positivity, plethysm and hyperbolicity of Siegel varieties in positive characteristic

TL;DR

This paper investigates the hyperbolicity of Siegel modular varieties in positive characteristic, introducing a new positivity concept for vector bundles and applying plethysm operations to establish key properties.

Contribution

It introduces the $( D)$-ampleness notion for vector bundles in characteristic $p$ and demonstrates its effectiveness for automorphic bundles on Siegel varieties.

Findings

Establishment of $( D)$-ampleness for many automorphic vector bundles

Application of plethysm operations to study hyperbolicity

Generalization of positivity results for the Hodge line bundle

Abstract

We study hyperbolicity properties of the moduli space of polarized abelian varieties (also known as the Siegel modular variety) in characteristic $p$. Our method uses the plethysm operation for Schur functors as a key ingredient and requires a new positivity notion for vector bundles in characteristic $p$ called $(\varphi,D)$-ampleness. Generalizing what was known for the Hodge line bundle, we also show that many automorphic vector bundles on the Siegel modular variety are $(\varphi,D)$-ample.