Hyperbolicity of varieties supporting a variation of Hodge structure

TL;DR

This paper extends previous work on the hyperbolicity of varieties with a variation of Hodge structure, utilizing curvature properties of period domains without relying on asymptotic Hodge metric behavior.

Contribution

It generalizes earlier results by proving hyperbolicity properties using only curvature of period domains, avoiding asymptotic Hodge metric analysis.

Findings

Varieties with a variation of Hodge structure exhibit hyperbolicity.

The proof relies solely on curvature properties of period domains.

No asymptotic Hodge metric results are used in the proof.

Abstract

We generalize former results of Zuo and the first author showing some hyperbolicity properties of varieties supporting a variation of Hodge structure. Our proof only uses the special curvature properties of period domains. In particular, in contrast to the former approaches, it does not use any result on the asymptotic behaviour of the Hodge metric.