An algebraic approach to the hyperbolicity of moduli stacks of Calabi-Yau varieties

TL;DR

This paper introduces an algebraic method to establish hyperbolicity properties of moduli stacks of Calabi-Yau varieties by leveraging reduction to positive characteristic and nonabelian Hodge theory, bypassing complex analytic techniques.

Contribution

It presents a novel algebraic approach to hyperbolicity of Calabi-Yau moduli stacks using positive characteristic reduction and nonabelian Hodge theory.

Findings

Hyperbolicity results proven algebraically.

Reduction to positive characteristic is effective.

Utilizes nonabelian Hodge theory in positive characteristic.

Abstract

The moduli stacks of Calabi-Yau varieties are known to enjoy several hyperbolicity properties. The best results have so far been proven using sophisticated analytic tools such as complex Hodge theory. Although the situation is very different in positive characteristic (e.g. the moduli stack of principally polarized abelian varieties of dimension at least 2 contains rational curves), we explain in this note how one can prove many hyperbolicity results by reduction to positive characteristic, relying ultimately on the nonabelian Hodge theory in positive characteristic developed by Ogus and Vologodsky.