Brody hyperbolicity of base spaces of certain families of varieties

TL;DR

This paper proves that certain base spaces of smooth families of minimal varieties of general type are Brody hyperbolic, meaning they do not contain Zariski dense entire curves, which has implications for the hyperbolicity of related moduli stacks.

Contribution

It establishes Brody hyperbolicity for base spaces of families of varieties of general type with maximal variation, answering a special case of a question by Viehweg and Zuo.

Findings

Base spaces of smooth families of minimal varieties of general type are Brody hyperbolic.

Moduli stacks of polarized varieties of this type are Brody hyperbolic.

Results extend to two-dimensional bases with families admitting a good minimal model.

Abstract

We prove that quasi-projective base spaces of smooth families of minimal varieties of general type with maximal variation do not admit Zariski dense entire curves. We deduce the fact that moduli stacks of polarized varieties of this sort are Brody hyperbolic, answering a special case of a question of Viehweg and Zuo. For two-dimensional bases, we show analogous results in the more general case of families of varieties admitting a good minimal model.