On the hyperbolicity of base spaces for maximally variational families of smooth projective varieties

TL;DR

This paper proves that base spaces of certain smooth projective families are pseudo Kobayashi hyperbolic, confirming conjectures relating hyperbolicity, log general type, and moduli space properties, thus advancing understanding of hyperbolic geometry in algebraic geometry.

Contribution

It establishes the pseudo Kobayashi hyperbolicity of base spaces for maximally variational families, confirming conjectures by Viehweg-Zuo and generalizing previous results.

Findings

Base spaces are pseudo Kobayashi hyperbolic, aligning with the Lang conjecture.

Proves Brody hyperbolicity of moduli spaces of polarized manifolds with semi-ample canonical bundle.

Establishes Kobayashi hyperbolicity of base spaces for effectively parametrized families of minimal projective manifolds.

Abstract

For smooth families with maximal variation, whose general fibers have semi-ample canonical bundle, the generalized Viehweg hyperbolicity conjecture states that the base spaces of such families are of log general type. This deep conjecture was recently proved by Popa-Schnell using the theory of Hodge modules and a theorem by Campana-P\u{a}un. In this paper we prove that those base spaces are pseudo Kobayashi hyperbolic, as predicted by the Lang conjecture: any complex quasi-projective manifold is pseudo Kobayashi hyperbolic if it is of log general type. As a consequence, we prove the Brody hyperbolicity of moduli spaces of polarized manifolds with semi-ample canonical bundle. This proves a conjecture by Viehweg-Zuo in 2003. We also prove the Kobayashi hyperbolicity of base spaces of effectively parametrized families of minimal projective manifolds of general type. This generalizes previous work by To-Yeung, in which they further assumed that these families are canonically polarized.