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

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-Păun. 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.

arXiv comment34 pages, final version merges arXiv:1809.05891; title changed. To appear in JEMS

References in corpus (12)

• On the isotriviality of families of projective manifolds over curves
• Minimal models and boundedness of stable varieties
• Bergman kernels and subadjunction
• Positivity of relative canonical bundles for families of canonically polarized manifolds
• Viehweg's hyperbolicity conjecture is true over compact bases
• Positivity properties of the bundle of logarithmic tensors on compact Kähler manifolds
• Pseudo Kobayashi hyperbolicity of base spaces of families of minimal projective manifolds with maximal variation
• Moduli of canonically polarized manifolds, higher order Kodaira-Spencer maps, and an analogy to Calabi-Yau manifolds
• A note on Lang's conjecture for quotients of bounded domains
• Hyperbolicity of bases of log Calabi-Yau families
• On subvarieties of singular quotients of bounded domains
• Fujita decomposition over higher dimensional base

Cited by in corpus (3)

• The Shafarevich conjecture revisited: Finiteness of pointed families of polarized varieties
• Hyperbolicity of bases of log Calabi-Yau families
• Big Picard theorem for moduli spaces of polarized manifolds