On the hyperbolicity of base spaces for maximally variational families of smooth projective varieties (with an appendix by Dan Abramovich) (Q2135432)

Summary: For maximal variational smooth families of projective manifolds whose general fibers have semi-ample canonical bundle, the Viehweg hyperbolicity conjecture states that the base spaces of such families are of log-general type. This deep conjecture was recently proved by Campana-Păun and was later generalized by Popa-Schnell. 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 2003 conjecture by \textit{E. Viehweg} and \textit{K. Zuo} [Duke Math. J. 118, No. 1, 103--150 (2003; Zbl 1042.14010)]. We also prove the Kobayashi hyperbolicity of base spaces for effectively parametrized families of minimal projective manifolds of general type. This generalizes previous work by To-Yeung, who further assumed that these families are canonically polarized.