Hyperbolicity of the base of an admissible family of log canonical stable minimal models

We investigate the stratified hyperbolicity properties of Birkar’s moduli stack of log canonical (lc) stable minimal models. The main technical result is a construction of Viehweg-Zuo’s system of Higgs sheaves associated with an admissible family of lc stable minimal models, using the theory of degenerations of Hodge structure and non-abelian Hodge theory.