Stratified Hyperbolicity of the moduli stack of stable minimal models, II: Big Picard Theorem and the stratified Brody hyperbolicity

Abstract: This is the second paper on the global geometry of Birkar's moduli of stable minimal models (e.g., the KSBA moduli stack). We introduces a birationally admissible stratification of the Deligne-Mumford stack of stable minimal models, such that the universal family over each stratum admits a simple normal crossing log birational model. The main result of this paper is to show that for each stratum $S$, the pair $(\overline{S},\partial S)$ satisfies the Big Picard theorem. In particular, we show that each stratum $S$ of the moduli stack is Borel hyperbolic and Brody hyperbolic.