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

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 $(\bar{S},\mathrm{\partial }S)$ satisfies the Big Picard theorem. In particular, we show that each stratum $S$ of the moduli stack is Borel hyperbolic and Brody hyperbolic.