Stratified Hyperbolicity of the moduli stack of stable minimal models, I

Abstract: In this paper, we introduce a birationally admissible stratification on the Deligne-Mumford stack of stable minimal models (e.g., the KSBA moduli stack), such that the universal family over each stratum admits a simple normal crossing log birational model. We further demonstrate that each stratum is hyperbolic in the sense that every schematic generically finite covering of any closed substack is of logarithmic general type. This provides a partial answer to C.Birkar's question regarding the global geometry of the moduli of stable minimal models.