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

TL;DR

This paper establishes that each stratum of the moduli stack of stable minimal models is Borel and Brody hyperbolic, satisfying the Big Picard theorem, thus advancing understanding of its global geometric properties.

Contribution

It introduces a stratification of the moduli stack ensuring each stratum satisfies the Big Picard theorem, revealing new hyperbolicity properties of the stack.

Findings

Each stratum satisfies the Big Picard theorem.

Strata are Borel hyperbolic.

Strata are Brody hyperbolic.

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.