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

TL;DR

This paper introduces a stratification of the moduli stack of stable minimal models, showing each stratum's hyperbolic properties and providing insights into the global geometry of these moduli spaces.

Contribution

It presents a birationally admissible stratification of the moduli stack with hyperbolic properties, advancing understanding of the moduli's global geometric structure.

Findings

Each stratum admits a simple normal crossing log birational model.

Every schematic generically finite covering of a closed substack is of logarithmic general type.

Provides partial answers to questions about the global geometry of the moduli of stable minimal models.

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.