Semi-toric and toroidal compactifications as log minimal models, and applications to weak K-moduli

TL;DR

This paper characterizes certain compactifications as log minimal models and applies these results to study weak K-moduli, providing new proofs and exploring generalizations related to moduli of polarized Enriques surfaces.

Contribution

It offers a new perspective on toroidal and semi-toric compactifications as log minimal models and applies this to weak K-moduli, including a novel proof of a key theorem.

Findings

Characterization of toroidal and semi-toric compactifications as log minimal models

Application to weak K-moduli compactifications with a new proof of a theorem

Discussion on generalizations and compatibility with log K-stability

Abstract

We give a characterization of toroidal (resp., semi-toric) compactifications due to Ash-Mumford-Rapoport-Tai (resp., Looijenga) as log minimal models and apply it to study weak K-moduli compactifications, giving a different proof to a theorem of Alexeev-Engel. We also discuss towards further generalization, in particular revisit Shah-Sterk compactification of moduli of polarized Enriques surfaces to show compatibility with log K-stability.