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

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.