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

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