On log minimality of weak K-moduli compactifications of Calabi-Yau varieties

TL;DR

This paper proves the log minimality of normalized weak K-moduli compactifications of Calabi-Yau varieties, providing new insights into their structure and including an appendix on Kulikov models and the MMP.

Contribution

It establishes the log minimality of normalized weak K-moduli compactifications under certain conditions, advancing understanding of Calabi-Yau moduli spaces.

Findings

Log minimality of normalized weak K-moduli compactifications

Conditions under which the minimality holds

Reconstruction of Kulikov models via MMP

Abstract

For moduli of polarized smooth K-trivial a.k.a., Calabi-Yau varieties in a general sense, we revisit a classical problem of constructing its "weak K-moduli" compactifications which parametrizes K-semistable (i.e., semi-log-canonical K-trivial) degenerations. Although weak K-moduli is not unique in general, they always contain a unique partial compactification (K-moduli). Our main theorem is the log minimality of their normalizations, under some conditions. Partially to confirm that known examples satisfy the conditions, we also include an appendix on the algebro-geometric reconstruction of Kulikov models via the MMP, which has been folklore at least but we somewhat strengthen.