The K-stability notion of polarized varieties (variety with ample line bundle datum) emerged in differential geometry to give the equivalent of existence of "canonical Kahler metric '', more precisely, cscK metric. Generally, now we start to understand that MMP-based birational geometry is quite compatible with K-stability, which sometimes gives certain refinements. I expect this interaction lasts for several more decades. This particular 10years witnessed big developments of the study of anticanonically polarized cases (automatically/equivalently KE on Fano) with increasing numbers of researchers, while for other general cases, we even can't answer many basic questions. My temporary plan is to explain some among the following mutually related topics.

1. to neatly explain/revisit O-Spotti-Sun (basic ideas of modern approach to Fano K-moduli, with examples illustration, one remark),

2. CM degree minimization conjecture,

3. arithmetic K-stability theory (Generalized Faltings height) / Non-archimedean aspect of K-stability

4. Non-compact version of K-polystability for log Calabi-Yau varieties (cf., https://arxiv.org/abs/2009.13876)

5. (Probably I do not choose: too old) NA Entropy term and SLC singularity

6. (Probably I do not choose: not directly related) recent joint work on Non-variety compactification of moduli of K3 surfaces

 

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