Optimal destabilizing centers and equivariant K-stability

We give an algebraic proof of the equivalence of equivariant K-semistability (resp. equivariant K-polystability) with geometric K-semistability (resp. geometric K-polystability). Along the way we also prove the existence and uniqueness of minimal optimal destabilizing centers on K-unstable log Fano pairs.

Automated summary

This study provides an algebraic proof of the equivalence between equivariant K-semistability (or equivariant K-polystability) and geometric K-semistability (or geometric K-polystability), and also proves the existence and uniqueness of minimal optimal destabilizing centers on K-unstable log Fano pairs.