We prove two new results on the
K
K
-polystability of
Q
\mathbb {Q}
-Fano varieties based on purely algebro-geometric arguments. The first one says that any
K
K
-semistable log Fano cone has a special degeneration to a uniquely determined
K
K
-polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to complete the proof of Donaldson-Sun’s conjecture which says that the metric tangent cone of any point appearing on a Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds depends only on the algebraic structure of the singularity. The second result says that for any log Fano variety with the torus action,
K
K
-polystability is equivalent to equivariant
K
K
-polystability, that is, to check
K
K
-polystability, it is sufficient to check special test configurations which are equivariant under the torus action.