Abstract:We prove two new results on the $K$-polystability of $\mathbb {Q}$-Fano varieties based on purely algebro-geometric arguments. The first one says that any $K$-semistable log Fano cone has a special degeneration to a uniquely determined $K$-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, $K$-polystability is equivalent to equivariant $K$-polystability, that is, to check $K$-polystability, it is sufficient to check special test configurations which are equivariant under the torus action.

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中文翻译：

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公制切锥的代数性和等变 K 稳定性

摘要：我们基于纯代数几何参数证明了 $\mathbb {Q}$-Fano 变体的 $K$-多稳定性的两个新结果。第一个说任何 $K$-semistable log Fano 锥都有一个特殊的退化为唯一确定的 $K$-polystable log Fano 锥。作为推论，我们将其与微分几何结果相结合以完成唐纳森-孙猜想的证明，即出现在 Kähler-Einstein Fano 流形的 Gromov-Hausdorff 极限上的任何点的度量切锥仅取决于代数奇点的结构。第二个结果表明，对于任何具有环面作用的 log Fano 变体，$K$-polystability 等价于等变 $K$-polystability，即检查 $K$-polystability，检查特殊的测试配置就足够了在环面作用下是等变的。

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