We show that among all Koll\'ar components obtained by plt blow ups of a klt singularity, there is at most one that is (log)-K-semistable. We achieve this by showing that if such a Koll\'ar component exists, it minimizes the normalized volume function introduced in [Li15a]. We also prove that for any klt singularity, the infimum of the normalized function is always approximated by a sequence of Koll\'ar components. Furthermore, if there is a minimizer, then we can choose this sequence of Koll\'ar components to approximate it in the space of valuations.

中文翻译：

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估值和 Kollár 组件的稳定性

我们表明，在通过 klt 奇点的 plt 爆炸获得的所有 Koll\'ar 分量中，至多有一个是 (log)-K-半稳定的。我们通过展示如果存在这样的 Koll\'ar 组件来实现这一点，它会最小化 [Li15a] 中引入的归一化体积函数。我们还证明，对于任何 klt 奇点，归一化函数的下界总是由 Koll\'ar 分量序列逼近。此外，如果有一个极小值，那么我们可以选择这个 Koll\'ar 分量序列在估值空间中近似它。