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We prove that among all Kollár 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 Kollár component exists, it uniquely minimizes the normalized volume function introduced in [Li18] among all divisorial valuations. Conversely, we show that any divisorial minimizer of the normalized volume function yields a K-semistable Kollár component. We also prove that for any klt singularity, the infimum of the normalized volume function is always approximated by the normalized volumes of Kollár components.
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Chi Li, Chenyang Xu, Stability of valuations and Kollár components. J. Eur. Math. Soc. 22 (2020), no. 8, pp. 2573–2627