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.
Cite this article
Chi Li, Chenyang Xu, Stability of valuations and Kollár components. J. Eur. Math. Soc. 22 (2020), no. 8, pp. 2573–2627DOI 10.4171/JEMS/972