JournalsjemsVol. 22, No. 8pp. 2573–2627

Stability of valuations and Kollár components

  • Chi Li

    Purdue University, West Lafayette, USA
  • Chenyang Xu

    Massachusetts Institute of Technology, Cambridge, USA
Stability of valuations and Kollár components cover

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Abstract

We prove that among all Kollár components obtained by plt blow ups of a klt singularity o(X,D)o \in (X,D), 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. 2573–2627

DOI 10.4171/JEMS/972