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Stability of valuations and Kollár components. (English) Zbl 1471.14076

The authors investigate klt singularities using tools from the theory of K-stability. This is natural because klt singularities are a local analog of Fano varieties. Let \(o\in (X,D)\) be a klt singularity, then a proper birational morphism \(\mu :Y\to X\) provides a Kollár component \(S\) if \(\mu\) is an isomorphism over \(X\setminus \{o\}\), \(\mu ^{-1}(o)=S\) is a \(\mathbb Q\)-Cartier prime divisor such that \((Y,S+\mu ^{-1}D)\) is plt and \(-S\) is \(\mu\)-ample. Such components always exist and have the structure of a (log-)Fano variety. The authors prove that amongst all Kollár components of a klt singularity \(o\in (X,D)\) there is at most one such component that is (log-)K-semistable. This component corresponds to the unique divisorial valuation that minimizes the normalized volume function introduced by Chi Li.

MSC:

14J17 Singularities of surfaces or higher-dimensional varieties
13A18 Valuations and their generalizations for commutative rings
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