Stability of Valuations and Kollár Components

2016/04/19 by Li, Chi, Xu, Chenyang · 4 citations
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1604.05398

Abstract

We prove that among all Kollár components obtained by plt blow ups of a klt singularity o ∈ (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 [Li15a] among all divisorial valuations. Conversely, we show 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 function is always approximated by the normalized volumes of Kollár components.

Cited by

• Optimal bounds for local volumes of threefold singularities
• On the uniform positivity of F-signature under reduction modulo p
• A valuative approach to the anticanonical minimal model program
• On volumes and the generic invariance of Fano type varieties

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• On boundedness of singularities and minimal log discrepancies of Kollár components
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