K-semistability is equivariant volume minimization

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This is a continuation of an earlier work in which we proposed a problem of minimizing normalized volumes over Q-Gorenstein Kawamata log terminal singularities. Here we consider its relation with K-semistability, which is an important concept in the study of Kähler-Einstein metrics on Fano varieties. In particular, we prove that for a Q-Fano variety V, the K-semistability of (V,-KV) is equivalent to the condition that the normalized volume is minimized at the canonical valuation ordV among all ℂ*-invariant valuations on the cone associated to any positive Cartier multiple of -KV. In this case, we show that ordV is the unique minimizer among all ℂ*-invariant quasimonomial valuations. These results allow us to give characterizations of K-semistability by using equivariant volume minimization, and also by using inequalities involving divisorial valuations over V.

Original languageEnglish (US)
Pages (from-to)3147-3218
Number of pages72
JournalDuke Mathematical Journal
Issue number16
StatePublished - Nov 1 2017
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


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