This is a continuation of an earlier work in which we proposed a problem of minimizing normalized volumes over -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 -Fano variety , the K-semistability of is equivalent to the condition that the normalized volume is minimized at the canonical valuation among all -invariant valuations on the cone associated to any positive Cartier multiple of . In this case, we show that is the unique minimizer among all -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 .
"K-semistability is equivariant volume minimization." Duke Math. J. 166 (16) 3147 - 3218, 1 November 2017. https://doi.org/10.1215/00127094-2017-0026