K-SEMISTABILITY IS EQUIVARIANT VOLUME MINIMIZATION

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 Kohler 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.