K-semistability is equivariant volume minimization

This is a continuation to the paper [arXiv:1511.08164] in which a problem of minimizing normalized volumes over $\mathbb{Q}$-Gorenstein klt singularities was proposed. Here we consider its relation with K-semistability, which is an important concept in the study of K"{a}hler-Einstein metrics on Fano varieties. In particular, we prove that for a $\mathbb{Q}$-Fano variety $V$, the K-semistability of $(V,-K_V)$ is equivalent to the condition that the normalized volume is minimized at the canonical valuation ${\mathrm{o}\mathrm{r}\mathrm{d}}_V$ among all ${\mathbb{C}}^{\ast }$-invariant valuations on the cone associated to any positive Cartier multiple of $-K_V$. In this case, it’s shown that ${\mathrm{o}\mathrm{r}\mathrm{d}}_V$ is the unique minimizer among all ${\mathbb{C}}^{\ast }$-invariant quasi-monomial 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 $V$.