On K-stability and the volume functions of $\mathbb{Q}$-Fano varieties

Abstract: We introduce a new effective stability named "divisorial stability" for Fano manifolds which is weaker than K-stability and is stronger than slope stability along divisors. We show that we can test divisorial stability via the volume function. As a corollary, we prove that the first coordinate of the barycenter of the Okounkov body of the anticanonical divisor is not bigger than one for any K\"ahler-Einstein Fano manifold. In particular, for toric Fano manifolds, the existence of K\"ahler-Einstein metrics is equivalent to divisorial semistability. Moreover, we find many non-K\"ahler-Einstein Fano manifolds of dimension three.