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 Kähler-Einstein Fano manifold. In particular, for toric Fano manifolds, the existence of Kähler-Einstein metrics is equivalent to divisorial semistability. Moreover, we find many non-Kähler-Einstein Fano manifolds of dimension three.

36 pages

References in corpus (4)

• Optimal bounds for the volumes of Kähler-Einstein Fano manifolds
• The volume of Kähler-Einstein Fano varieties and convex bodies
• Fano manifolds having (n-1,0)-type extremal rays with large Picard number
• Examples of K-unstable Fano manifolds with the Picard number one

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• A Note On Kähler-Ricci Flow on Fano Threefolds
• K-semistable Fano manifolds with the smallest alpha invariant
• On K-stability of ${\mathbb{P}}^3$ blown up along a quintic elliptic curve