K-semistability is equivariant volume minimization

Abstract

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 Kähler-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 ${\mathrm{ord}}_V$ 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 ${\mathrm{ord}}_V$ 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 $V$.

Accepted version

References in corpus (2)

• Sasaki-Einstein Manifolds and Volume Minimisation
• On K-stability and the volume functions of $\mathbb{Q}$-Fano varieties

Cited by in corpus (22)

• Optimal destabilizing centers and equivariant K-stability
• Existence of valuations with smallest normalized volume
• Reductivity of the automorphism group of K-polystable Fano varieties
• The volume of singular Kähler-Einstein Fano varieties
• Uniform K-stability and plt blowups of log Fano pairs
• Optimal destabilization of K-unstable Fano varieties via stability thresholds
• Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces
• K-stability of cubic threefolds
• Birational superrigidity and K-stability of singular Fano complete intersections
• Birational superrigidity and K-stability of Fano complete intersections of index one (with an appendix written jointly with Charlie Stibitz)
• The existence of the Kähler-Ricci soliton degeneration
• K-stability of birationally superrigid Fano varieties
• Product theorem for K-stability
• Delta-invariants for Fano varieties with large automorphism groups
• Greatest Lower Bounds on Ricci Curvature for Fano $T$-manifolds of Complexity $1$
• Higher Codimensional Alpha Invariants and Characterization of Projective Spaces
• A Note On Kähler-Ricci Flow on Fano Threefolds
• On K-stability of ${\mathbb{P}}^3$ blown up along a quintic elliptic curve
• Compactifications of moduli of del Pezzo surfaces via line arrangement and K-stability
• On local delta invariant of del Pezzo surfaces
• Local delta invariants of weak del Pezzo surfaces with the anti-canonical degree $\ge 5$
• K-moduli of Fano threefolds in family 3.10