K\"ahler-Einstein metrics and volume minimization

TL;DR

This paper proves that certain valuations minimize volume on Fano varieties with K"ahler-Einstein metrics, confirming conjectures on K-semistability and extending results to orbifold and logarithmic cases.

Contribution

It establishes volume minimization at the canonical valuation for degenerations to K"ahler-Einstein Fano varieties, confirming a key conjecture and generalizing previous results.

Findings

Normalized volume is minimized at the canonical valuation for degenerations to KE Fano varieties.

Confirms a conjecture on K-semistability characterization for smooth Fano manifolds.

Valuation from Reeb vector field minimizes volume in Sasaki-Einstein metrics.

Abstract

We prove that if a $\mathbb{Q}$-Fano variety $V$ specially degenerates to a K\"{a}hler-Einstein $\mathbb{Q}$-Fano variety $V$, then for any ample Cartier divisor $H=-r^{-1} K_V$ with $r\in \mathbb{Q}_{>0}$, the normalized volume $\widehat{\rm vol}(v)=A_{\mathcal{C}}^n(v)\cdot {\rm vol}(v)$ is globally minimized at the canonical valuation ${\rm ord}_V$ among all real valuations which are centered at the vertex of the affine cone $\mathcal{C}:=C(V,H)$. This is also generalized to the logarithmic and the orbifold setting. As a consequence, we complete the confirmation of a conjecture in [arXiv:1511.08164] on an equivalent characterization of K-semistability for any smooth Fano manifold. We also prove that the valuation associated to the Reeb vector field of a smooth Sasaki-Einstein metric minimizes $\widehat{\rm vol}$ over the corresponding K\"ahler cone. These results strengthen the minimization result of Martelli-Sparks-Yau [Martelli et al 08].