## K-semistability is equivariant volume minimization.(English)Zbl 1409.14008

Let $$V$$ be a complex projective manifold that is Fano, i.e. the anticanonical bundle $$-K_V$$ is ample. The problem of describing the situations when $$V$$ admits a special metric has received a lot of attention in the last years, culminating in the proof of the equivalence of $$K$$-stability and existence of Kähler-Einstein metrics by X. Chen et al. [J. Am. Math. Soc. 28, No. 1, 183–197 (2015; Zbl 1312.53096)] and G. Tian [Commun. Pure Appl. Math. 68, No. 7, 1085–1156 (2015; Zbl 1318.14038)]. In this paper the author proposes to describe $$K$$-semistability in terms of valuations: for a Fano manifold $$V$$, one considers the space of real valuations $$\text{Val}_{X,o}$$ on the affine cone $$X=C(V, -K_V)$$ with center the vertex $$o$$. On the space of valuations the normalized volume functional $$\widehat{\text{vol}}$$ is defined to be $$A_X(v)^n \text{vol}(v)$$ where $$A_X(v)$$ is the log-discrepancy of $$v$$ and $$\text{vol}(v)$$ its volume. In [Math. Z. 289, No. 1–2, 491–513 (2018; Zbl 1423.14025)] the author conjectured that the Fano manifold $$V$$ is $$K$$-semistable if and only if $$\widehat{\text{vol}}$$ is minimized at the canonical valuation $$\text{ord}_V$$. In this paper he proves that $$V$$ is $$K$$-semistable if and only if $$\widehat{\text{vol}}$$ is minimized among $$\mathbb C^*$$-invariant valuations. Moreover $$\text{ord}_V$$ is the unique minimizer among all $$\mathbb C^*$$-invariant quasimonomial valuations. We refer to the beautifully written paper for the definitions of these notions and a detailed description of the proofs. Combined with the author’s recent preprint with Y. Liu [“Kähler-Einstein metrics and volume minimization”, Preprint, arXiv:1602.05094], the results of this paper establish his conjecture.

### MSC:

 14B05 Singularities in algebraic geometry 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 13A18 Valuations and their generalizations for commutative rings 52A27 Approximation by convex sets

### Citations:

Zbl 1312.53096; Zbl 1318.14038; Zbl 1423.14025
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