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.


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
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