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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
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