A non-Archimedean approach to K-stability

By S. Boucksom, Mattias Jonsson. 2018

80 citations recorded by the source corpus.

We study K-stability properties of a smooth Fano variety X using non-Archimedean geometry, specifically the Berkovich analytification of X with respect to the trivial absolute value on the ground field. More precisely, we view K-semistability and uniform K-stability as conditions on the space of plurisubharmonic (psh) metrics on the anticanonical bundle of X. Using the non-Archimedean Calabi-Yau theorem and the Legendre transform, this allows us to give a new proof that K-stability is equivalent to Ding stability. By choosing suitable psh metrics, we also recover the valuative criterion of K-stability by Fujita and Li. Finally, we study the asymptotic Fubini-Study operator, which associates a psh metric to any graded filtration (or norm) on the anticanonical ring. Our results hold for arbitrary smooth polarized varieties, and suitable adjoint/twisted notions of K-stability and Ding stability. They do not rely on the Minimal Model Program.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bdc9e6345bd70ba8039da33e3e9eb243f9e541f6

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In Geometry and Topology

1. Chi Li
2. Mattias Jonsson
3. H. Blum
4. Yuchen Liu
5. S. Boucksom
6. Ziquan Zhuang
7. Robert J. Berman
8. Chenyang Xu

In Mathematical Physics

1. Chi Li
2. H. Blum
3. Mattias Jonsson
4. S. Boucksom
5. Yuchen Liu
6. G. Tian
7. Feng Wang
8. Dennis Eriksson

Citations in the corpus

• A Guided Tour to Normalized Volume (2018, background): “, YUCHEN LIU, AND CHENYANG XU related to the metric tangent cone (see section 5.2); while the valuation on K(S) yielding δ(S,B) is related to the the existence of twisted Kahler-Einstein metrics (see [BJ18]). For a log Fano pair (S,B) and a cone x ∈(X,D) = C(S,B;−r(KS + B)), if (S,B) is not K-semist”
• Geodesic rays and stability in the cscK problem (2020, background): “arieties πX : X → Csatisfying: 3 (i) There is a C∗-action on X such that πX is C∗-equivariant; (ii) There is a C∗-equivariant isomorphism X \X0 ∼= X×C∗. This is also called a test configuration of Xin [16]. A model Y is called dominating if there exists a C∗-equivariant birational morphism µY : Y → X”
• The Uniform Version of Yau–Tian–Donaldson Conjecture for Singular Fano Varieties (2019, methodology): “ non-Archimedean functional for test configurations: for functional F appearing in (51)-(55): FNA(φ (Z,L)) = F NA(Z,Q,L). We will need the valuative criterion for the uniform Ding-stability studied in [15, 30]. Let Zbe a projective variety with polarization L. For any divisorial valuation ordE over Z”
• Thresholds, valuations, and K-stability (2017, background): “ystability replaced by K-polystability) in [CDS15, Tia15]. For a general ample line bundle Lon a smooth complex projective variety, the stability threshold (L) detects Ding stability in the sense of [BJ18b] and has the following analytic interpretation.2 Let (L) be the greatest Ricci lower bound, i.”
• Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry (2018, result): “dean case, the main examples are K = Qp, Cp, or fields of formal Laurent series and their completed algebraic closure, but the trivially valued case is also important for the study of K-stability (see [BJ18b]). Let X be a geometrically reduced projective scheme over K, Lbe a line bundle on X, and φ,ψ”
• Uniqueness of K-polystable degenerations of Fano varieties (2018, background): “at βX,∆(E) ≥ εAX,∆(E)(−KX− ∆)n for all divisors Eover X. 14 HAROLDBLUMANDCHENYANGXU The equivalence of the above definition with the original definitions was addressed in [Fuj16b,Fuj17a,Li17] (see also [BJ18]) and the arguments rely on the special degeneration theory of [LX14]. In Corollary 4.3, we wi”