A variational approach to the Yau–Tian–Donaldson conjecture

By Robert J. Berman, S. Boucksom, Mattias Jonsson. Journal of The American Mathematical Society, 2015

247 citations recorded by the source corpus.

We give a variational proof of a version of the Yau–Tian–Donaldson conjecture for twisted Kähler–Einstein currents, and use this to express the greatest (twisted) Ricci lower bound in terms of a purely algebro-geometric stability threshold. Our approach does not involve a continuity method or the Cheeger–Colding–Tian theory, and uses instead pluripotential theory and valuations. Along the way, we study the relationship between geodesic rays and non-Archimedean metrics.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d21172d7cc4f4743d832711a07d8b6888a07aee0

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In Applied Mathematics

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

In Geometry and Topology

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

In Mathematical Physics

1. Chi Li
2. H. Blum
3. Ziquan Zhuang
4. Mattias Jonsson
5. S. Boucksom
6. Yuchen Liu
7. Thibaut Delcroix
8. Jiyuan Han

Citations in the corpus

• Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs (2015, background): “r the log Fano case, that is, (X;B) is klt and L:= K (X;B) is ample. We can then consider stability with respect to the non-Archimedean Ding functional DNA B on H NA dened in x7.7. Namely, following [BBJ15] (see also Fujita [Fuj15b, Fuj16]) we say that (X;B) is Ding semistable if DNA B 0, and unifor”
• Kähler–Einstein metrics and volume minimization (2016, result): “;L)/Cof (X,D;−δ−1KX). Remark 2.3. For special test configurations, log-CM weight coincides with logDing invariant: log-CM(X,D;L) = −δn+1L¯n+1 (n+1)(−KX−D)n = log-Ding(X,D;L). (3) By the recent work of [BBJ15], log-Ding-semistability is equivalent to log-Ksemistability. Notice that since the work in [”
• A non-Archimedean approach to K-stability (2018, methodology): “In the Fano case, the answer is also ‘yes’ by [BBJ15, Fuj16].”
• K-stability of Fano spherical varieties (2016, methodology): “alled special test configurations. Other proofs of the Yau-Tian-Donaldson conjecture were obtained by Tian [Tia15], Datar and Székelyhidi [DS16], Chen, Sun and Wang [CSW], Berman, Boucksom and Jonsson [BBJ]. The work of Datar and Székelyhidi 2010 Mathematics Subject Classification. 14M27,32Q26,32Q20. ”
• On positivity of the CM line bundle on K-moduli spaces (2019, background): “heorem 3.6]), we know the log Fano pairs parametrized by C-valued points of Msm,Kps n,v,c all admit weak conical Ka¨hler-Einstein metrics, thus they are reduced uniformly K-stable by [Li19] (see also [BBJ15]). Therefore, we conclude by Theorem 7.9. AppendixA. Reducedδ-invariants In this section, we ”
• The metric geometry of singularity types (2019, result): “n [DL18, Section 4] the first and last author carried out a detailed analysis of the space of Lpgeodesic rays in the Ka¨hler case. For similar flavour results in the non-Archimedean context we refer to [BBJ18] and references therein. Given that our main focus here is on the space of singularity types,”
• Thresholds, valuations, and K-stability (2017, background): “and Odaka [FO18]; what is new are the direct implications. The notion of uniform K-stability was introduced in [BHJ17, Der16]. As a special case of the Yau{Tian{Donaldson conjecture, it was proved in [BBJ15] that a Fano manifold X without nontrivial vector elds is uniformly K-stable i X admits a Kah”
• Positivity of the CM line bundle for families of K-stable klt Fano varieties (2020, background): “Moduli applications [118] [8,71,76] [35] Volume and slope inequalities [122] [9,95]”