Paper · Aug 5, 2016

K-Stability of Fano spherical varieties

Thibaut Delcroix

math.AGarXiv:1608.01852v2
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Abstract

We prove a criterion for K-stability of a -Fano spherical variety with respect to equivariant special test configurations, in terms of its moment polytope and some combinatorial data associated to the open orbit. Combined with the equivariant version of the Yau-Tian-Donaldson conjecture for Fano manifolds proved by Datar and Székelyhidi, it yields a criterion for the existence of a Kähler-Einstein metric on a spherical Fano manifold. The results hold also for modified K-stability and existence of Kähler-Ricci solitons.

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Submitted
Aug 5, 2016
Last revised
Nov 26, 2019 (v2)
Subjects
math.AG, math.DG
Journal
Annales Scientifiques de l'ENS, 2020, Tome 53, fascicule 3, pages 615-662
DOI
10.24033/asens.2430
Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryStability & uniquenessKähler–Einstein & K-stability

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Coupled Kähler-Ricci solitons on toric Fano manifolds (Jakob Hultgren, 2017)7Hultgren 2017 Coupled complex Monge-Ampère equations on Fano horosymmetric manifolds (Thibaut Delcroix and Jakob Hultgren, 2018)3Delcroix 2018 Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons (Yi Yao, 2019)6Yao 2019 Examples of K-unstable Fano manifolds (Thibaut Delcroix, 2019)2Delcroix 2019 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)4Guo 2020 On the Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci soliton equations (Jiyuan Han and Chi Li, 2020)1Han 2020 Kähler-Ricci flow on mathbf G-spherical Fano manifolds (Feng Wang and Xiaohua Zhu, 2023)5Wang 2023 Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein (Yasufumi Nitta, 2026)8Nitta 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci soliton equations

    Jiyuan Han and Chi Li

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Tian Donaldson”, “Yau Tian”, “Donaldson conjecture”

  2. Examples of K-unstable Fano manifolds

    Thibaut Delcroix

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Same author Thibaut Delcroix · Shared terms “Fano manifolds”

  3. Coupled complex Monge-Ampère equations on Fano horosymmetric manifolds

    Thibaut Delcroix and Jakob Hultgren

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Same author Thibaut Delcroix · Shared terms “combinatorial data”, “Fano”

  4. On the Kahler-Ricci flow on Fano manifolds

    Bin Guo, Duong H. Phong and Jacob Sturm

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Fano manifolds”

  5. Kähler-Ricci flow on -spherical Fano manifolds

    Feng Wang and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “spherical Fano”, “spherical variety”, “Fano manifolds”

  6. Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons

    Yi Yao

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Tian Donaldson”, “Yau Tian”, “test configurations”

  7. Coupled Kähler-Ricci solitons on toric Fano manifolds

    Jakob Hultgren

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Fano manifolds”, “Donaldson conjecture”, “Tian Donaldson”

  8. Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein

    Yasufumi Nitta

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Fano manifolds”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.