Paper · Aug 26, 2019

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

Yi Yao

math.DGarXiv:1908.09518v2
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Abstract

Mabuchi solitons generalize Kähler-Einstein metrics on Fano manifolds, which constitute a Yau-Tian-Donaldson type correspondence with relative Ding stability. Comparing with Kähler-Ricci solitons, there is a distinct necessary condition for the existence. We show this condition can be implied by the uniformly relative Ding stability. For this we study the inner product of -actions on equivariant test-configurations and obtain an integration formula over the total space. To analyze the uniform stability, by adapting Okounkov body construction to the setting of torus action, we give a convex-geometry description for the reduced non-Archimedean J-functionals.

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Submitted
Aug 26, 2019
Last revised
Jan 30, 2022 (v2)
Subjects
math.DG
DOI
10.1007/s12220-021-00858-z

Comments: 40 pages, 1 figure. Final version. To appear in J. Geom. Anal

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 K-Stability of Fano spherical varieties (Thibaut Delcroix, 2016)2Delcroix 2016 Coupled Kähler-Ricci solitons on toric Fano manifolds (Jakob Hultgren, 2017)6Hultgren 2017 Equivariant calculus on μ-character and μK-stability of polarized schemes (Eiji Inoue, 2020)4Inoue 2020 On the Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci soliton equations (Jiyuan Han and Chi Li, 2020)1Han 2020 Basis divisors and balanced metrics (Yanir A. Rubinstein, Gang Tian and Kewei Zhang, 2020)5Rubinstein 2020 Weighted K-stability and coercivity with applications to extremal Kahler and Sasaki metrics (Vestislav Apostolov, Simon Jubert and Abdellah Lahdili, 2021)7Apostolov 2021 Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type (Yasuhiro Nakagawa and Satoshi Nakamura, 2022)3Nakagawa 2022 Deformations of Fano manifolds with weighted solitons (Akito Futaki, 2023)8Futaki 2023
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 “Ding stability”, “Mabuchi solitons”, “Tian Donaldson”

  2. K-Stability of Fano spherical varieties

    Thibaut Delcroix

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

  3. Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type

    Yasuhiro Nakagawa and Satoshi Nakamura

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

  4. Equivariant calculus on -character and K-stability of polarized schemes

    Eiji Inoue

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler geometry · Shared terms “test configurations”, “equivariant”, “relative”

  5. Basis divisors and balanced metrics

    Yanir A. Rubinstein, Gang Tian and Kewei Zhang

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler geometry · Shared terms “Ding stability”, “Donaldson”, “Fano”

  6. Coupled Kähler-Ricci solitons on toric Fano manifolds

    Jakob Hultgren

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

  7. Weighted K-stability and coercivity with applications to extremal Kahler and Sasaki metrics

    Vestislav Apostolov, Simon Jubert and Abdellah Lahdili

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Mabuchi”, “test configurations”, “total space”

  8. Deformations of Fano manifolds with weighted solitons

    Akito Futaki

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

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.