Paper · Mar 29, 2021

The existence of the Kähler-Ricci soliton degeneration

Harold Blum, Yuchen Liu, Chenyang Xu, Ziquan Zhuang

math.AGarXiv:2103.15278v2
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Abstract

We prove an algebraic version of the Hamilton-Tian Conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a reduced uniformly Ding stable triple, which admits a Kähler-Ricci soliton when the ground field .

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Submitted
Mar 29, 2021
Last revised
Jun 16, 2023 (v2)
Subjects
math.AG, math.DG
Journal
Forum Math. Pi 11 (2023), Paper No. e9, 28 pp
DOI
10.1017/fmp.2023.5

Comments: Final version

Ricci solitonsKähler–Ricci solitonsKähler geometryKähler–Einstein & K-stability

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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 Metric geometry of normal Kähler spaces, energy properness, and existence of canonical metrics (Tamás Darvas, 2016)2Darvas 2016 The Kähler-Ricci flow and optimal degenerations (Ruadhaí Dervan and Gábor Székelyhidi, 2016)7Dervan 2016 Compactness of Kahler-Ricci solitons on Fano manifolds (Bin Guo, Duong H. Phong, Jian Song et al., 2018)8Guo 2018 Existence of Kahler-Ricci solitons on smoothable Q-Fano varities (Yan Li, 2019)6Li 2019 Lower bound of modified K-energy on a Fano manifold with degeneration for Kähler-Ricci solitons (Liang Zhang, 2021)4Zhang 2021 Kähler-Ricci solitons on Fano threefolds with non-trivial moduli (Minghao Miao and Linsheng Wang, 2023)1Miao 2023 Optimal Degenerations of K-unstable Fano threefolds (Minghao Miao and Linsheng Wang, 2024)5Miao 2024 K-stability of special Gushel-Mukai manifolds (Yuchen Liu and Linsheng Wang, 2024)3Liu 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Kähler-Ricci solitons on Fano threefolds with non-trivial moduli

    Minghao Miao and Linsheng Wang

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler geometry · Shared terms “log Fano”, “Fano pairs”, “precisely”

  2. Metric geometry of normal Kähler spaces, energy properness, and existence of canonical metrics

    Tamás Darvas

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

  3. K-stability of special Gushel-Mukai manifolds

    Yuchen Liu and Linsheng Wang

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler geometry · Same author Yuchen Liu · Shared terms “Fano”

  4. Lower bound of modified -energy on a Fano manifold with degeneration for Kähler-Ricci solitons

    Liang Zhang

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “degeneration”, “Fano”

  5. Optimal Degenerations of K-unstable Fano threefolds

    Minghao Miao and Linsheng Wang

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler geometry · Shared terms “degeneration”, “Hamilton Tian”, “Tian conjecture”

  6. Existence of Kahler-Ricci solitons on smoothable Q-Fano varities

    Yan Li

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

  7. The Kähler-Ricci flow and optimal degenerations

    Ruadhaí Dervan and Gábor Székelyhidi

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

  8. Compactness of Kahler-Ricci solitons on Fano manifolds

    Bin Guo, Duong H. Phong, Jian Song et al.

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

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) 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) Kähler–Einstein & K-stability A Fano manifold carries a Kähler–Einstein metric exactly when it is K-polystable, an algebraic condition (Chen–Donaldson–Sun, Tian). 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.