Paper · Aug 27, 2019

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

Yan Li

math.DGarXiv:1908.10091
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Abstract

In this article we prove the existence of Kahler-Ricci solitons on smoothable, K-stable Q-Fano varieties. We also investigate the behavior of twisted Kahler-Ricci solitons in the Gromov-Hausdorff topology under this smoothing family.

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Submitted
Aug 27, 2019
Subjects
math.DG
Ricci solitonsKähler–Ricci solitonsKähler geometryLimit spaces & RCDKähler–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 The moduli space of Fano manifolds with Kähler-Ricci solitons (Eiji Inoue, 2018)4Inoue 2018 Compactness of Kahler-Ricci solitons on Fano manifolds (Bin Guo, Duong H. Phong, Jian Song et al., 2018)1Guo 2018 Singular limits of Kähler-Ricci flow on Fano G-manifolds (Yan Li, Gang Tian and Xiaohua Zhu, 2018)6Li 2018 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)2Guo 2020 The existence of the Kähler-Ricci soliton degeneration (Harold Blum, Yuchen Liu, Chenyang Xu et al., 2021)5Blum 2021 Horosymmetric limits of Kähler-Ricci flow on Fano G-manifolds (Gang Tian and Xiaohua Zhu, 2022)8Tian 2022 A Note On Kähler-Ricci Flow on Fano Threefolds (Minghao Miao and Gang Tian, 2022)7Miao 2022 Kähler-Ricci flow on mathbf G-spherical Fano manifolds (Feng Wang and Xiaohua Zhu, 2023)3Wang 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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, Limit spaces & RCD · Shared terms “Fano”, “Hausdorff topology”, “Gromov Hausdorff”

  2. On the Kahler-Ricci flow on Fano manifolds

    Bin Guo, Duong H. Phong and Jacob Sturm

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

  3. Kähler-Ricci flow on -spherical Fano manifolds

    Feng Wang and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Limit spaces & RCD, Kähler geometry · Shared terms “Fano”, “Gromov Hausdorff”

  4. The moduli space of Fano manifolds with Kähler-Ricci solitons

    Eiji Inoue

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Limit spaces & RCD · Shared terms “Fano”, “Gromov Hausdorff”

  5. The existence of the Kähler-Ricci soliton degeneration

    Harold Blum, Yuchen Liu, Chenyang Xu et al.

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

  6. Singular limits of Kähler-Ricci flow on Fano -manifolds

    Yan Li, Gang Tian and Xiaohua Zhu

    Topics Kähler–Einstein & K-stability, Kähler geometry · Same author Yan Li · Shared terms “Fano”

  7. A Note On Kähler-Ricci Flow on Fano Threefolds

    Minghao Miao and Gang Tian

    Topics Kähler–Ricci solitons, Limit spaces & RCD, Kähler geometry · Shared terms “Fano”, “Gromov Hausdorff”

  8. Horosymmetric limits of Kähler-Ricci flow on Fano -manifolds

    Gang Tian and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Limit spaces & RCD, Kähler geometry · Shared terms “Fano”, “Gromov Hausdorff”

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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.