Paper · May 9, 2023

Kähler-Ricci flow on -spherical Fano manifolds

Feng Wang, Xiaohua Zhu

math.DGarXiv:2305.05366
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Abstract

We prove that the Gromov-Hausdorff limit of Kähler-Ricci flow on a -spherical Fano manifold is a -spherical -Fano variety , which admits a (singular) Kähler-Ricci soliton. Moreover, the -spherical variety structure of can be constructed as a center of torus -degeneration of induced by an element in the Lie algebra of Cartan torus of .

Ricci flowKähler–Ricci flowRicci solitonsKähler–Ricci solitonsKähler geometryLimit spaces & RCD

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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 2021, 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 Kähler-Ricci flow for deformed complex structures (Gang Tian, Liang Zhang and Xiaohua Zhu, 2021)4Tian 2021 Lower bound of modified K-energy on a Fano manifold with degeneration for Kähler-Ricci solitons (Liang Zhang, 2021)5Zhang 2021 Horosymmetric limits of Kähler-Ricci flow on Fano G-manifolds (Gang Tian and Xiaohua Zhu, 2022)1Tian 2022 A Note On Kähler-Ricci Flow on Fano Threefolds (Minghao Miao and Gang Tian, 2022)3Miao 2022 Quantization of the Kähler-Ricci flow and optimal destabilizer for a Fano manifold (Tomoyuki Hisamoto, 2024)6Hisamoto 2024 Laplace comparison on Kähler Ricci flow and convergence (Gang Tian, Qi S. Zhang, Zhenlei Zhang et al., 2025)7Tian 2025 Weighted volume comparison and monotonicity for L^p-bound of Bakry-Émery Ricci curvature (Jintao Ye and Xiaohua Zhu, 2026)8Ye 2026 Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds (Liding Huang and Xiaohua Zhu, 2026)2Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Horosymmetric limits of Kähler-Ricci flow on Fano -manifolds

    Gang Tian and Xiaohua Zhu PDF

    Topics Kähler–Ricci solitons, Kähler–Ricci flow, Limit spaces & RCD · Same author Xiaohua Zhu · Shared terms “Fano”, “Cartan torus”, “Hausdorff limit”

  2. Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds

    Liding Huang and Xiaohua Zhu PDF

    Topics Kähler–Ricci solitons, Kähler–Ricci flow, Kähler geometry · Same author Xiaohua Zhu · Shared terms “Fano manifold”

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

    Minghao Miao and Gang Tian PDF

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

  4. Kähler-Ricci flow for deformed complex structures

    Gang Tian, Liang Zhang and Xiaohua Zhu PDF

    Topics Kähler–Ricci solitons, Kähler–Ricci flow, Kähler geometry · Same author Xiaohua Zhu · Shared terms “Fano manifold”, “Lie”

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

    Liang Zhang PDF

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

  6. Quantization of the Kähler-Ricci flow and optimal destabilizer for a Fano manifold

    Tomoyuki Hisamoto PDF

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

  7. Laplace comparison on Kähler Ricci flow and convergence

    Gang Tian, Qi S. Zhang, Zhenlei Zhang et al. PDF

    Topics Kähler–Ricci flow, Kähler geometry · Same author Xiaohua Zhu · Shared terms “Fano manifold”

  8. Weighted volume comparison and monotonicity for -bound of Bakry-Émery Ricci curvature

    Jintao Ye and Xiaohua Zhu PDF

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same author Xiaohua Zhu

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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) 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)

In the guidebook

  1. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  2. 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.

  3. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

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