Paper · Jan 2, 2024

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

Tomoyuki Hisamoto

math.DGarXiv:2401.01153
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Abstract

For a Fano manifold, We consider the geometric quantization of the Kähler-Ricci flow and the associated entropy functional. Convergence to the original flow and entropy is established. It is also possible to formulate the finite-dimensional analogue of the optimal degeneration for the anti-canonical polarization.

Ricci flowKähler–Ricci flowKähler geometry

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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 Horosymmetric limits of Kähler-Ricci flow on Fano G-manifolds (Gang Tian and Xiaohua Zhu, 2022)2Tian 2022 The modified Kähler-Ricci flow, II (Haotian Wu and Zhou Zhang, 2022)8Wu 2022 Kähler-Ricci flow on mathbf G-spherical Fano manifolds (Feng Wang and Xiaohua Zhu, 2023)1Wang 2023 L^∞ estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation (Wangjian Jian and Yalong Shi, 2023)6Jian 2023 A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds (Wangjian Jian, Jian Song and Gang Tian, 2023)4Jian 2023 Two Comments on "Laplace comparison on Kahler Ricci flow and convergence (arXiv:2509.14820v1)" (Xiuxiong Chen and Bing Wang, 2025)5Chen 2025 Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds (Liding Huang and Xiaohua Zhu, 2026)7Huang 2026 Kähler-Ricci Flow is Symplectic Deflation plus an Exact Symplectic Deformation (Randall R. Van Why, 2026)3Van Why 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Kähler-Ricci flow on -spherical Fano manifolds

    Feng Wang and Xiaohua Zhu PDF

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

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

    Gang Tian and Xiaohua Zhu PDF

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

  3. Kähler-Ricci Flow is Symplectic Deflation plus an Exact Symplectic Deformation

    Randall R. Van Why PDF

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

  4. A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds

    Wangjian Jian, Jian Song and Gang Tian PDF

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

  5. Two Comments on "Laplace comparison on Kahler Ricci flow and convergence (arXiv:2509.14820v1)"

    Xiuxiong Chen and Bing Wang PDF

    Topics Kähler–Ricci flow, Kähler geometry

  6. estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation

    Wangjian Jian and Yalong Shi PDF

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

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

    Liding Huang and Xiaohua Zhu PDF

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

  8. The modified Kähler-Ricci flow, II

    Haotian Wu and Zhou Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry

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) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.