Paper · Oct 4, 2026

Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds

Liding Huang, Xiaohua Zhu

math.DGarXiv:2610.05201
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Abstract

We first define an extended Mabuchi functional for a class of generalized Kähler metrics of symplectic type on a toric Fano manifold , and prove it is proper. Then by showing the monotonicity of functional along the normalized generalized Kähler-Ricci flow for any initial -invariant generalized Kähler metric with its symplectic form in , we prove that the flow converges to a Kähler-Ricci soliton. The result confirms a conjecture of Apostolov-Streets-Ustinovskiy.

Ricci flowKähler–Ricci flowRicci 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 2021, plus the canon.

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  1. Kähler-Ricci flow on -spherical Fano manifolds

    Feng Wang and Xiaohua Zhu PDF

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

  2. Kähler-Ricci flow for deformed complex structures

    Gang Tian, Liang Zhang and Xiaohua Zhu PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler–Ricci flow · Same author Xiaohua Zhu · Shared terms “Fano manifold”

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

    Gang Tian and Xiaohua Zhu PDF

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

  4. Toric geometry of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al. PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “generalized Kähler”, “symplectic type”

  5. Shrinking Kähler-Ricci solitons on toric fano fibrations

    Tristan C. Collins, Genggeng Huang, Freid Tong et al. PDF

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

  6. Formal structure of scalar curvature in generalized Kähler geometry

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Shared terms “generalized Kähler”, “symplectic type”, “Mabuchi”

  7. Variational structure and uniqueness of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy PDF

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

  8. 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”

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. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  3. 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.