Paper · Oct 17, 2026

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

Randall R. Van Why

math.DGarXiv:2610.24825
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Abstract

We demonstrate that Kähler-Ricci flow on a Fano surface decomposes as a symplectic deflation along a plurianticanonical divisor plus an exact symplectic deformation in the divisor complement. We apply this decomposition to the study of certain projective bundles. We establish the optimal decay rate in the setting of Song, Székelyhidi, and Weinkove for a family of toric metrics which match the cohomological profile of Kähler-Ricci flow. In doing so, we identify a -obstruction to proving the optimal decay rate in the setting of ruled toric surfaces. We present these ideas as a new symplectic-topological approach to the study of Kähler-Ricci flow.

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Submitted
Oct 17, 2026
Subjects
math.DG, math.AG, math.SG
Ricci flowKähler–Ricci flowKähler geometry

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 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 The modified Kähler-Ricci flow, II (Haotian Wu and Zhou Zhang, 2022)8Wu 2022 Geometric flows from unified string theories (Duong H. Phong, 2023)3Phong 2023 Quantization of the Kähler-Ricci flow and optimal destabilizer for a Fano manifold (Tomoyuki Hisamoto, 2024)1Hisamoto 2024 Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle (Jiawei Liu and Xi Zhang, 2024)4Liu 2024 Two Comments on "Laplace comparison on Kahler Ricci flow and convergence (arXiv:2509.14820v1)" (Xiuxiong Chen and Bing Wang, 2025)5Chen 2025 Kähler-Ricci Flow: from divisors to cusps (Eleonora Di Nezza, Vincent Guedj and Chinh H. Lu, 2026)2Di Nezza 2026 Geometric smoothing by the Kähler-Ricci Flow (Eleonora Di Nezza, Vincent Guedj and Chinh Hoang Lu, 2026)7Di Nezza 2026 Failure of the local decay formula for Arnold multiplicities under twisted Kähler–Ricci flow (Xiangsen Qin, 2026)6Qin 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Quantization of the Kähler-Ricci flow and optimal destabilizer for a Fano manifold

    Tomoyuki Hisamoto

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

  2. Kähler-Ricci Flow: from divisors to cusps

    Eleonora Di Nezza, Vincent Guedj and Chinh H. Lu

    Topics Kähler–Ricci flow, Kähler geometry

  3. Geometric flows from unified string theories

    Duong H. Phong

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

  4. Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle

    Jiawei Liu and Xi Zhang

    Topics Kähler–Ricci flow, Kähler geometry

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

    Xiuxiong Chen and Bing Wang

    Topics Kähler–Ricci flow, Kähler geometry

  6. Failure of the local decay formula for Arnold multiplicities under twisted Kähler–Ricci flow

    Xiangsen Qin

    Topics Kähler–Ricci flow, Kähler geometry

  7. Geometric smoothing by the Kähler-Ricci Flow

    Eleonora Di Nezza, Vincent Guedj and Chinh Hoang Lu

    Topics Kähler–Ricci flow, Kähler geometry

  8. The modified Kähler-Ricci flow, II

    Haotian Wu and Zhou Zhang

    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.