Paper · Jul 9, 2026

Geometric smoothing by the Kähler-Ricci Flow

Eleonora Di Nezza, Vincent Guedj, Chinh Hoang Lu

math.DGarXiv:2607.08325
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Abstract

We study the geometric regularization of a positive closed current by the (twisted) Kähler-Ricci flow on a compact Kähler manifold. We conjecture that the local Arnold multiplicities linearly decrease to zero, while the flow produces complete Kähler metrics in the Zariski open subset of points that have small Lelong numbers. We prove this conjecture in complex dimension 1 and provide several partial results in higher dimension.

Comments: 30 pages

Ricci flowKähler–Ricci flowKähler geometry

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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.

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)7Wu 2022 Quantization of the Kähler-Ricci flow and optimal destabilizer for a Fano manifold (Tomoyuki Hisamoto, 2024)6Hisamoto 2024 Immortal solutions of the Kähler-Ricci flow (Valentino Tosatti, 2024)5Tosatti 2024 Regularizing property of the twisted conical Kähler-Ricci flow (Jiawei Liu, Shiyu Zhang 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)8Chen 2025 Kähler-Ricci flows coming out of metric spaces (Alix Deruelle, Vincent Guedj, Henri Guenancia et al., 2025)3Deruelle 2025 Kähler-Ricci Flow: from divisors to cusps (Eleonora Di Nezza, Vincent Guedj and Chinh H. Lu, 2026)1Di Nezza 2026 Failure of the local decay formula for Arnold multiplicities under twisted Kähler–Ricci flow (Xiangsen Qin, 2026)2Qin 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: from divisors to cusps

    Eleonora Di Nezza, Vincent Guedj and Chinh H. Lu PDF

    Topics Kähler–Ricci flow, Kähler geometry · Same authors Eleonora Di Nezza, Vincent Guedj, Chinh Hoang Lu · Shared terms “geometric regularization”, “Zariski open”, “closed current”

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

    Xiangsen Qin PDF

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “Arnold multiplicities”, “local Arnold”, “Zariski”

  3. Kähler-Ricci flows coming out of metric spaces

    Alix Deruelle, Vincent Guedj, Henri Guenancia et al. PDF

    Topics Kähler–Ricci flow, Kähler geometry · Same author Vincent Guedj · Shared terms “compact Kähler”

  4. Regularizing property of the twisted conical Kähler-Ricci flow

    Jiawei Liu, Shiyu Zhang and Xi Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “closed current”, “Lelong numbers”, “positive closed”

  5. Immortal solutions of the Kähler-Ricci flow

    Valentino Tosatti PDF

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

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

    Tomoyuki Hisamoto PDF

    Topics Kähler–Ricci flow, Kähler geometry

  7. The modified Kähler-Ricci flow, II

    Haotian Wu and Zhou Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry

  8. 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

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.