Paper · Jun 14, 2026

Kähler-Ricci Flow: from divisors to cusps

Eleonora Di Nezza, Vincent Guedj, Chinh H. Lu

math.DGarXiv:2606.15816
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Abstract

We study the geometric regularization of positive closed currents by the Kähler-Ricci flow on compact Kähler manifolds. In a previous work of ours, it was shown that the Kähler-Ricci flow immediately smoothes out such a current when it has zero Lelong numbers. We study here the case when has divisorial singularities, showing that the flow gradually replaces the latter by Poincaré type ones, providing an approximation of by complete Kähler metrics with bounded curvature in a Zariski open set.

Ricci flowKähler–Ricci flowSingularitiesKähler 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 The modified Kähler-Ricci flow, II (Haotian Wu and Zhou Zhang, 2022)6Wu 2022 Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle (Jiawei Liu and Xi Zhang, 2024)4Liu 2024 Regularizing property of the twisted conical Kähler-Ricci flow (Jiawei Liu, Shiyu Zhang and Xi Zhang, 2024)3Liu 2024 Kähler-Ricci flows coming out of metric spaces (Alix Deruelle, Vincent Guedj, Henri Guenancia et al., 2025)2Deruelle 2025 Weak transcendental base-point freeness and diameter lower bounds for the Kähler-Ricci flow (Junsheng Zhang, 2025)7Zhang 2025 Geometric smoothing by the Kähler-Ricci Flow (Eleonora Di Nezza, Vincent Guedj and Chinh Hoang Lu, 2026)1Di Nezza 2026 Failure of the local decay formula for Arnold multiplicities under twisted Kähler–Ricci flow (Xiangsen Qin, 2026)5Qin 2026 Kähler-Ricci Flow is Symplectic Deflation plus an Exact Symplectic Deformation (Randall R. Van Why, 2026)8Van Why 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Geometric smoothing by the Kähler-Ricci Flow

    Eleonora Di Nezza, Vincent Guedj and Chinh Hoang Lu PDF

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

  2. 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”

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

    Jiawei Liu, Shiyu Zhang and Xi Zhang PDF

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

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

    Jiawei Liu and Xi Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities

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

    Xiangsen Qin PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “divisorial”, “Zariski”

  6. The modified Kähler-Ricci flow, II

    Haotian Wu and Zhou Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities

  7. Weak transcendental base-point freeness and diameter lower bounds for the Kähler-Ricci flow

    Junsheng Zhang PDF

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

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

    Randall R. Van Why PDF

    Topics Kähler–Ricci flow, Kähler geometry

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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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.