Paper · Sep 17, 2026

Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II

Wangjian Jian, Jian Song

math.DGarXiv:2609.20513
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Abstract

The analytic minimal model program proposed in seeks to describe the birational transitions and fibre collapsing of the Kähler–Ricci flow through the geometry of its singularities. A fundamental conjecture of predicts Type I bounds for the scalar curvature and diameter of every fibre contratced by the limiting cohomological class at the finite singular time. This paper is a continuation of that establishes the Type I conjecture for collapsing solutions on Fano bundles with arbitrary smooth Fano fibres. If the fibre admits a smooth Kähler-Einstein metric, the Type I bound holds for the full curvature tensor.

Comments: comments are welcome

Ricci flowKähler–Ricci flowEinstein metricsSingularities(Non)collapsingKähler geometryScalar curvatureKähler–Einstein & K-stabilityAnalytic minimal model program

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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 L^∞ estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation (Wangjian Jian and Yalong Shi, 2023)8Jian 2023 Finite time singularities of the Kähler-Ricci flow (Wangjian Jian, Jian Song and Gang Tian, 2023)2Jian 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)5Jian 2023 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)7Hallgren 2023 Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow (Alexander Bednarek, 2025)6Bednarek 2025 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles (Wangjian Jian and Jian Song, 2026)1Jian 2026 Type I estimates for the Kahler-Ricci flow I (Wangjian Jian and Jian Song, 2026)3Jian 2026 Type I estimates for the Kähler-Ricci flow II (Wangjian Jian and Jian Song, 2026)4Jian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles

    Wangjian Jian and Jian Song PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, (Non)collapsing · Same authors Wangjian Jian, Jian Song · Shared terms “Fano bundles”, “time singularity”, “finite time”

  2. Finite time singularities of the Kähler-Ricci flow

    Wangjian Jian, Jian Song and Gang Tian PDF

    Topics Kähler–Ricci flow, (Non)collapsing, Scalar curvature · Same authors Wangjian Jian, Jian Song · Shared terms “time singularity”, “finite time”, “fibre collapsing”

  3. Type I estimates for the Kahler-Ricci flow I

    Wangjian Jian and Jian Song PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Scalar curvature · Same authors Wangjian Jian, Jian Song · Shared terms “time singularity”, “finite time”

  4. Type I estimates for the Kähler-Ricci flow II

    Wangjian Jian and Jian Song PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Scalar curvature · Same authors Wangjian Jian, Jian Song · Shared terms “finite time”, “diameter”, “limiting”

  5. 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, Scalar curvature, Kähler geometry · Same authors Wangjian Jian, Jian Song · Shared terms “JST”, “Fano”, “diameter”

  6. Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow

    Alexander Bednarek PDF

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “Fano”, “finite time”, “diameter”

  7. Geometric regularity of blow-up limits of the Kähler-Ricci flow

    Max Hallgren, Wangjian Jian, Jian Song et al. PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same authors Wangjian Jian, Jian Song · Shared terms “limiting”

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

    Wangjian Jian and Yalong Shi PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Same author Wangjian Jian · Shared terms “Fano”

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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)

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. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  3. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.