Paper · Sep 2, 2026

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

Wangjian Jian, Jian Song

math.DGarXiv:2609.02878
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Abstract

We study collapsing finite-time singularities of the unnormalized Kähler–Ricci flow on Fano bundles arising in the analytic minimal model program. For a Fano bundle , we prove a maximal splitting theorem by the Kähler-Ricci flow that every tangent space at a fixed limiting point splits globally as . If the fibre has complex dimension one, we prove a global Type-I bound for the full curvature tensor and show that the ambient and intrinsic diameters of every fibre are uniformly comparable to . Furthermore, every tangent flow at any fixed limiting point is the round shrinking cylinder .

Comments: Comments are welcome

Ricci flowKähler–Ricci flowSingularities(Non)collapsingKähler geometryAnalytic 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 Finite time singularities of the Kähler-Ricci flow (Wangjian Jian, Jian Song and Gang Tian, 2023)3Jian 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)6Hallgren 2023 Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program (Longteng Chen, Max Hallgren and Lucas Lavoyer, 2026)7Chen 2026 Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I (Charles Cifarelli, Ronan Conlon, Max Hallgren et al., 2026)8Cifarelli 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II (Wangjian Jian and Jian Song, 2026)1Jian 2026 Type I estimates for the Kahler-Ricci flow I (Wangjian Jian and Jian Song, 2026)2Jian 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 II

    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. Type I estimates for the Kahler-Ricci flow I

    Wangjian Jian and Jian Song PDF

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

  3. Finite time singularities of the Kähler-Ricci flow

    Wangjian Jian, Jian Song and Gang Tian PDF

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

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

    Wangjian Jian and Jian Song PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · 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, Kähler geometry, Singularities · Same authors Wangjian Jian, Jian Song · Shared terms “Fano”, “diameter”, “finite time”

  6. 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”

  7. Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program

    Longteng Chen, Max Hallgren and Lucas Lavoyer PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, (Non)collapsing · Shared terms “time singularity”, “tangent flow”, “complex dimension”

  8. Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I

    Charles Cifarelli, Ronan Conlon, Max Hallgren et al. PDF

    Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Shared terms “time singularity”, “finite time”, “collapsing finite”

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) 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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

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.