Paper · May 24, 2026

Singular Kähler–Ricci shrinkers and polarized Fano fibrations

Max Hallgren, Junsheng Zhang

math.DGarXiv:2605.25213v2
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Abstract

We prove that any singular Kähler–Ricci shrinker arising as a noncollapsed limit of Kähler–Ricci flows admits a natural structure of a polarized Fano fibration. We also show that it is simply connected, has unique tangent cones at every point, and is a smooth orbifold outside a subset of complex codimension at least three. As an application, we prove a new long-time pseudolocality theorem for almost-selfsimilar Kähler–Ricci flows.

Comments: 63 pages. v2: Title changed and a new section added to strengthen our previous results. Comments welcome!

Ricci flowKähler–Ricci flowRicci solitons(Non)collapsingKähler geometryAnalytic minimal model program

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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 Kähler-Ricci shrinkers and Fano fibrations (Song Sun and Junsheng Zhang, 2024)2Sun 2024 Expanding Soliton Models for Kähler-Ricci Flow Near Conical Singularities (Longteng Chen, Max Hallgren and Lucas Lavoyer, 2026)8Chen 2026 Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers (Yu Li and Junsheng Zhang, 2026)5Li 2026 Compactness and Rigidity of Complete Kähler Ricci Shrinkers (Tongxin Xu and Zhenlei Zhang, 2026)7Xu 2026 Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program (Longteng Chen, Max Hallgren and Lucas Lavoyer, 2026)6Chen 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)3Cifarelli 2026 A Matsushima theorem, uniqueness and K-polystability for Kähler-Ricci shrinkers (Carlos Esparza and Junsheng Zhang, 2026)4Esparza 2026 Scalar curvature bounds and tangent cones for Kähler–Ricci shrinkers (Yu Li and Junsheng Zhang, 2026)1Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Scalar curvature bounds and tangent cones for Kähler–Ricci shrinkers

    Yu Li and Junsheng Zhang PDF

    Topics Analytic minimal model program, Kähler geometry, Ricci solitons · Same author Junsheng Zhang · Shared terms “Ricci shrinkers”, “polarized Fano”, “tangent cone”

  2. Kähler-Ricci shrinkers and Fano fibrations

    Song Sun and Junsheng Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry, Ricci solitons · Same author Junsheng Zhang · Shared terms “Fano fibration”, “Ricci shrinkers”, “polarized Fano”

  3. 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 · Same authors Max Hallgren, Junsheng Zhang

  4. A Matsushima theorem, uniqueness and K-polystability for Kähler-Ricci shrinkers

    Carlos Esparza and Junsheng Zhang PDF

    Topics Analytic minimal model program, Kähler geometry, Ricci solitons · Same author Junsheng Zhang · Shared terms “Ricci shrinkers”, “polarized Fano”, “Fano fibration”

  5. Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers

    Yu Li and Junsheng Zhang PDF

    Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Same author Junsheng Zhang · Shared terms “Ricci shrinkers”, “Fano fibration”, “noncollapsed”

  6. 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 · Same author Max Hallgren · Shared terms “Ricci shrinkers”, “noncollapsed”

  7. Compactness and Rigidity of Complete Kähler Ricci Shrinkers

    Tongxin Xu and Zhenlei Zhang PDF

    Topics Analytic minimal model program, Kähler geometry, Ricci solitons · Shared terms “Ricci shrinkers”, “polarized Fano”, “Fano fibration”

  8. Expanding Soliton Models for Kähler-Ricci Flow Near Conical Singularities

    Longteng Chen, Max Hallgren and Lucas Lavoyer PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Same author Max Hallgren

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) (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.