Paper · Jun 17, 2025

On Sun-Zhang's theory of Fano fibrations – weighted volumes, moduli and bubbling Fano fibrations

Yuji Odaka

math.AGarXiv:2506.14671v3
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Abstract

We revisit the recent theory of Sun-Zhang on general Fano fibration (germs) which emerged from the study of non-compact Kahler-Ricci soliton metrics, primarily from an algebro-geometric perspective. In addition to reviewing the existing framework, we present new results, conjectures, and remarks. These include methods for computing weighted volumes via (restricted) volumes, Laplace transforms, and incomplete Gamma-functions, and a conjectural algebro-geometric construction ("bubbling") of Fano fibration with asymptotically conical base from degenerating Fano fibration.

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Submitted
Jun 17, 2025
Last revised
Sep 27, 2025 (v3)
Subjects
math.AG, math.DG

Comments: v3: final vers, 34 pages. For a special volume of PAMQ in honor of Prof. Caucher Birkar

Ricci solitonsKähler–Ricci solitonsKähler geometryBoundaries & non-compactAnalytic minimal model program

Its neighbourhood

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 Kähler-Ricci shrinkers and Fano fibrations (Song Sun and Junsheng Zhang, 2024)5Sun 2024 Uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons (Carlos Esparza, 2025)3Esparza 2025 Shrinking gradient Kähler-Ricci solitons are simply-connected (Carlos Esparza, 2025)2Esparza 2025 K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers (Charles Cifarelli and Carlos Esparza, 2025)4Cifarelli 2025 Fano Fibrations and Twisted Kähler-Einstein Metrics I (Alexander Bednarek, 2025)8Bednarek 2025 Singular Kähler–Ricci shrinkers and polarized Fano fibrations (Max Hallgren and Junsheng Zhang, 2026)6Hallgren 2026 Shrinking Kähler-Ricci solitons on toric fano fibrations (Tristan C. Collins, Genggeng Huang, Freid Tong et al., 2026)1Collins 2026 Scalar curvature bounds and tangent cones for Kähler–Ricci shrinkers (Yu Li and Junsheng Zhang, 2026)7Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Shrinking Kähler-Ricci solitons on toric fano fibrations

    Tristan C. Collins, Genggeng Huang, Freid Tong et al.

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

  2. Shrinking gradient Kähler-Ricci solitons are simply-connected

    Carlos Esparza

    Topics Analytic minimal model program, Kähler–Ricci solitons, Kähler geometry · Shared terms “Sun Zhang”, “Fano fibration”

  3. Uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons

    Carlos Esparza

    Topics Analytic minimal model program, Kähler–Ricci solitons, Boundaries & non-compact · Shared terms “Sun Zhang”, “Fano fibration”, “asymptotically conical”

  4. K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers

    Charles Cifarelli and Carlos Esparza

    Topics Analytic minimal model program, Boundaries & non-compact, Kähler geometry · Shared terms “Fano fibration”, “Sun Zhang”, “asymptotically conical”

  5. Kähler-Ricci shrinkers and Fano fibrations

    Song Sun and Junsheng Zhang

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Shared terms “Fano fibration”, “weighted volume”

  6. Singular Kähler–Ricci shrinkers and polarized Fano fibrations

    Max Hallgren and Junsheng Zhang

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

  7. Scalar curvature bounds and tangent cones for Kähler–Ricci shrinkers

    Yu Li and Junsheng Zhang

    Topics Analytic minimal model program, Boundaries & non-compact, Kähler geometry · Shared terms “Fano fibration”

  8. Fano Fibrations and Twisted Kähler-Einstein Metrics I

    Alexander Bednarek

    Topics Analytic minimal model program, Kähler geometry · Shared terms “Fano fibration”, “Zhang”, “compact Kähler”

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 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) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.