Paper · Oct 2, 2026

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

Carlos Esparza, Junsheng Zhang

math.DGarXiv:2610.03592
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Abstract

For noncompact smooth Kähler-Ricci shrinkers, we prove a Matsushima theorem identifying the centralizer of the soliton field with the complexification of its Killing centralizer, uniqueness up to biholomorphism on a fixed complex manifold, and K-polystability of the associated polarized Fano fibration.

Comments: 35 pages

Ricci solitonsKähler geometryStability & uniquenessBoundaries & non-compactKähler–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 Kähler-Ricci shrinkers and Fano fibrations (Song Sun and Junsheng Zhang, 2024)6Sun 2024 Uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons (Carlos Esparza, 2025)4Esparza 2025 Shrinking gradient Kähler-Ricci solitons are simply-connected (Carlos Esparza, 2025)5Esparza 2025 K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers (Charles Cifarelli and Carlos Esparza, 2025)1Cifarelli 2025 Gromov-Hausdorff limits of immortal Kähler-Ricci flows (Man-Chun Lee, Valentino Tosatti and Junsheng Zhang, 2026)8Lee 2026 Singular Kähler–Ricci shrinkers and polarized Fano fibrations (Max Hallgren and Junsheng Zhang, 2026)3Hallgren 2026 Compactness and Rigidity of Complete Kähler Ricci Shrinkers (Tongxin Xu and Zhenlei Zhang, 2026)7Xu 2026 Scalar curvature bounds and tangent cones for Kähler–Ricci shrinkers (Yu Li and Junsheng Zhang, 2026)2Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers

    Charles Cifarelli and Carlos Esparza PDF

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Boundaries & non-compact · Same author Carlos Esparza · Shared terms “polystability”, “Ricci shrinkers”, “polarized Fano”

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

    Yu Li and Junsheng Zhang PDF

    Topics Analytic minimal model program, Boundaries & non-compact, Kähler geometry · Same author Junsheng Zhang · Shared terms “Ricci shrinkers”, “polarized Fano”, “Fano fibration”

  3. Singular Kähler–Ricci shrinkers and polarized Fano fibrations

    Max Hallgren 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”

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

    Carlos Esparza PDF

    Topics Analytic minimal model program, Boundaries & non-compact, Kähler geometry · Same author Carlos Esparza · Shared terms “complex manifold”, “biholomorphism”, “polarized Fano”

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

    Carlos Esparza PDF

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

  6. Kähler-Ricci shrinkers and Fano fibrations

    Song Sun and Junsheng Zhang PDF

    Topics Boundaries & non-compact, Kähler geometry, Stability & uniqueness · Same author Junsheng Zhang · Shared terms “Ricci shrinkers”, “Fano fibration”, “polarized Fano”

  7. Compactness and Rigidity of Complete Kähler Ricci Shrinkers

    Tongxin Xu and Zhenlei Zhang PDF

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

  8. Gromov-Hausdorff limits of immortal Kähler-Ricci flows

    Man-Chun Lee, Valentino Tosatti and Junsheng Zhang PDF

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler geometry · Same author Junsheng Zhang

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 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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.