Paper · Mar 6, 2025

Shrinking gradient Kähler-Ricci solitons are simply-connected

Carlos Esparza

math.DGarXiv:2503.05838
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Abstract

We show that smooth polarized Fano fibrations have no nontrivial finite covers. Using results by Sun-Zhang and Wylie, it follows that shrinking Kähler-Ricci solitons are simply-connected.

Comments: 3 pages. Comments welcome

Ricci solitonsKähler–Ricci solitonsKähler 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 A new complete two-dimensional shrinking gradient Kähler-Ricci soliton (Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al., 2022)7Bamler 2022 Uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons (Carlos Esparza, 2025)1Esparza 2025 On Sun-Zhang's theory of Fano fibrations – weighted volumes, moduli and bubbling Fano fibrations (Yuji Odaka, 2025)5Odaka 2025 K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers (Charles Cifarelli and Carlos Esparza, 2025)3Cifarelli 2025 Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones (Ronan J. Conlon and Alix Deruelle, 2026)6Conlon 2026 Shrinking Kähler-Ricci solitons on toric fano fibrations (Tristan C. Collins, Genggeng Huang, Freid Tong et al., 2026)2Collins 2026 Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton (Ronan J. Conlon and Alix Deruelle, 2026)4Conlon 2026 A Matsushima theorem, uniqueness and K-polystability for Kähler-Ricci shrinkers (Carlos Esparza and Junsheng Zhang, 2026)8Esparza 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons

    Carlos Esparza PDF

    Topics Analytic minimal model program, Kähler–Ricci solitons, Kähler geometry · Same author Carlos Esparza · Shared terms “shrinking gradient”, “gradient Kähler”, “Sun Zhang”

  2. Shrinking Kähler-Ricci solitons on toric fano fibrations

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

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

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

    Charles Cifarelli and Carlos Esparza PDF

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

  4. Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton

    Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “shrinking gradient”, “gradient Kähler”

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

    Yuji Odaka PDF

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

  6. Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones

    Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “shrinking gradient”, “gradient Kähler”, “shrinking Kähler”

  7. A new complete two-dimensional shrinking gradient Kähler-Ricci soliton

    Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al. PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “shrinking gradient”, “gradient Kähler”

  8. 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 Carlos Esparza · Shared terms “polarized Fano”, “Fano fibration”

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) Analytic minimal model program On a projective variety the Kähler–Ricci flow runs until its Kähler class stops being Kähler, a time read off from cohomology. Start with: Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type (Tsuji, 1988)

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.