Paper · Dec 3, 2024

From Kähler Ricci solitons to Calabi-Yau Kähler cones

Vestislav Apostolov, Abdellah Lahdili, Eveline Legendre

math.DGarXiv:2412.02564v2
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Abstract

We show that if is a smooth Fano manifold which caries a Kähler Ricci soliton, then the canonical cone of the product of with a complex projective space of sufficiently large dimension is a Calabi–Yau cone. This can be seen as an asymptotic version of a conjecture by Mabuchi and Nikagawa. This result is obtained by the openness of the set of weight functions over the momentum polytope of a given smooth Fano manifold, for which a -soliton exists. We discuss other ramifications of this approach, including a Licherowicz type obstruction to the existence of a Kähler Ricci soliton and a Fujita type volume bound for the existence of a -soliton.

Comments: Final version

Ricci solitonsKähler–Ricci solitonsRicci-flat & Calabi–YauKähler geometryKähler–Einstein & K-stability

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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 Variational structure and uniqueness of generalized Kähler-Ricci solitons (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2021)4Apostolov 2021 The generalized Kähler Calabi-Yau problem (Vestislav Apostolov, Xin Fu, Jeffrey Streets et al., 2022)6Apostolov 2022 Deformations of Fano manifolds with weighted solitons (Akito Futaki, 2023)1Futaki 2023 Hamiltonian 2-forms and new explicit Calabi–Yau metrics and gradient steady Kähler–Ricci solitons on ℂⁿ (Vestislav Apostolov and Charles Cifarelli, 2023)2Apostolov 2023 Formal structure of scalar curvature in generalized Kähler geometry (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2024)3Apostolov 2024 Explicit complete Ricci-flat metrics and Kähler-Ricci solitons on direct sum bundles (Charles Cifarelli, 2024)7Cifarelli 2024 The asymptotic behavior of the steady gradient Kähler-Ricci soliton of the Taub-NUT type of Apostolov and Cifarelli (Daheng Min, 2024)8Min 2024 Toric geometry of generalized Kähler-Ricci solitons (Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al., 2025)5Apostolov 2025
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  1. Deformations of Fano manifolds with weighted solitons

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    Topics Ricci-flat & Calabi–Yau, Kähler–Einstein & K-stability, Kähler–Ricci solitons · Shared terms “Fano manifold”, “Yau cone”, “weight functions”

  2. Hamiltonian -forms and new explicit Calabi–Yau metrics and gradient steady Kähler–Ricci solitons on

    Vestislav Apostolov and Charles Cifarelli PDF

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci solitons, Kähler geometry · Same author Vestislav Apostolov

  3. Formal structure of scalar curvature in generalized Kähler geometry

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Same author Vestislav Apostolov · Shared terms “Mabuchi”, “Calabi”, “obstruction”

  4. Variational structure and uniqueness of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same author Vestislav Apostolov

  5. Toric geometry of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al. PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same author Vestislav Apostolov

  6. The generalized Kähler Calabi-Yau problem

    Vestislav Apostolov, Xin Fu, Jeffrey Streets et al. PDF

    Topics Ricci-flat & Calabi–Yau, Kähler geometry · Same author Vestislav Apostolov

  7. Explicit complete Ricci-flat metrics and Kähler-Ricci solitons on direct sum bundles

    Charles Cifarelli PDF

    Topics Ricci-flat & Calabi–Yau, Kähler–Einstein & K-stability, Kähler–Ricci solitons · Shared terms “Fano manifold”

  8. The asymptotic behavior of the steady gradient Kähler-Ricci soliton of the Taub-NUT type of Apostolov and Cifarelli

    Daheng Min PDF

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci solitons, Kähler geometry

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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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)

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.