Paper · Sep 1, 2025

Toric geometry of generalized Kähler-Ricci solitons

Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets, Yury Ustinovskiy

math.DGarXiv:2509.01639v2
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Abstract

We establish a local equivalence between toric steady Kähler-Ricci solitons and -type toric generalized Kähler-Ricci solitons (GKRS). Under natural global conditions we show this equivalence extends to complete GKRS, yielding a general construction of new examples in all dimensions. We show that in four dimensions, all GKRS are either described by the generalized Kähler Gibbons-Hawking ansatz, or have split tangent bundle, or are -type toric. This yields a local classification in four dimensions, together with a conjecturally exhaustive construction of complete symplectic-type examples.

Comments: Bibliography error corrected

Ricci solitonsKähler–Ricci solitonsKähler geometry

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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)1Apostolov 2021 On geometry of steady toric Kähler-Ricci solitons (Yury Ustinovskiy, 2022)3Ustinovskiy 2022 The generalized Kähler Calabi-Yau problem (Vestislav Apostolov, Xin Fu, Jeffrey Streets et al., 2022)5Apostolov 2022 Hamiltonian 2-forms and new explicit Calabi–Yau metrics and gradient steady Kähler–Ricci solitons on ℂⁿ (Vestislav Apostolov and Charles Cifarelli, 2023)7Apostolov 2023 Generalized almost-Kähler-Ricci solitons (Michael Albanese, Giuseppe Barbaro and Mehdi Lejmi, 2024)4Albanese 2024 Formal structure of scalar curvature in generalized Kähler geometry (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2024)2Apostolov 2024 From Kähler Ricci solitons to Calabi-Yau Kähler cones (Vestislav Apostolov, Abdellah Lahdili and Eveline Legendre, 2024)8Apostolov 2024 Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds (Liding Huang and Xiaohua Zhu, 2026)6Huang 2026
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  1. 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 authors Vestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”

  2. Formal structure of scalar curvature in generalized Kähler geometry

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    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Vestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”, “symplectic type”

  3. On geometry of steady toric Kähler-Ricci solitons

    Yury Ustinovskiy PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same author Yury Ustinovskiy · Shared terms “steady Kähler”

  4. Generalized almost-Kähler-Ricci solitons

    Michael Albanese, Giuseppe Barbaro and Mehdi Lejmi PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same author Giuseppe Barbaro · Shared terms “symplectic”

  5. The generalized Kähler Calabi-Yau problem

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

    Topics Kähler geometry · Same authors Vestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”

  6. Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds

    Liding Huang and Xiaohua Zhu PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “generalized Kähler”, “symplectic type”

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

    Vestislav Apostolov and Charles Cifarelli PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same author Vestislav Apostolov · Shared terms “steady Kähler”, “yielding”

  8. From Kähler Ricci solitons to Calabi-Yau Kähler cones

    Vestislav Apostolov, Abdellah Lahdili and Eveline Legendre PDF

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

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)

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.