Paper · May 8, 2017

On the classification of Kahler-Ricci solitons on Gorenstein del Pezzo surfaces

Jacob Cable, Hendrik Süß

math.AGarXiv:1705.02920v2
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Abstract

We give a classification of all pairs (X,v) of Gorenstein del Pezzo surfaces X and vector fields v which are K-stable in the sense of Berman-Nystrom and therefore are expected to admit a Kahler-Ricci solition. Moreover, we provide some new examples of Fano threefolds admitting a Kahler-Ricci soliton.

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Submitted
May 8, 2017
Last revised
Mar 10, 2018 (v2)
Subjects
math.AG, math.DG
Journal
European Journal of Mathematics (2018) 4: 137-161
DOI
10.1007/s40879-017-0204-y

Comments: 21 pages, ancillary files containing calculations in SageMath; minor corrections

Ricci solitonsKähler–Ricci solitons3-manifoldsKähler geometryKähler–Einstein & K-stability

Its neighbourhood

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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 2016, 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 Existence of Kahler-Ricci solitons on smoothable Q-Fano varities (Yan Li, 2019)7Li 2019 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)8Guo 2020 The existence of the Kähler-Ricci soliton degeneration (Harold Blum, Yuchen Liu, Chenyang Xu et al., 2021)5Blum 2021 A Note On Kähler-Ricci Flow on Fano Threefolds (Minghao Miao and Gang Tian, 2022)6Miao 2022 Log del Pezzo ℂ^*-surfaces, Kähler-Einstein metrics, Kähler-Ricci solitons and Sasaki-Einstein metrics (Daniel Hättig, Jürgen Hausen and Hendrik Süß, 2023)1Hättig 2023 Kähler-Ricci solitons on Fano threefolds with non-trivial moduli (Minghao Miao and Linsheng Wang, 2023)3Miao 2023 Optimal Degenerations of K-unstable Fano threefolds (Minghao Miao and Linsheng Wang, 2024)4Miao 2024 K-stability of special Gushel-Mukai manifolds (Yuchen Liu and Linsheng Wang, 2024)2Liu 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Log del Pezzo -surfaces, Kähler-Einstein metrics, Kähler-Ricci solitons and Sasaki-Einstein metrics

    Daniel Hättig, Jürgen Hausen and Hendrik Süß

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler geometry · Same author Hendrik Süß · Shared terms “Del Pezzo”, “Gorenstein”, “admitting”

  2. K-stability of special Gushel-Mukai manifolds

    Yuchen Liu and Linsheng Wang

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, 3-manifolds · Shared terms “Del Pezzo”, “threefold”, “Fano”

  3. Kähler-Ricci solitons on Fano threefolds with non-trivial moduli

    Minghao Miao and Linsheng Wang

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, 3-manifolds · Shared terms “Fano threefold”, “admitting”

  4. Optimal Degenerations of K-unstable Fano threefolds

    Minghao Miao and Linsheng Wang

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, 3-manifolds · Shared terms “Fano threefold”, “admitting”

  5. The existence of the Kähler-Ricci soliton degeneration

    Harold Blum, Yuchen Liu, Chenyang Xu et al.

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler geometry · Shared terms “Fano”

  6. A Note On Kähler-Ricci Flow on Fano Threefolds

    Minghao Miao and Gang Tian

    Topics Kähler–Ricci solitons, 3-manifolds, Kähler geometry · Shared terms “Fano threefold”

  7. Existence of Kahler-Ricci solitons on smoothable Q-Fano varities

    Yan Li

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler geometry · Shared terms “Fano”

  8. On the Kahler-Ricci flow on Fano manifolds

    Bin Guo, Duong H. Phong and Jacob Sturm

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler geometry · Shared terms “Fano”, “admitting”

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.