Paper · Jun 6, 2023

Log del Pezzo -surfaces, Kähler-Einstein metrics, Kähler-Ricci solitons and Sasaki-Einstein metrics

Daniel Hättig, Jürgen Hausen, Hendrik Süß

math.AGarXiv:2306.03796
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Abstract

We consider two classes of non-toric log del Pezzo -surfaces: on the one side the 1/3-log canonical ones and on the other side those of Picard number one and Gorenstein index at most 65. In each of the two classes we figure out the surfaces admitting a Kähler-Einstein metric, a Kähler-Ricci soliton and those allowing a Sasaki-Einstein metric on the link of their anticanonical cone. We encounter examples that admit a Kähler-Ricci soliton but no Sasaki-Einstein cone link metric.

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Submitted
Jun 6, 2023
Subjects
math.AG, math.DG

Comments: 27 pages

Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryKähler–Einstein & K-stabilitySasakian & contact geometry

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 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 Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type (Yasuhiro Nakagawa and Satoshi Nakamura, 2022)4Nakagawa 2022 Deformations of Fano manifolds with weighted solitons (Akito Futaki, 2023)5Futaki 2023 K-stability of special Gushel-Mukai manifolds (Yuchen Liu and Linsheng Wang, 2024)2Liu 2024 On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five (Shu-Cheng Chang, Fengjiang Li, Chien Lin et al., 2024)1Chang 2024 The weighted Hermite–Einstein equation (Michael Hallam and Abdellah Lahdili, 2024)6Hallam 2024 The Kähler-Ricci soliton on bounded pseudoconvex domains (Zehao Sha, 2024)8Sha 2024 On Kähler-Einstein Currents (Yifan Chen, Shih-Kai Chiu, Max Hallgren et al., 2025)7Chen 2025 On weighted extremal Kähler metrics (Akito Futaki, 2026)3Futaki 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five

    Shu-Cheng Chang, Fengjiang Li, Chien Lin et al.

    Topics Sasakian & contact geometry, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Sasaki Einstein”, “log del”, “Del Pezzo”

  2. K-stability of special Gushel-Mukai manifolds

    Yuchen Liu and Linsheng Wang

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

  3. On weighted extremal Kähler metrics

    Akito Futaki

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

  4. Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type

    Yasuhiro Nakagawa and Satoshi Nakamura

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

  5. Deformations of Fano manifolds with weighted solitons

    Akito Futaki

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

  6. The weighted Hermite–Einstein equation

    Michael Hallam and Abdellah Lahdili

    Topics Sasakian & contact geometry, Kähler–Ricci solitons, Einstein metrics · Shared terms “Sasaki”, “admitting”

  7. On Kähler-Einstein Currents

    Yifan Chen, Shih-Kai Chiu, Max Hallgren et al.

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics

  8. The Kähler-Ricci soliton on bounded pseudoconvex domains

    Zehao Sha

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… 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.