Paper · Apr 17, 2023

Weighted extremal metrics on blowups

Michael Hallam

math.DGarXiv:2304.08338v3
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Abstract

We show that if a compact Kähler manifold admits a weighted extremal metric for the action of a torus, so too does its blowup at a relatively stable point that is fixed by both the torus action and the extremal field. This generalises previous results on extremal metrics by Arezzo–Pacard–Singer and Székelyhidi to many other canonical metrics, including extremal Sasaki metrics, deformations of Kähler–Ricci solitons and -cscK metrics. In a sequel to this paper, we use this result to study the weighted K-stability of weighted extremal manifolds.

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Submitted
Apr 17, 2023
Last revised
May 7, 2025 (v3)
Subjects
math.DG

Comments: 50 pages, minor corrections, final version

Ricci solitonsKähler–Ricci solitonsSingularitiesKähler geometryStability & uniquenessKähler–Einstein & K-stabilitySasakian & contact geometry

Its neighbourhood

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 Moment map, convex function and extremal point (King Leung Lee, Jacob Sturm and Xiaowei Wang, 2022)7Lee 2022 Gradient Shrinking Sasaki-Ricci Solitons on Sasakian Manifolds of Dimension Up to Seven (Der-Chen Chang, Shu-Cheng Chang, Yingbo Han et al., 2022)5Chang 2022 Deformations of Fano manifolds with weighted solitons (Akito Futaki, 2023)6Futaki 2023 Log del Pezzo ℂ^*-surfaces, Kähler-Einstein metrics, Kähler-Ricci solitons and Sasaki-Einstein metrics (Daniel Hättig, Jürgen Hausen and Hendrik Süß, 2023)8Hättig 2023 On the Existence of Weighted-cscK Metrics (Jiyuan Han and Yaxiong Liu, 2024)3Han 2024 The weighted Hermite–Einstein equation (Michael Hallam and Abdellah Lahdili, 2024)2Hallam 2024 On weighted extremal Kähler metrics (Akito Futaki, 2026)1Futaki 2026 Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein (Yasufumi Nitta, 2026)4Nitta 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On weighted extremal Kähler metrics

    Akito Futaki

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

  2. The weighted Hermite–Einstein equation

    Michael Hallam and Abdellah Lahdili

    Topics Sasakian & contact geometry, Kähler–Ricci solitons, Kähler geometry · Same author Michael Hallam · Shared terms “torus action”, “Sasaki”

  3. On the Existence of Weighted-cscK Metrics

    Jiyuan Han and Yaxiong Liu

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

  4. Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein

    Yasufumi Nitta

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

  5. Gradient Shrinking Sasaki-Ricci Solitons on Sasakian Manifolds of Dimension Up to Seven

    Der-Chen Chang, Shu-Cheng Chang, Yingbo Han et al.

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

  6. Deformations of Fano manifolds with weighted solitons

    Akito Futaki

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

  7. Moment map, convex function and extremal point

    King Leung Lee, Jacob Sturm and Xiaowei Wang

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

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

    Daniel Hättig, Jürgen Hausen and Hendrik Süß

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

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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)

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