Paper · May 3, 2025

Remarks on Singular Kähler-Einstein Metrics

Max Hallgren, Gábor Székelyhidi

math.DGarXiv:2505.01943
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Abstract

We study two different natural notions of singular Kähler-Einstein metrics on normal complex varieties. In the setting of singular Ricci flat Kähler cone metrics that arise as non-collapsed limits of sequences of Kähler-Einstein metrics or Kähler-Ricci flows, we show that an a priori weaker notion is equivalent to the stronger one introduced by Eyssidieux-Guedj-Zeriahi, and in particular the underlying variety has log terminal singularities in this case. Our method applies to more general singular Kähler-Einstein spaces as well, assuming that they define RCD spaces.

Comments: 18 pages

Ricci flowKähler–Ricci flowEinstein metricsRicci-flat & Calabi–YauSingularities(Non)collapsingKähler geometryWeak & singular flowsLimit spaces & RCDKähler–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 K-stability of Gorenstein Fano group compactifications with rank two (Jae-Hyouk Lee, Kyeong-Dong Park and Sungmin Yoo, 2022)8Lee 2022 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)3Hallgren 2023 Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces (Ye-Won Luke Cho and Young-Jun Choi, 2024)6Cho 2024 On Kähler-Einstein Currents (Yifan Chen, Shih-Kai Chiu, Max Hallgren et al., 2025)1Chen 2025 An ε-Regularity Theorem for Non-collapsed Ricci Flow (Harry Fluck and Max Hallgren, 2025)4Fluck 2025 Fano Fibrations and Twisted Kähler-Einstein Metrics I (Alexander Bednarek, 2025)7Bednarek 2025 Singular Kähler–Ricci shrinkers and polarized Fano fibrations (Max Hallgren and Junsheng Zhang, 2026)2Hallgren 2026 Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I (Charles Cifarelli, Ronan Conlon, Max Hallgren et al., 2026)5Cifarelli 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On Kähler-Einstein Currents

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

    Topics Kähler–Einstein & K-stability, Einstein metrics, Limit spaces & RCD · Same authors Max Hallgren, Gábor Székelyhidi · Shared terms “singular Kähler”

  2. Singular Kähler–Ricci shrinkers and polarized Fano fibrations

    Max Hallgren and Junsheng Zhang PDF

    Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Same author Max Hallgren · Shared terms “singular Kähler”

  3. Geometric regularity of blow-up limits of the Kähler-Ricci flow

    Max Hallgren, Wangjian Jian, Jian Song et al. PDF

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same author Max Hallgren · Shared terms “sequence”

  4. An -Regularity Theorem for Non-collapsed Ricci Flow

    Harry Fluck and Max Hallgren PDF

    Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Same author Max Hallgren · Shared terms “collapsed”, “sequence”

  5. Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I

    Charles Cifarelli, Ronan Conlon, Max Hallgren et al. PDF

    Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Same author Max Hallgren

  6. Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces

    Ye-Won Luke Cho and Young-Jun Choi PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “singular Kähler”, “terminal singularities”, “log terminal”

  7. Fano Fibrations and Twisted Kähler-Einstein Metrics I

    Alexander Bednarek PDF

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

  8. K-stability of Gorenstein Fano group compactifications with rank two

    Jae-Hyouk Lee, Kyeong-Dong Park and Sungmin Yoo PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, 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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) 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) 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)

In the guidebook

  1. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  2. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  3. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.