Paper · Feb 13, 2025

On Kähler-Einstein Currents

Yifan Chen, Shih-Kai Chiu, Max Hallgren, Gábor Székelyhidi, Tat Dat Tô, Freid Tong

math.DGarXiv:2502.09825
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Abstract

We show that a general class of singular Kähler metrics with Ricci curvature bounded below define Kähler currents. In particular the result applies to singular Kähler-Einstein metrics on klt pairs, and an analogous result holds for Kähler-Ricci solitons. In addition we show that if a singular Kähler-Einstein metric can be approximated by smooth metrics on a resolution whose Ricci curvature has negative part that is bounded uniformly in for , then the metric defines an RCD space.

Comments: 20 pages

Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryLimit spaces & RCDKähler–Einstein & K-stabilityAnalytic minimal model program

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  1. Remarks on Singular Kähler-Einstein Metrics

    Max Hallgren and Gábor Székelyhidi 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 Analytic minimal model program, Kähler geometry, Ricci solitons · Same author Max Hallgren · Shared terms “singular Kähler”

  3. Shrinking Kähler-Ricci solitons on toric fano fibrations

    Tristan C. Collins, Genggeng Huang, Freid Tong et al. PDF

    Topics Analytic minimal model program, Kähler–Ricci solitons, Kähler geometry · Same author Freid Tong

  4. Fano Fibrations and Twisted Kähler-Einstein Metrics I

    Alexander Bednarek PDF

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “singular Kähler”

  5. The Kähler-Ricci soliton on bounded pseudoconvex domains

    Zehao Sha PDF

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

  6. Scalar Curvature, Volumes and the Bergman Kernel

    Bo-Yong Chen, Yuanpu Xiong and Liyou Zhang PDF

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “singular Kähler”, “analogous”

  7. Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow

    Alexander Bednarek PDF

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Einstein metrics

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

    Daniel Hättig, Jürgen Hausen and Hendrik Süß PDF

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

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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. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.