Paper · Oct 15, 2026

Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class

math.DGarXiv:2610.21934

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Abstract

Let $X$ be a compact Kähler manifold with semi-ample canonical bundle. For any Kähler class $[ω]$ , Jian–Shi–Song proved that there exists a unique cscK metric in the class $2 π c_{1} (K_{X}) + ε [ω]$ when $ε \in (0, ε_{0}]$ , using the properness criterion of Chen–Cheng. We prove that the cscK metrics in $2 π c_{1} (K_{X}) + ε [ω]$ converge, as $ε ↓ 0$ , in the Gromov–Hausdorff topology to the metric completion of the twisted Kähler-Einstein metric on the canonical model of $X$ as conjectured by Jian–Shi–Song. Based on the technique developed in this paper, we can also give a new proof the theorem of Lee–Tosatti–Zhang on the Gromov-Hausdorff convergence of the normalized Kähler–Ricci flow on $X$ .

Comments: 41 pages; comments are welcome

Ricci flow Kähler–Ricci flow Einstein metrics Kähler geometry Scalar curvature Limit spaces & RCD Kähler–Einstein & K-stability Analytic minimal model program

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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) 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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5A.6
Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry
The Kähler–Ricci flow, and when to come back to it.
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.
9B.4
Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds
Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class

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