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

Ye-Won Luke Cho; Young-Jun Choi

Paper · Jan 8, 2024

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

Ye-Won Luke Cho, Young-Jun Choi

math.DGarXiv:2401.03935v2

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Abstract

We prove the continuity of bounded solutions to complex Monge-Ampère equations on reduced, locally irreducible compact Kähler spaces. This in particular implies that any singular Kähler-Einstein potentials constructed in and are continuous. We also provide an affirmative answer to a conjecture in by showing that a resolution of any compact normal Kähler space satisfies the continuous approximation property. Finally, we settle the continuity of the potentials of the weak Kähler-Ricci flows on compact Kähler varieties with log terminal singularities.

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Submitted: Jan 8, 2024
Last revised: Aug 14, 2025 (v2)
Subjects: math.DG, math.AG, math.CV

Comments: 20 pages. Comments are welcome. The final version is to appear in Math. Ann

Ricci flow Kähler–Ricci flow Einstein metrics Singularities Kähler geometry Kähler–Einstein & K-stability Analytic minimal model program

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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 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) 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)

In the guidebook

5A.6
Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry
The Kähler–Ricci flow, and when to come back to it.
11B.4
Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.
11B.6
Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities
The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

[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] 11B.4
Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

[3] 11B.6
Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities
The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

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

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