Paper · Dec 26, 2025

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

math.DGarXiv:2512.21910

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Abstract

This is the second of two papers studying both the geometric structure of Fano fibrations and the application to Kähler-Ricci flows developing a singularity in finite time. We assume that the Kähler-Ricci flow on a compact Kähler manifold has a rational initial metric and develops a singularity in finite time such that the manifold admits a Fano fibration structure. Moreover, it is assumed that the volume form of the flow collapses uniformly at the rate of $C^{- 1} (T - t)^{n - m} Ω \leq ω (t)^{n} \leq C (T - t)^{n - m} Ω$ . Under this setting, a diameter bound is obtained in any compact set away from singular fibres and the diameter of the fibres is proven to collapse at the optimal rate $T - t$ . Furthermore, several precise $C^{0}$ -estimates are proven for the potential of the complex Monge-Ampere flow which involve the potentials of singular twisted Kähler-Einstein metrics on the base variety from part I. Finally, in the case of Kähler-Einstein Fano fibres, we deduce Type I scalar curvature in any compact set away from singular fibres and globally for a submersion.

Comments: 45 pages

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

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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.

Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other

Fano Fibrations and Twisted Kähler-Einstein Metrics I
Alexander Bednarek2025 PDF

Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Same author Alexander Bednarek · Shared terms “twisted Kähler”, “Fano fibration”, “flow developing”
Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II
Wangjian Jian and Jian Song2026 PDF

Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “Fano”, “finite time”, “diameter”
On Miyaoka-Yau Inequalities and Weil-Petersson Metrics
Alexander Bednarek2026 PDF

Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Same author Alexander Bednarek · Shared terms “Fano”, “compact Kähler”
Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces
Ye-Won Luke Cho and Young-Jun Choi2024 PDF

Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “complex Monge”, “Monge Ampère”, “compact Kähler”
$L^{\infty}$ estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation
Wangjian Jian and Yalong Shi2023 PDF

Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “Einstein Fano”, “Monge Ampère”
Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities
Alexander Bednarek2024 PDF

Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Alexander Bednarek · Shared terms “finite time”, “rational”
Type I estimates for the Kähler-Ricci flow II
Wangjian Jian and Jian Song2026 PDF

Topics Analytic minimal model program, Kähler–Ricci flow, Scalar curvature · Shared terms “diameter bound”, “finite time”, “compact Kähler”
Scalar Curvature, Volumes and the Bergman Kernel
Bo-Yong Chen, Yuanpu Xiong and Liyou Zhang2026 PDF

Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “compact Kähler”

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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.

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

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