Failure of the local decay formula for Arnold multiplicities under twisted Kähler–Ricci flow

Xiangsen Qin

Paper · Aug 31, 2026

Failure of the local decay formula for Arnold multiplicities under twisted Kähler–Ricci flow

Xiangsen Qin

math.DGarXiv:2608.30302v2

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Abstract

Di Nezza–Guedj–Lu asked whether local Arnold multiplicities under maximal weak twisted Kähler–Ricci flow with fixed cohomology class satisfy $λ (φ_{t}, x) = max {λ (φ_{0}, x) - t, 0}$ . We construct counterexamples with analytic initial singularities in every dimension $n \geq 2$ . On the Hirzebruch surface $F_{e}$ , $e \geq 2$ , the initial current $[S] + (e + 1) [F]$ yields $λ (φ_{t}, x) = 1/2 - t / e$ for $x \in S ∖ F$ and $0 < t < e /2$ , where $S$ is the negative section and $F$ a fibre. Products with smooth stationary factors give higher-dimensional examples. The Zariski negative part of ${S} + {F}$ , the reduced support class, forces extra divisor mass. For pure divisorial data with simple normal crossings support, a smooth semipositive representative of the reduced support class's positive part gives exact short-time formulas. On one surface we construct flows with identical initial current germs, background data and global initial Arnold multiplicity, but different local rates.

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Submitted: Aug 31, 2026
Last revised: Oct 26, 2026 (v2)
Subjects: math.DG, math.AP, math.CV

Ricci flow Kähler–Ricci flow Singularities Kähler geometry

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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) 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) 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

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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.
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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.

Failure of the local decay formula for Arnold multiplicities under twisted Kähler–Ricci flow

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