Failure of the local decay formula for Arnold multiplicities under twisted Kähler--Ricci flow
Abstract
Di Nezza--Guedj--Lu asked whether local Arnold multiplicities under maximal weak twisted Kähler--Ricci flow with fixed cohomology class satisfy λ(φt,x)=max{λ(φ0,x)−t,0}. We construct counterexamples with analytic initial singularities in every dimension n≥2. On the Hirzebruch surface 𝔽e, e≥2, the initial current [S]+(e+1)[F] yields λ(φt,x)=1/2−t/e for x∈ 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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BibTeX
@misc{qin2026failure,
title = {{Failure of the local decay formula for Arnold multiplicities under twisted Kähler--Ricci flow}},
author = {Xiangsen Qin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.30302},
eprint = {2608.30302},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30302},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Failure of the local decay formula for Arnold multiplicities under twisted Kähler--Ricci flow
AU - Xiangsen Qin
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2608.30302
UR - https://arxiv.org/abs/2608.30302
AB - 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≥2. On the Hirzebruch surface 𝔽_e, e≥2, the initial current [S]+(e+1)[F] yields λ(φ_t,x)=1/2−t/e for x∈ 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.
ER -