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 $\lambda ({\phi }_t,x)=\max \{\lambda ({\phi }_0,x)-t,0\}$. We construct counterexamples with analytic initial singularities in every dimension $n\ge 2$. On the Hirzebruch surface ${\mathbb{F}}_e$, $e\ge 2$, the initial current $[S]+(e+1)[F]$ yields $\lambda ({\phi }_t,x)=1/2-t/e$ for $x\in S\setminus F$ 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.