Failure of the Proposed Local Decay Formula for Local Arnold Multiplicities under Twisted K\"ahler--Ricci Flow

Abstract

Let $\lambda(u,x)$ be the local Arnold multiplicity of a quasi-plurisubharmonic function $u$. Di Nezza--Guedj--Lu asked whether every maximal weak solution $\varphi_t$ of the twisted K\"ahler--Ricci flow satisfies $\lambda(\varphi_t,x)=\max\{\lambda(\varphi_0,x)-t,0\}$. We give counterexamples on the Hirzebruch surface $\mathbb F_e=\mathbb P_{\mathbb P^1} (\mathcal O_{\mathbb P^1}\oplus\mathcal O_{\mathbb P^1}(-e))$, $e\ge2$. Let $S$ be its negative section and $F_1,\ldots,F_k$ be distinct fibres. If $a,b_i>0$, $\sum_i b_i>ea$, and the initial current is $a[S]+\sum_i b_i[F_i]$, then $\lambda(\varphi_t,x)=a-\min\{k/e,1\}t$ for $x\in S\setminus\bigcup_iF_i$ and $0<t<\min\{a,b_1,\ldots,b_k\}$. Thus the formula fails for $k<e$; the slower decay is forced by the Zariski negative part of the residual class. Under explicit SNC and positivity hypotheses, we also prove an exact surface formula for pure divisorial data. Consequences include nonlocality with fixed background data and explicit multiplier ideals. Products with smooth factors give counterexamples in every complex dimension $n\ge2$.