Pome36EC36A38FABProposition 5.18.12.36EC36A38FAB Let $X$ be a compact Kähler manifold and $\theta $ be a closed real smooth $(1,1)$-form on $X$. Consider $\varphi \in \mathrm{PSH}(X,\theta )$. Let $(\varphi _ j)_ j$ and $(\psi _ j)_ j$ be two quasi-equisingular approximations of $\varphi $. Then for any $\epsilon \in (0,1)$ and any $j>0$, we can find $k_0>0$ such that for any $k\geq k_0$, we have
Proof. The final assertion of the comparison theorem for analytic approximants applies to the two given quasi-equisingular approximations. Its hypotheses are precisely compact Kähler $X$, a smooth closed background $\theta$, a $\theta$-plurisubharmonic limit, and the multiplier-ideal inclusions in the definition of quasi-equisingularity. For every fixed $j$ and $0<\varepsilon<1$ it yields $\psi_k\preceq(1-\varepsilon)\varphi_j$ for every sufficiently large $k$. The index is allowed to depend on $j$ and $\varepsilon$. In particular, no global pointwise ordering of two unrelated approximation sequences is being assumed.
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