Corollary 18.1.14.F1CD22EF1353 Let $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$ and $(\epsilon _ j)_ j$ be a decreasing sequence in $\mathbb {R}_{\geq 0}$ with limit $0$. Fix a Kähler form $\omega $ on $X$. Consider a decreasing sequence $(\varphi _ j)_{j>0}$, where each $\varphi _ j\in \mathrm{PSH}(X,\theta +\epsilon _ j\omega )$ is a potential with analytic singularities. Assume that $\varphi =\inf _ j \varphi _ j$. Then the following are equivalent:
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$\varphi _ j\xrightarrow {d_ S}P_{\theta }[\varphi ]_{\mathcal{I}}$, and
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$(\varphi _ j)_ j$ is a quasi-equisingular approximation of $\varphi $.
Proof.
By corollary 16.9.36 and example 18.1.11, we may replace $\theta $ by $\theta +C\omega $ for some large constant $C>0$ and assume that $\varphi ,\varphi _ j\in \mathrm{PSH}(X,\theta -\omega )$ for all $j\geq 1$.
Here the limit in 1 is unchanged by this replacement. Indeed, $P_{\theta }[\varphi ]_{\mathcal I}$ is $\mathcal I$-model, hence $\mathcal I$-good by example 18.1.11. This implies
\[ P_{\theta +C\omega }\bigl [P_{\theta }[\varphi ]_{\mathcal I}\bigr ]=P_{\theta +C\omega }\bigl [P_{\theta }[\varphi ]_{\mathcal I}\bigr ]_{\mathcal I}=P_{\theta +C\omega }[\varphi ]_{\mathcal I}. \]
Thus the old and new targets are $P$-equivalent in the enlarged class, and proposition 16.9.6, corollary 16.9.36 shows that condition 1 is unchanged.
2 $\implies $ 1. This is already proved in the proof of theorem 18.1.2.
1 $\implies $ 2. This follows from theorem 16.9.59.
$\square$
Pome
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