The same definition in the compact Kähler setting. Decrease of the approximants makes the multiplier-ideal condition at one index equivalent to the condition at all sufficiently large indices.
Definition 6.17.5.165F88FAE668 Let $\theta $ be a smooth closed real $(1,1)$-form on $X$ and $\varphi \in \mathrm{PSH}(X,\theta )$. A quasi-equisingular approximation of $\varphi $ is a sequence $(\Phi _ j)_{j\geq 1}$ of quasi-plurisubharmonic functions with analytic singularities, decreasing pointwise to $\varphi $, such that $\theta +\mathrm{dd}^{\mathrm{c}}\Phi _ j\geq -\varepsilon _ j\omega $ for some numbers $\varepsilon _ j\downarrow 0$, and such that for all $0<\lambda <\lambda '$
\[ \mathcal I(\lambda '\Phi _ j)\subseteq \mathcal I(\lambda \varphi ) \quad \text{for every sufficiently large }j. \]
Read the alternative statement and proof
Definition 6.17.5.165F88FAE668 Let $X$ be a compact Kähler manifold and $\theta $ be a closed real smooth $(1,1)$-form on $X$. Consider $\varphi \in \mathrm{PSH}(X,\theta )$. A sequence $(\varphi _ j)_{j\in \mathbb {Z}_{>0}}$ in $\mathrm{QPSH}(X)$ is quasi-equisingular approximation of $\varphi $ if
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$\varphi _ j$ has analytic singularities for each $j$;
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$\varphi _ j$ is decreasing with limit $\varphi $;
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there is a decreasing sequence $\epsilon _ j\geq 0$ with limit $0$ and a Kähler form $\omega $ on $X$ such that $\varphi _ j\in \mathrm{PSH}(X,\theta +\epsilon _ j\omega )$;
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for each $\lambda '>\lambda >0$, there is $j>0$ such that
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