Pome9348BF57481BTheorem 5.18.11.9348BF57481B Let $X$ be a compact Kähler manifold and $\theta $ be a closed real smooth $(1,1)$-form on $X$. Then any $\varphi \in \mathrm{PSH}(X,\theta )$ admits a quasi-equisingular approximation $(\varphi _ j)_{j\in \mathbb {Z}_{>0}}$.
Moreover, we can guarantee that for all $j>0$, $\varphi _ j$ has gentle analytic singularities of type $\left(2^{-j},\mathcal{I}(2^ j\varphi )\right)$.
Proof. Apply the gentle quasi-equisingular approximation theorem to $(X,\theta,\varphi)$. It constructs approximants of type $(2^{-j},\mathcal I(2^j\varphi))$, with a curvature error tending to zero, convergence in $L^1$, and the required multiplier-ideal inclusions. These are the defining conditions of a quasi-equisingular approximation in this section. Its approximants have gentle analytic singularities, so they also supply the additional regularity asserted here. The powers $2^j$ fix a subsequence of Bergman approximations; they impose no extra assumption on $\varphi$.
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