Definition 5.18.10FFAA7E68EEC1

Definition 5.18.10.FFAA7E68EEC1 Let $\mathcal{I}\subseteq \mathcal O_ X$ be a coherent ideal sheaf and $c\in \mathbb {Q}_{>0}$. A function $\varphi \in \mathrm{QPSH}(X)$ is said to have gentle analytic singularities (of type $(c,\mathcal{I})$) if

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   $\varphi $ has analytic singularities of type $(c,\mathcal{I})$;

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   $\mathrm{e}^{{\varphi }/{c}}\colon X\to \mathbb {R}_{\geq 0}$ is a smooth function;

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   there is a proper bimeromorphic morphism $\pi \colon \widetilde{X}\to X$ from a Kähler manifold $\widetilde{X}$ and an effective $\mathbb {Z}$-divisor $D$ on $\widetilde{X}$ such that one can write $\pi ^*\varphi $ locally as

   \[ \pi ^*\varphi =c\log |g|^2+h, \]

   where $g$ is a local equation of the divisor $D$ and $h$ is smooth.