Theorem 5.36.5 (Demailly regularization)88038BD901A0

Theorem 5.36.5 (Demailly regularization).88038BD901A0 Let $\alpha $ be a smooth real $(1,1)$-form and let $u\in \mathrm{PSH}(X,\alpha )$. There are numbers $\delta _ j\downarrow 0$ and functions $\widetilde u_ j\in \mathrm{PSH}(X,\alpha +\delta _ j\omega )$ decreasing to $u$ with the following properties. Each $\widetilde u_ j$ is smooth outside an analytic pole set, and there is a modification $\pi _ j:Y_ j\to X$, obtained by a finite sequence of blowups with smooth centers, such that locally on $Y_ j$

Here the $s_\ell $ are local holomorphic defining functions of divisors, the $a_\ell >0$ are real numbers, and $b$ is a smooth real function. The divisors may be taken to have simple normal crossings.