Theorem 6.17.6080EF8C715CF

Proof. Put $\Phi _ j=G_{2^ j}+2D/2^ j$, with $D$ as in lemma 6.17.3. The doubling inequality gives

so the sequence is decreasing, and by lemma 6.17.3 its pointwise and $L^1$ limit is $\varphi $. Adding a constant changes neither the curvature, nor the analytic type, nor the Lelong numbers. Moreover $\exp (2^ j\Phi _ j)=\mathrm e^{2D}R_{2^ j}$ is smooth, and the remainder on a log resolution changes only by a constant. Hence $\Phi _ j$ has gentle analytic singularities of type $(2^{-j},\mathcal I(2^ j\varphi ))$ by lemma 6.17.1(1), and $\theta +\mathrm{dd}^{\mathrm{c}}\Phi _ j\geq -c'2^{-j}\omega $.

It remains to prove the inclusions of multiplier ideals. Let $0<\lambda <\lambda '$. The function $\exp (2^ j(\Phi _ j-\varphi ))=\mathrm e^{2D}\mathrm e^{2^ j(G_{2^ j}-\varphi )}$ is integrable on $X$ by lemma 6.17.3. If $j$ is so large that $2^ j\geq \lambda \lambda '/(\lambda '-\lambda )$, then lemma 6.17.4, with $u=\varphi $, $v=\Phi _ j$ and $m=2^ j$, gives $\mathcal I(\lambda '\Phi _ j)_ x\subseteq \mathcal I(\lambda \varphi )_ x$ at every point $x\in X$. Since this threshold for $j$ depends only on $\lambda $ and $\lambda '$, the inclusion holds as an inclusion of sheaves on $X$ for all such $j$. $\square$