Proposition 6.18.2649F62FDFBAD

Proof. Choose $0<\eta <\varepsilon $ and $\delta >0$ so small that $(1-\eta )/(1+\delta )\geq 1-\varepsilon $. Fix $t\geq t_0$, where $t_0$ is given by lemma 6.18.1 for $v=\Phi $ and $\eta $. By the hypothesis on $(\Psi _ k)$, applied with $\lambda =t$ and $\lambda '=t(1+\delta )$, and since $\varphi \leq \Phi $,

for every sufficiently large $k$. Fix such a $k$ and choose local generators $g_ i$ of $\mathcal I(t(1+\delta )\Psi _ k)$ near a point of $X$. They belong to $\mathcal I(t\Phi )$, so the first part of lemma 6.18.1 gives $\log |g_ i|^2\leq t(1-\eta )\Phi +O(1)$ near the point, and hence $\log \sum _ i|g_ i|^2\leq t(1-\eta )\Phi +O(1)$. Its second part, applied to $u=\Psi _ k$ and $t(1+\delta )$, then gives

The upper semicontinuous function $\Phi $ is bounded above on the compact space $X$, say by $M$. Since $\Phi -M\leq 0$ and $(1-\eta )/(1+\delta )\geq 1-\varepsilon $, we have $\frac{1-\eta }{1+\delta }(\Phi -M)\leq (1-\varepsilon )(\Phi -M)$, and therefore $\Psi _ k\leq (1-\varepsilon )\Phi +O(1)$ near every point. By compactness $\Psi _ k\preceq (1-\varepsilon )\Phi $; the constant may depend on $k$.

For the last assertion, recall from definition 6.17.5 that each $\Phi _ j$ is quasi-plurisubharmonic with analytic singularities and that $\varphi \leq \Phi _ j$, since the sequence decreases to $\varphi $; and $(\Psi _ k)$ satisfies the hypothesis on the inclusions of multiplier ideals. Hence the first part applies with $\Phi =\Phi _ j$. $\square$