PomeBA0E41A3C821Corollary 6.19.3.BA0E41A3C821 Let $v$ be a quasi-plurisubharmonic function with analytic singularities on a compact complex manifold $X$. For every $\varepsilon >0$ there is $k_0>0$ such that for every real $k\geq k_0$
Proof. Apply lemma 6.18.1 with $\eta =\varepsilon /(1+\varepsilon )$ and $t=k(1+\varepsilon )$, so that $t(1-\eta )=k$. The lemma provides one threshold $t_0$ for all germs on $X$; put $k_0=t_0/(1+\varepsilon )$. For $k\geq k_0$, every germ $f\in \mathcal I(k(1+\varepsilon )v)$ satisfies $\log |f|^2\leq kv+O(1)$ near its base point, that is, $f\in \mathcal I_\infty (kv)$. In dimension zero all these ideals are the unit ideal. $\square$
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