Lemma 22.1.92A8896460B129

Proof. After replacing $L$ with a sufficiently high power, we may assume that $L$ is a line bundle. Take a very ample line bundle $H$ on $X$. By Siu's uniform global generation theorem [Siu98], [Theorem 6.27, Dem12] there exists $b>0$ large enough so that $H^{b}\otimes L^ k\otimes \mathcal{I}(k\varphi )$ is globally generated for all $k>0$. Let $\{ s_ i\} _ i$ be a finite set of global sections that generate the sheaf $H^{b}\otimes L^ k\otimes \mathcal{I}(k\varphi )$. Then

It follows that $v\mapsto -k^{-1}v\left(\mathcal{I}(k\varphi ) \right)$ lies in $\mathcal{H}^{\mathrm{gf}}_{\mathbb {Q}}((L+\frac{b}{k}H)^{\mathrm{An}})$. Using (22.1.81), we conclude that $\varphi ^{\mathrm{An}}\in \mathrm{PSH}(L^{\mathrm{An}})$. $\square$