Corollary 22.1.1009D2A9B228E44

Proof. Observe that the extension is unique if it exists by theorem 22.1.72. We may assume that $\Gamma _{\max }=0$ without loss of generality.

When $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$, our assertion follows from proposition 22.1.98 and proposition 22.1.84.

In general, fix an ample line bundle $H$ on $X$ and a Kähler form $\omega \in c_1(H)$. Then we know that

for any $m\in \mathbb {Z}_{>0}$. Therefore, $\Gamma ^{\mathrm{An}}\in \mathrm{PSH}(L^{\mathrm{An}})$ by proposition 22.1.74. $\square$