Theorem 21.13.5346AFD20390FF

Proof. The proof is similar to that of theorem 21.12.13, but due to its importance, we give the details. Again, the correspondence between 2 and 3 follows easily from proposition 21.11.38.

Given $\varphi $, we can construct $F_{\varphi }$ in 2 as explained earlier in (21.13.26). Conversely, suppose that $F\in \mathcal{P}(N_{\mathbb {R}},P_ D)$, then $F\preceq F_{V_{\theta }}$ by theorem 21.13.45. Then

by 21.12.7. Since $F\preceq F_{V_{\theta }}$, we see that $\operatorname{Trop}^*(F-F_{\theta })$ is bounded from above. It follows that Grauert–Remmert's extension theorem theorem 5.14.10 is applicable, and this function extends to a unique $\theta $-psh function $\varphi $. The uniqueness of the extension guarantees that $\varphi \in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )$.

The two maps are inverse to each other by construction. $\square$