Theorem 21.12.138B5B73657BF7

Proof. The bijection between 2 and 3 is the classical Legendre duality. Given $F$ as in 2, we construct $G=F^*$ and vice versa, see proposition 21.11.38.

The map from 1 to 2 is given as follows: Given $\varphi \in \mathrm{PSH}_{\mathrm{tor}}(X,\omega )$, since $\varphi $ is $T_ c$-invariant, we can find $f\colon N_{\mathbb {R}}\rightarrow [-\infty ,\infty )$ such that

We then define $F=f+F_0$. Then $\operatorname{Trop}^* F\in \mathrm{PSH}(T(\mathbb {C}))$. By 21.12.7, $F(n)$ is finite for any $n\in N_{\mathbb {R}}$ and $F$ is convex. Moreover, $F\preceq \operatorname{Supp}_ P$ since $\varphi $ is bounded from above on the compact space $X$ and $F_0\preceq \operatorname{Supp}_ P$.

Conversely, given a map $F\in \mathcal{P}(N_{\mathbb {R}},P)$, then

Since $F\preceq \operatorname{Supp}_ P$ and $F_0\sim \operatorname{Supp}_ P$, the difference $F-F_0$ is bounded from above. Hence the corresponding local psh potentials are locally bounded from above near $X\setminus T(\mathbb {C})$. It follows from theorem 5.14.10 that this function can be extended uniquely to an $\omega $-psh function on $X$. The uniqueness of the extension guarantees its $T_ c$-invariance.

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