PomeLemma 22.1.107.AB6A7C1FD159 Let $\phi \in \mathcal{H}^{\mathrm{gf}}_{\mathbb {Q}}(L^{\mathrm{An}})$. Then there is a piecewise linear curve $A$ in $\mathrm{PSH}(X,\theta )$ with $\phi =A^{\mathrm{An}}$. In particular, $\phi $ is in the image of (22.1.102).
Lemma 22.1.107AB6A7C1FD159
Proof. Let us write
C18150DD4B94
22.1.108
\begin{equation} \phi =\frac{1}{m}\max _{i=1,\ldots ,M} \left(\log |s_ i|+\lambda _ i\right), \end{equation}
where $m\in \mathbb {Z}_{>0}$, $s_{1},\ldots , s_ M$ are a finite number of sections of $L^ m$ and $\lambda _1,\ldots ,\lambda _ M\in \mathbb {Q}$.
Write $I_{\tau }$ for the set of $i$ such that $m^{-1}\lambda _ i=\tau $. We denote the finitely many $\tau $ so that $I_{\tau }$ is non-empty as $\tau _0> \cdots > \tau _ N$. For each $i=0,\ldots ,N$, we write
\[ A_{\tau _ i}=\frac{1}{m}\max _{j\in I_{\tau _ i}} \log |s_ j|^2_{h^ m}. \]
We define $A$ as the piecewise linear curve associated with the $A_{\tau _ i}$'s and the $\tau _ i$'s. For every $v\in X^{\mathrm{An}}$,
\[ A^{\mathrm{An}}(v)=\max _{i=0,\ldots ,N}\left(A_{\tau _ i}^{\mathrm{An}}(v)+\tau _ i\right) =\frac{1}{m}\max _{j=1,\ldots ,M}\left(\log |s_ j|(v)+\lambda _ j\right)=\phi (v), \]
where we used the convention $\log |s_ j|(v)=-v(s_ j)$.
The final assertion follows from 22.1.89 and theorem 22.1.72. $\square$
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