Proposition 22.1.6.2CB4FA8703B1 There is a natural injective map
\[ \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}\hookrightarrow \mathrm{PSH}^{\mathrm{NA}}(X,\theta ),\quad \Gamma \mapsto \left(P_{\theta +\omega }[\Gamma ]_{\mathcal{I}}\right)_{\omega \in \textup{Käh}(X)}. \]
Proof.
The map is well-defined by the compatibility of the transition maps. It suffices to argue its injectivity. Suppose that $\Gamma ,\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ and
\[ P_{\theta +\omega }[\Gamma ]_{\mathcal{I}}=P_{\theta +\omega }[\Gamma ']_{\mathcal{I}} \]
for some Kähler form $\omega $ on $X$. Then $\Gamma _{\max }=\Gamma '_{\max }$ and for any $\tau <\Gamma _{\max }$, we have
\[ \Gamma _{\tau }\sim _{\mathcal{I}}\Gamma '_{\tau } \]
by proposition 16.8.17. It follows again from proposition 16.8.17 that
\[ \Gamma _{\tau }=\Gamma '_{\tau }. \]
This completes the proof.
$\square$
Pome
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