Lemma 17.15.6.5BE2CC04AAD0 Let $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$ and $\Gamma ' \in \mathrm{TC}(X,\theta ')_{>0}$. Then we have $\Gamma +\Gamma '\in \mathrm{TC}(X,\theta +\theta ')_{>0}$. In particular, for any $\tau <(\Gamma +\Gamma ')_{\max }$,
\[ \sup _{\delta \in \mathbb {R}} \left(\Gamma _\delta +\Gamma '_{\tau -\delta }\right)\in \mathrm{PSH}(X,\theta +\theta '). \]
This potential is $\mathcal{I}$-good if $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ and $\Gamma ' \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta ')_{>0}$.
In particular, (17.15.5) in definition 17.15.4 makes sense.
Proof.
Let
\[ \eta _{\tau }=\sup _{\delta \in \mathbb {R}} \left(\Gamma _\delta +\Gamma '_{\tau -\delta }\right)=\sup _{\tau -\Gamma '_{\max }<\delta <\Gamma _{\max }}\left(\Gamma _\delta +\Gamma '_{\tau -\delta }\right) \]
for all $\tau \in \mathbb {R}$. Set
\[ Z=\left\{ x\in X: \Gamma _{\upsilon }(x)=-\infty \, \forall \upsilon \in \mathbb {R}\right\} \cup \left\{ x\in X: \Gamma '_{\upsilon }(x)=-\infty \, \forall \upsilon \in \mathbb {R}\right\} . \]
It follows from proposition 21.11.37 that for any $x\in X\setminus Z$, we have
\[ \eta ^*_ t(x)=\Gamma ^*_ t(x)+\Gamma '^*_ t(x) \]
for all $t>0$. The same trivially holds when $x\in Z$, so the equation holds everywhere. In particular, by corollary 21.11.26 and proposition 5.14.15, we have
\[ \eta _{\tau }=\left(\Gamma ^*+\Gamma '^*\right)^*_{\tau }\in \mathrm{PSH}(X,\theta +\theta ') \]
when $\tau <\Gamma _{\max }+\Gamma '_{\max }$.
We now check that $\Gamma +\Gamma '$ is a test curve. The family $\tau \mapsto \eta _\tau $ is decreasing, and the monotonicity of the $P$-envelope proposition 16.4.2 3 gives the decreasingness of $\Gamma +\Gamma '$. If $\tau =(1-s)\tau _0+s\tau _1$, then for arbitrary $\delta _0,\delta _1\in \mathbb {R}$ and $\delta =(1-s)\delta _0+s\delta _1$,
\[ \Gamma _\delta +\Gamma '_{\tau -\delta }\geq (1-s)(\Gamma _{\delta _0}+\Gamma '_{\tau _0-\delta _0})+s(\Gamma _{\delta _1}+\Gamma '_{\tau _1-\delta _1}). \]
Taking suprema and using proposition 16.4.2 gives the concavity of $\Gamma +\Gamma '$.
Fix $\tau <\Gamma _{\max }+\Gamma '_{\max }$. Choose $\delta \in \mathbb {R}$ with $\delta <\Gamma _{\max }$ and $\tau -\delta <\Gamma '_{\max }$. Then
\[ \eta _\tau \geq \Gamma _\delta +\Gamma '_{\tau -\delta }. \]
By 17.12.6, $\Gamma _\delta \in \mathrm{PSH}(X,\theta )_{>0}$ and $\Gamma '_{\tau -\delta }\in \mathrm{PSH}(X,\theta ')_{>0}$. Hence $\Gamma _\delta +\Gamma '_{\tau -\delta }$ has positive $(\theta +\theta ')$-mass by the multi-linearity of non-pluripolar products proposition 13.11.4 6. It follows from the monotonicity theorem theorem 13.13.11 that $\eta _\tau \in \mathrm{PSH}(X,\theta +\theta ')_{>0}$. By corollary 16.3.20,
\[ (\Gamma +\Gamma ')_\tau =P_{\theta +\theta '}[\eta _\tau ] \]
is a model potential. Finally, $(\Gamma +\Gamma ')_{-\infty }\in \mathrm{PSH}(X,\theta +\theta ')$ by proposition 5.14.2; since it dominates $(\Gamma +\Gamma ')_\tau $ for any $\tau <\Gamma _{\max }+\Gamma '_{\max }$, it has positive mass by another application of the monotonicity theorem theorem 13.13.11. Thus $\Gamma +\Gamma '\in \mathrm{TC}(X,\theta +\theta ')_{>0}$.
Next, assume that $\Gamma $ and $\Gamma '$ are $\mathcal{I}$-model. Fix $\tau <\Gamma _{\max }+\Gamma '_{\max }$. Then for each $\delta \in \mathbb {R}$ such that $\delta <\Gamma _{\max }$ and $\tau -\delta <\Gamma '_{\max }$, we know that $\Gamma _\delta \in \mathrm{PSH}(X,\theta )_{>0}$ and $\Gamma '_{\tau -\delta }\in \mathrm{PSH}(X,\theta ')_{>0}$ by 17.12.6. It follows from example 18.1.11 that $\Gamma _\delta $ and $\Gamma '_{\tau -\delta }$ are both $\mathcal{I}$-good, hence so is $\Gamma _\delta +\Gamma '_{\tau -\delta }\in \mathrm{PSH}(X,\theta +\theta ')_{>0}$ by proposition 18.1.17. These sums are bounded above by $0$, so $\eta _{\tau }$ is $\mathcal{I}$-good by proposition 18.1.19. Since $(\Gamma +\Gamma ')_\tau =P_{\theta +\theta '}[\eta _\tau ]$, we conclude that $\Gamma +\Gamma '$ is $\mathcal{I}$-model.
$\square$
Pome
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