PomeProposition 20.2.53.A2BD7EA3B4B4 Assume that $n\geq 2$. Let $\varphi ,\psi \in \mathrm{QPSH}(X)$. Assume that $\varphi \preceq _ P \psi $. Then quasi-every $H\in \Lambda $ is connected and smooth, and $\varphi |_ H\preceq _ P \psi |_ H$.
Proposition 20.2.53A2BD7EA3B4B4
Proof. Thanks to 16.8.13, we may replace $\varphi $ by $\varphi \lor \psi $ and assume that $\varphi \sim _ P \psi $. It suffices to show that $\varphi |_ H\sim _ P \psi |_ H$ for quasi-every $H\in \Lambda $.
Take a smooth closed real $(1,1)$-form $\theta $ on $X$ so that $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. It suffices to compare $\varphi $ and $\psi $ with $P_{\theta }[\varphi ]$, so without loss of generality, we may assume that $\psi $ is a model potential in $\mathrm{PSH}(X,\theta )_{>0}$. Up to adding a constant to $\varphi $, we may then assume that $\varphi \leq \psi $. It follows from 13.13.30 that we can find a sequence $(\eta _ j)_ j$ in $\mathrm{PSH}(X,\theta )_{>0}$ such that
\[ j^{-1}\eta _ j+\left(1-j^{-1}\right)\psi \leq \varphi \]
for all $j\geq 2$. By corollary 20.2.49, 20.2.51, we can find a co-pluripolar set $\Lambda '\subseteq \Lambda $ such that any $H\in \Lambda '$ satisfies:
402CADA317DA$H$ is smooth and connected;
001701E8EE1D$\eta _ j|_ H\in \mathrm{PSH}(H,\theta |_ H)_{>0}$ for all $j\geq 2$ and $\psi |_ H\in \mathrm{PSH}(H,\theta |_ H)_{>0}$.
Therefore, taking proposition 16.4.2 into account, we arrive at
\[ j^{-1} P_{\theta |_ H}\bigl [\eta _ j|_ H\bigr ]+\left(1-j^{-1}\right)P_{\theta |_ H}\bigl [\psi |_ H\bigr ]\leq P_{\theta |_ H}\bigl [\varphi |_ H\bigr ] \]
for all $j\geq 2$. The normalized envelopes $P_{\theta |_ H}[\eta _ j|_ H]$ are uniformly bounded in $L^1$, hence
\[ j^{-1}P_{\theta |_ H}[\eta _ j|_ H]\xrightarrow {L^1} 0. \]
We conclude that
\[ P_{\theta |_ H}[\psi |_ H]\leq P_{\theta |_ H}[\varphi |_ H] \]
and hence $\psi |_ H\preceq _ P \varphi |_ H$. $\square$
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