Pome

Proposition 16.8.298C181599D520

Proposition 16.8.29.8C181599D520 Let $\varphi ,\psi ,\varphi ',\psi '\in \mathrm{PSH}(X,\theta )$ for some closed smooth real $(1,1)$-form $\theta $ on $X$. Assume that

\[ \varphi \sim _ P \varphi ',\quad \psi \sim _ P\psi ',\quad \varphi '\land \psi '\in \mathrm{PSH}(X,\theta )_{>0}. \]

Then

\[ \varphi \land \psi \in \mathrm{PSH}(X,\theta )_{>0},\quad \varphi \land \psi \sim _ P \varphi '\land \psi '. \]

Proof. We first observe that $\varphi ,\varphi ',\psi ,\psi '\in \mathrm{PSH}(X,\theta )_{>0}$ by assumption. Let

\[ \phi \coloneqq P_{\theta }[\varphi ]=P_{\theta }[\varphi '],\quad \gamma =P_{\theta }[\psi ]=P_{\theta }[\psi ']. \]

Since $\varphi '\preceq \phi $ and $\psi '\preceq \gamma $, after adding a common constant to $\varphi '$ and $\psi '$, we may assume that $\varphi '\leq \phi $ and $\psi '\leq \gamma $. Then $\varphi '\land \psi '\leq \phi \land \gamma $, so the assumption $\varphi '\land \psi '\in \mathrm{PSH}(X,\theta )_{>0}$ implies $\phi \land \gamma \in \mathrm{PSH}(X,\theta )_{>0}$ by monotonicity. It follows from 16.3.32 that

\[ \int _ X \theta _{\varphi '\land \psi '}^ n=\int _ X \theta _{\phi \land \gamma }^ n. \]

Next, we apply 16.3.32 again to conclude that $\varphi \land \psi \in \mathrm{PSH}(X,\theta )$ and

\[ \int _ X \theta _{\varphi \land \psi }^ n=\int _ X \theta _{\phi \land \gamma }^ n=\int _ X \theta _{\varphi '\land \psi '}^ n>0. \]

Since $\varphi \land \psi \preceq \phi \land \gamma $ and the two potentials have the same positive mass, theorem 16.3.15 gives $P_{\theta }[\varphi \land \psi ]=\phi \land \gamma $. The same argument gives $P_{\theta }[\varphi '\land \psi ']=\phi \land \gamma $, hence $\varphi \land \psi \sim _ P\varphi '\land \psi '$. $\square$

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