Pome

Proposition 16.8.27C42706880DED

Proposition 16.8.27.C42706880DED Let $\varphi ,\psi ,\varphi ',\psi '\in \mathrm{QPSH}(X)$. Assume that

\[ \varphi \preceq _ P \psi ,\quad \varphi '\preceq _ P \psi '. \]

Then

\[ \varphi +\varphi '\preceq _ P \psi +\psi '. \]

The same holds with $\preceq _{\mathcal{I}}$ in place of $\preceq _ P$.

Proof. Take a Kähler form $\omega $ on $X$ such that $\varphi ,\psi ,\varphi ',\psi '\in \mathrm{PSH}(X,\omega )_{>0}$. The statement for $\preceq _{\mathcal{I}}$ is a simple consequence of proposition 5.16.8. We only need to handle the case of $\preceq _{P}$.

Step 1. We first show that

\[ P_{\omega }[\varphi ]+P_{\omega }[\varphi ']\sim _ P \varphi +\varphi '. \]

In fact, we clearly have

\[ P_{\omega }[\varphi ]+P_{\omega }[\varphi ']\succeq \varphi +\varphi '. \]

So by proposition 16.8.7, it suffices to show that they have the same mass. We compute

\[ \begin{aligned} & \int _ X \Bigl (2\omega +\mathrm{dd}^{\mathrm{c}}P_{\omega }[\varphi ]+\mathrm{dd}^{\mathrm{c}}P_{\omega }[\varphi ']\Bigr )^ n\\ =& \sum _{j=0}^ n \binom {n}{j} \int _ X \Bigl (\omega +\mathrm{dd}^{\mathrm{c}}P_{\omega }[\varphi ]\Bigr )^ j\wedge \Bigl (\omega +\mathrm{dd}^{\mathrm{c}}P_{\omega }[\varphi ']\Bigr )^{n-j}\\ =& \sum _{j=0}^ n \binom {n}{j} \int _ X \omega _{\varphi }^ j\wedge \omega _{\varphi '}^{n-j}\\ =& \int _ X \left(2\omega +\mathrm{dd}^{\mathrm{c}}\varphi +\mathrm{dd}^{\mathrm{c}}\varphi '\right)^ n, \end{aligned} \]

where we applied proposition 16.3.10 to the mixed products on the third line.

Step 2. By Step 1, we may assume that $\varphi ,\psi ,\varphi ',\psi '$ are all model potentials, after replacing them by their $P_{\omega }$-envelopes. So $\varphi \leq \psi $ and $\varphi '\leq \psi '$. Our assertion follows. $\square$

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