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Lemma 18.1.822BB587E13F8

Lemma 18.1.8.22BB587E13F8 Let $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$ for some smooth closed real $(1,1)$-form $\theta $ on $X$. Take a Kähler form $\omega $ on $X$. Then the following are equivalent:

  1. 9755050D9C5E

    $P_{\theta }[\varphi ]=P_{\theta }[\varphi ]_{\mathcal{I}}$;

  2. D166FB4A811A

    $P_{\theta +\omega }[\varphi ]=P_{\theta +\omega }[\varphi ]_{\mathcal{I}}$.

Proof. 1 $\implies $ 2. By theorem 18.1.2, we can find a sequence $(\varphi _ j)_ j$ in $\mathrm{PSH}(X,\theta )$ with analytic singularities such that $\varphi _ j\xrightarrow {d_{S,\theta }}\varphi $. By corollary 16.9.36, we have $\varphi _ j\xrightarrow {d_{S,\theta +\omega }}\varphi $. Therefore, by theorem 18.1.2 again, 2 holds.

2 $\implies $ 1. Suppose that 1 fails, so that

\[ \int _ X (\theta +\mathrm{dd}^{\mathrm{c}}\varphi )^ n< \int _ X (\theta +\mathrm{dd}^{\mathrm{c}}P_{\theta }[\varphi ]_{\mathcal{I}})^ n. \]

Since $\varphi \preceq P_{\theta }[\varphi ]_{\mathcal I}$, the monotonicity theorem theorem 13.13.11 gives

\[ \int _ X \theta _{\varphi }^ i\wedge \omega ^{n-i}\leq \int _ X \theta _{P_{\theta }[\varphi ]_{\mathcal I}}^ i\wedge \omega ^{n-i} \]

for all $i=0,\ldots ,n$, and the inequality is strict for $i=n$. It follows that

\[ \begin{aligned} \int _ X (\theta +\omega +\mathrm{dd}^{\mathrm{c}}\varphi )^ n =& \sum _{i=0}^ n\binom {n}{i}\int _ X \theta _{\varphi }^ i\wedge \omega ^{n-i}\\ <& \sum _{i=0}^ n\binom {n}{i}\int _ X \theta _{P_{\theta }[\varphi ]_{\mathcal{I}}}^ i\wedge \omega ^{n-i}\\ =& \int _ X (\theta +\omega +\mathrm{dd}^{\mathrm{c}}P_{\theta }[\varphi ]_{\mathcal{I}})^ n \\ \leq & \int _ X (\theta +\omega +\mathrm{dd}^{\mathrm{c}}P_{\theta +\omega }[\varphi ]_{\mathcal{I}})^ n. \end{aligned} \]

So 2 fails as well. $\square$

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