Pome

Corollary 16.9.363EB846B21817

Corollary 16.9.36.3EB846B21817 Let $(\varphi _ i)_{i\in I}$ be a net in $\mathrm{PSH}(X,\theta )$ and $\varphi \in \mathrm{PSH}(X,\theta )$. Let $\omega $ be a closed smooth positive $(1,1)$-form on $X$. Then the following are equivalent:

  1. 9F20E828AA4A

    $\varphi _ i\xrightarrow {d_{S,\theta }}\varphi $;

  2. 952942BD613A

    $\varphi _ i\xrightarrow {d_{S,\theta +\omega }}\varphi $.

Proof. 1 $\implies $ 2. It suffices to show that for each $j=0,\ldots ,n$, we have

\[ \begin{split} 2\int _ X (\theta +\omega )_{\varphi _ i\lor \varphi }^ j\wedge (\theta +\omega )_{V_{\theta +\omega }}^{n-j}-\int _ X (\theta +\omega )_{\varphi _ i}^ j\wedge (\theta +\omega )_{V_{\theta +\omega }}^{n-j}\\ -\int _ X (\theta +\omega )_{\varphi }^ j\wedge (\theta +\omega )_{V_{\theta +\omega }}^{n-j}\to 0. \end{split} \]

Note that this quantity is a linear combination of terms of the following form:

\[ \begin{split} 2\int _ X \theta _{\varphi _ i\lor \varphi }^ r\wedge \omega ^{j-r}\wedge (\theta +\omega )_{V_{\theta +\omega }}^{n-j}-\int _ X \theta _{\varphi _ i}^ r\wedge \omega ^{j-r}\wedge (\theta +\omega )_{V_{\theta +\omega }}^{n-j}\\ -\int _ X \theta _{\varphi }^ r\wedge \omega ^{j-r}\wedge (\theta +\omega )_{V_{\theta +\omega }}^{n-j}, \end{split} \]

where $r=0,\ldots ,j$. By theorem 16.9.30, it suffices to show that $\varphi \lor \varphi _ i \xrightarrow {d_ S}\varphi $. But this follows from corollary 16.9.33.

2 $\implies $ 1. From the direction we already proved, for each $C\geq 1$, we have

\[ \varphi _ i\xrightarrow {d_{S,\theta +C\omega }}\varphi . \]

By theorem 16.9.30, it follows that

\[ \lim _{i\in I}\int _ X (\theta +C\omega )_{\varphi _ i}^ j\wedge \theta _{V_{\theta }}^{n-j}=\int _ X (\theta +C\omega )_{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j} \]

for all $j=0,\ldots ,n$. It follows that

B509AA3C2576 16.9.37
\begin{equation} \lim _{i\in I}\int _ X \theta _{\varphi _ i}^ j\wedge \theta _{V_{\theta }}^{n-j}=\int _ X \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}. \end{equation}

By corollary 16.9.33, it remains to show that $\varphi _ i\lor \varphi \xrightarrow {d_{S,\theta }}\varphi $. By corollary 16.9.33 again, we know that $\varphi _ i\lor \varphi \xrightarrow {d_{S,\theta +\omega }}\varphi $. So it suffices to apply (16.9.37) to $\varphi _ i\lor \varphi $ instead of $\varphi _ i$, and we conclude by 16.9.7. $\square$

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