Pome

Proposition 18.28.740DFF36A22512

Proposition 18.28.74.0DFF36A22512 Suppose that $\varphi \in \mathrm{PSH}(X,\theta )$ is such that $\theta _{\varphi }$ is a Kähler current. Let $(K,v)$ be a weighted subset of $X$ and $\mu \in \mathrm{BM}(K,v)$. Then

7EA15066EE63 18.28.75
\begin{equation} \beta ^ k_{v,\varphi ,\mu } \rightharpoonup \theta _{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}^ n\quad \textup{as } k\to \infty . \end{equation}

Proof. Let $\tau $ be the weak limit of a subsequence of $\beta ^ k_{v,\varphi ,\mu }$. We claim that

CBCF54FCD801 18.28.76
\begin{equation} \tau \leq \theta _{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}^ n. \end{equation}

Observe that this claim implies the conclusion. In fact, by theorem 18.1.46, we have equality of the total masses, so equality holds in (18.28.76). As $\tau $ is an arbitrary cluster point of the sequence $(\beta ^ k_{v,\varphi ,\mu })_ k$, we get (18.28.75).

It remains to prove (18.28.76). Let $(\varphi _ j)$ be a quasi-equisingular approximation of $\varphi $ in $\mathrm{PSH}(X,\theta )$. We may assume that $\theta _{\varphi _ j}$ is a Kähler current for all $j\geq 1$. By corollary 18.1.14, we know that

\[ \varphi _ j\xrightarrow {d_ S}P_{\theta }[\varphi ]_{\mathcal{I}}. \]

The quasi-equisingular approximation is decreasing, so applying the decreasing case of proposition 18.28.35 to the limit $P_{\theta }[\varphi ]_{\mathcal{I}}$, and using the definition of the relative $\mathcal{I}$-envelope, we get

\[ P_{\theta ,K}\left[\varphi _ j\right]_{\mathcal{I}}(v)\searrow P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v). \]

In particular,

FD0D14628CD8 18.28.77
\begin{equation} \lim _{j\to \infty } \int _ X \theta _{P_{\theta ,K}\left[\varphi _ j\right]_{\mathcal{I}}(v)}^ n= \int _ X \theta _{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}^ n. \end{equation}

Observe that

\[ \beta ^ k_{v,\varphi ,\mu } \leq \beta ^ k_{v,\varphi _ j,\mu } \]

for any $k \geq 1$. As $\mu \in \mathrm{BM}(K,v)$, by proposition 18.28.73,

\[ \tau \leq \theta _{P_{\theta ,K}[\varphi _ j]_{\mathcal{I}}(v)}^ n, \]

for any $j \geq 1$ fixed. By (18.28.77) and the semicontinuity theorem theorem 13.13.9, we have

\[ \theta _{P_{\theta ,K}\left[\varphi _ j\right]_{\mathcal{I}}(v)}^ n\rightharpoonup \theta _{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}^ n \]

as $j\to \infty $. Therefore, (18.28.76) follows. $\square$

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