Pome

Theorem 18.28.49EA3E101DCA5E

Theorem 18.28.49.EA3E101DCA5E Suppose that $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then

A6F9F2DF4A0D 18.28.50
\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 of theorem 18.28.49. By 18.28.15 and proposition 16.6.11, the three potentials below have the same multiplier ideals. Hence

\[ \begin{aligned} \mathrm{H}^0\left(X,L^ k \otimes T \otimes \mathcal{I}(k\varphi )\right) & = \mathrm{H}^0\left(X,L^ k \otimes T \otimes \mathcal I(kP_{\theta }[\varphi ]_{\mathcal{I}})\right)\\ & = \mathrm{H}^0\left(X,L^ k \otimes T \otimes \mathcal I\left(kP_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)\right)\right). \end{aligned} \]

This allows us to replace $\varphi $ with $P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)$.

By 13.13.47, there exists a sequence $(\varphi _ j)_ j$ in $\mathrm{PSH}(X,\theta )$ such that $\varphi _ j\nearrow \varphi $ a.e. and $\theta _{\varphi _ j}$ is a Kähler current for each $j\geq 1$. This gives

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

Let $\tau $ be the weak limit of a subsequence of $(\beta ^ k_{v,\varphi ,\mu })_ k$. Then by proposition 18.28.74,

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

By proposition 18.28.35 and corollary 13.13.12, we have

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

Hence,

8FD19D2B07C7 18.28.78
\begin{equation} \theta _{P_{\theta ,K} [\varphi ]_{\mathcal{I}}(v)}^ n\leq \tau . \end{equation}

A comparison of total masses using (18.28.48) gives that equality holds in (18.28.78). As $\tau $ is arbitrary, we obtain (18.28.50). $\square$

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