Lemma 20.2.54

Lemma 20.2.54.A172685DB2AD Assume that $n\geq 2$. Let $\theta $ be a closed smooth $(1,1)$-form on $X$ representing a big cohomology class and $(\varphi _ j)_ j$ be a decreasing sequence in $\mathrm{PSH}(X,\theta )$. Assume that $\varphi \in \mathrm{PSH}(X,\theta )$ and $\varphi _ j\xrightarrow {d_ S}\varphi $. Furthermore, assume that $\varphi \leq \varphi _ j$ for each $j>0$. Then quasi-every $H\in \Lambda $ is connected and smooth, $\varphi _ j|_ H\not\equiv -\infty $ for all $j\geq 1$, $\varphi |_ H\not\equiv -\infty $, and

\[ \varphi _ j|_ H\xrightarrow {d_ S}\varphi |_ H. \]

Proof. By corollary 16.9.36, we may assume that $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$.

Using 13.13.30, we can find a decreasing sequence $(\epsilon _ j)_ j$ in $(0,1)$ with limit $0$ and $\eta _ j\in \mathrm{PSH}(X,\theta )_{>0}$ such that $\eta _ j\leq \varphi _ j$ and

\[ \epsilon _ j\eta _ j+(1-\epsilon _ j)\varphi _ j\leq \varphi . \]

By corollary 20.2.49, 20.2.51, we can find a co-pluripolar set $\Lambda '\subseteq \Lambda $ such that any $H\in \Lambda '$ satisfies:

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$H$ is smooth and connected;
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$\eta _ j|_ H\in \mathrm{PSH}(H,\theta |_ H)_{>0}$ for all $j\geq 1$ and $\varphi |_ H\in \mathrm{PSH}(H,\theta |_ H)_{>0}$.

Therefore, taking proposition 16.4.2 into account, we arrive at

\[ \epsilon _ j P_{\theta |_ H}\bigl [\eta _ j|_ H\bigr ]+\left(1-\epsilon _ j\right)P_{\theta |_ H}\bigl [\varphi _ j|_ H\bigr ]\leq P_{\theta |_ H}\bigl [\varphi |_ H\bigr ]. \]

Since the normalized envelopes $P_{\theta |_ H}[\eta _ j|_ H]$ are relatively compact in $L^1$, we have $\epsilon _ jP_{\theta |_ H}[\eta _ j|_ H]\to 0$ in $L^1$. Letting $j\to \infty $, we get

\[ \lim _{j\to \infty } P_{\theta |_ H}\bigl [\varphi _ j|_ H\bigr ]\leq P_{\theta |_ H}\bigl [\varphi |_ H\bigr ]. \]

The reverse inequality is trivial, so we get an equality here. By theorem 13.13.11 and proposition 16.4.4, we conclude that

\[ \lim _{j\to \infty }\int _ H \Bigl (\theta |_ H+\mathrm{dd}^{\mathrm{c}}\varphi _ j|_ H\Bigr )^{n-1}=\int _ H \Bigl (\theta |_ H+\mathrm{dd}^{\mathrm{c}}\varphi |_ H\Bigr )^{n-1}. \]

Therefore, using corollary 16.9.14, we conclude that $\varphi _ j|_ H\xrightarrow {d_ S}\varphi |_ H$. $\square$

Lemma 20.2.54`A172685DB2AD`

Comments

Lemma 20.2.54A172685DB2AD

Comments

Lemma 20.2.54`A172685DB2AD`