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Lemma 20.2.51CB547C0AF98F

Lemma 20.2.51.CB547C0AF98F Assume that $n\geq 2$. Let $T$ be a closed positive $(1,1)$-current on $X$ with $\int _ X T^ n>0$. Then quasi-every $H\in \Lambda $ is connected and smooth, $T|_ H$ is well-defined and satisfies

\[ \int _ H T|_ H^{n-1}>0. \]

Proof. Write $T=\theta _{\varphi }$ for some smooth closed real $(1,1)$-form $\theta $ on $X$ and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Thanks to 13.13.47, we can find $\psi \in \mathrm{PSH}(X,\theta )$ such that $\theta _{\psi }$ is a Kähler current and $\psi \leq \varphi $. By corollary 20.2.49 and the connectedness assumption on $\Lambda $, we can find a co-pluripolar set $\Lambda '\subseteq \Lambda $ such that each $H\in \Lambda '$ satisfies:

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    $H$ is smooth and connected;

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    the restriction $\psi |_ H$ is not identically $-\infty $.

Since $\theta _\psi $ is a Kähler current, its restriction to such an $H$ has positive non-pluripolar mass. As $\psi |_ H\leq \varphi |_ H$, theorem 13.13.11 gives

\[ \int _ H(\theta |_ H+\mathrm{dd}^{\mathrm{c}}\varphi |_ H)^{n-1} \geq \int _ H(\theta |_ H+\mathrm{dd}^{\mathrm{c}}\psi |_ H)^{n-1}>0. \]

We conclude the proof. $\square$

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