PomeCorollary 20.2.52.3AEE64F8D403 Assume that $n\geq 2$. Let $T$ be a closed positive $(1,1)$-current on $X$ with $\operatorname{vol}T>0$. Then quasi-every $H\in \Lambda $ is connected and smooth, and $\operatorname{Tr}_ H^{\{ T\} |_ H}(T)$ is well-defined and has positive volume.
Corollary 20.2.523AEE64F8D403
Proof. Write $T=\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ and set $\psi =P_{\theta }[\varphi ]_{\mathcal{I}}$. Then $\psi \sim _{\mathcal{I}}\varphi $ and
\[ \int _ X \theta _{\psi }^{n}=\operatorname{vol}T>0. \]
By 20.2.51, for quasi-every $H\in \Lambda $, the hypersurface $H$ is connected and smooth, $\psi |_ H$ is well-defined, and
\[ \int _ H \left(\theta |_ H+\mathrm{dd}^{\mathrm{c}}\psi |_ H\right)^{n-1}>0. \]
For such $H$, corollary 20.2.50 gives $\operatorname{Tr}_ H(\psi )\sim _{\mathcal{I}}\psi |_ H$. Since $\varphi \sim _{\mathcal{I}}\psi $, proposition 20.2.21 also gives $\operatorname{Tr}_ H(\varphi )\sim _ P\operatorname{Tr}_ H(\psi )$. The assertion now follows from example 20.2.17. $\square$
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