PomeProposition 18.1.36.BB7441317AAF Let $T_1,\ldots ,T_ n$ be closed positive $(1,1)$-currents on $X$. Then
\[ \operatorname{vol}(T_1,\ldots ,T_ n)\geq \prod _{i=1}^ n(\operatorname{vol}T_ i)^{1/n}. \]
Proposition 18.1.36BB7441317AAF
Proof. We may assume that $\operatorname{vol}T_ i>0$ for each $i=1,\ldots ,n$ since there is nothing to prove otherwise. In this case, we need to show that
\[ \begin{split} & \int _ X \left(\theta _1+\mathrm{dd}^{\mathrm{c}}P_{\theta _1}[\varphi _1]_{\mathcal{I}}\right)\wedge \cdots \wedge \left(\theta _ n+\mathrm{dd}^{\mathrm{c}}P_{\theta _ n}[\varphi _ n]_{\mathcal{I}}\right)\\ \geq & \prod _{i=1}^ n\left(\int _ X \left(\theta _ i+\mathrm{dd}^{\mathrm{c}}P_{\theta _ i}[\varphi _ i]_{\mathcal{I}} \right)^ n \right)^{1/n}. \end{split} \]
This is a special case of theorem 13.13.5. $\square$
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