PomeProposition 19.4.33.86E922DD5140 Let $(L', h')$ be another Hermitian big line bundle on $X$. Then
\[ \Delta _{\nu }(L,h)+\Delta _{\nu }(L', h')\subseteq \Delta _{\nu }(L\otimes L',h\otimes h'). \]
Proposition 19.4.3386E922DD5140
Proof. Take a Hermitian metric $h_0'$ on $L'$ and let $\theta '=c_1(L',h_0')$. We identify $h'$ with $\varphi '\in \mathrm{PSH}(X,\theta ')$. Then we need to show
70AAFA1F54AB
19.4.34
\begin{equation} \Delta _{\nu }(\theta ,\varphi )+\Delta _{\nu }(\theta ',\varphi ')\subseteq \Delta _{\nu }(\theta +\theta ',\varphi +\varphi '). \end{equation}
We observe that
\[ P_{\theta }[\varphi ]_{\mathcal{I}}+P_{\theta '}[\varphi ']_{\mathcal{I}}\sim _{\mathcal{I}} \varphi +\varphi '. \]
Thus, after replacing $\varphi $ and $\varphi '$ by their $\mathcal{I}$-envelopes, in view of proposition 19.4.28, we may assume that $\varphi $ and $\varphi '$ are $\mathcal{I}$-good.
By theorem 18.1.2, we can find sequences $(\varphi _ j)_ j$ and $(\varphi '_ j)_ j$ in $\mathrm{PSH}(X,\theta )_{>0}$ and $\mathrm{PSH}(X,\theta ')_{>0}$ respectively such that
9C52F43A7B75$\varphi _ j$ and $\varphi '_ j$ both have analytic singularities for all $j\geq 1$, and
3F2CA81FC88F$\varphi _ j\xrightarrow {d_ S}\varphi $, $\varphi '_ j\xrightarrow {d_ S}\varphi '$.
Then $\varphi _ j+\varphi '_ j\in \mathrm{PSH}(X,\theta +\theta ')_{>0}$ and $\varphi _ j+\varphi '_ j\xrightarrow {d_ S} \varphi +\varphi '$ by theorem 16.9.45. Thus, by theorem 19.4.30, we may assume that $\varphi $ and $\varphi '$ both have analytic singularities. Taking a birational resolution, we may further assume that they have log singularities. By 19.4.23, we reduce to the case without singularities, in which case the result is just proposition 19.4.13. $\square$
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