Proposition 18.1.17AC437613D2C1

Proof. Take a closed real smooth $(1,1)$-form $\theta $ on $X$ such that $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. It follows from theorem 18.1.2 that there are sequences $(\varphi _ j)_ j,(\psi _ j)_ j$ in $\mathrm{PSH}(X,\theta )$ with analytic singularities such that $\varphi _ j\xrightarrow {d_ S}\varphi $ and $\psi _ j\xrightarrow {d_ S}\psi $.

By theorem 16.9.45, proposition 16.9.42, we have

By proposition 5.18.2, the potentials $\varphi _ j+\psi _ j$ and $\varphi _ j\lor \psi _ j$ have analytic singularities. So $\varphi +\psi $ and $\varphi \lor \psi $ are $\mathcal{I}$-good by theorem 18.1.2.

On the other hand, 16.9.41 gives

Here the convergence is understood in the scaled class. So by theorem 18.1.2, in order to show that $\lambda \varphi $ is $\mathcal{I}$-good, it suffices to show that $\lambda \varphi _ j$ is $\mathcal{I}$-good. In other words, we may assume that $\varphi $ has analytic singularities. We may further assume that $\varphi \leq 0$.

In this case, take a decreasing sequence of rational numbers $\lambda _ j\searrow \lambda $. Take a sufficiently large Kähler form $\omega $ on $X$ so that $\lambda _ j\varphi ,\lambda \varphi \in \mathrm{PSH}(X,\omega )_{>0}$ for any $j$. Then observe that

Therefore, $\lambda _ j\varphi \xrightarrow {d_ S} \lambda \varphi $ by corollary 16.9.12. Each $\lambda _ j \varphi $ has analytic singularities, and hence is $\mathcal{I}$-good by example 18.1.10. Therefore, $\lambda \varphi $ is $\mathcal{I}$-good by theorem 18.1.2. $\square$