PomeCE8C594C13B9Let $X$ be a connected compact Kähler manifold of dimension $n$.
We show that $\mathcal{I}$-goodness is preserved by a number of natural operations.Proposition 18.1.17.AC437613D2C1 Let $\varphi ,\psi \in \mathrm{QPSH}(X)$ be $\mathcal{I}$-good and $\lambda >0$. Then the following potentials are all $\mathcal{I}$-good:
DF6905DB50A6$\varphi +\psi $;
0771DD6805B8$\varphi \lor \psi $;
51229C9B0C0E$\lambda \varphi $.
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. $\square$
Example 18.1.18.3689F6A0A9EC Let $L$ be a pseudo-effective line bundle on $X$. Elementary metrics on $L$ are defined in definition 16.8.20. Let $h$ be an elementary metric on $L$. Then $\mathrm{dd}^{\mathrm{c}}h$ is $\mathcal{I}$-good.
This is a direct consequence of proposition 18.1.17 and example 18.1.10, and the fact that $\mathcal{I}$-goodness depends only on the $P$-singularity type.
Proposition 18.1.19.FD679E0ECD3D Let $(\varphi _ j)_{j\in I}$ be a non-empty family of $\mathcal{I}$-good potentials in $\mathrm{PSH}(X,\theta )$ for some closed real smooth $(1,1)$-form $\theta $ on $X$. Assume that this family is locally uniformly bounded from above. Then $\operatorname *{sup}_{j\in I}{}\! {}^{*} \varphi _ j$ is $\mathcal{I}$-good.
Proof. After adding a Kähler form to $\theta $, we may assume that $\varphi _ j\in \mathrm{PSH}(X,\theta )_{>0}$ for all $j\in I$.
When $I$ is finite, this result follows from proposition 18.1.17. When $I$ is infinite, by Choquet's lemma proposition 5.14.3, we may assume that $I=\mathbb {Z}_{>0}$. Replacing $\varphi _ j$ by $\varphi _1\lor \cdots \lor \varphi _ j$ and using proposition 18.1.17, we may assume that the sequence $(\varphi _ j)_ j$ is increasing. In this case, as shown in corollary 16.9.12,
Therefore, $\operatorname *{sup}_{i\in \mathbb {Z}_{>0}}{}\! {}^{*} \varphi _ i$ is $\mathcal{I}$-good by corollary 18.1.9. $\square$
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