18.1.1 The notion of $\mathcal{I}$-good singularities466586A864A3

Proof. 1 $\implies $ 2. By theorem 16.9.30, we have

We may therefore assume that $\int _ X \theta _{\varphi _ j}^ n>0$ for all $j \geq 1$. It follows from proposition 16.6.22 that

for any $j\geq 1$. Using theorem 16.9.57, we conclude (18.1.3).

2 $\iff $ 3. By definition,

Moreover, $\varphi -\sup _ X\varphi \leq P_{\theta }[\varphi ]_{\mathcal I}$. Thus, if 2 holds, theorem 16.3.15 gives $P_{\theta }[\varphi ]_{\mathcal I}\leq P_{\theta }[\varphi ]$; the reverse inequality follows from proposition 16.6.18. Conversely, if 3 holds, then 2 follows from proposition 16.3.10.

3 $\implies $ 1. Note that the condition in 1 characterizes the closure of analytic singularities in $\mathrm{PSH}(X,\theta )$.

Step 1. We first assume that $\theta _{\varphi }$ is a Kähler current. We will prove the following more general result in this case: Without assuming 3, $P_{\theta }[\varphi ]_{\mathcal{I}}$ always lies in the closure of analytic singularities.

Let $(\varphi _ j)_ j$ be a quasi-equisingular approximation of $\varphi $ in $\mathrm{PSH}(X,\theta )$. We will show that $\varphi _ j\xrightarrow {d_ S}P_{\theta }[\varphi ]_{\mathcal{I}}$. Let

Since $\varphi \leq \varphi _ j$ for all $j>0$, we have

for all $j$, and hence

In particular, $\int _ X\theta _\psi ^ n>0$.

Now proposition 16.9.6, proposition 16.4.4 and corollary 16.9.14 give

Moreover, observe that $\psi $ is $\mathcal{I}$-model by proposition 16.6.29 and proposition 16.6.22. Hence, if $\varphi \sim _{\mathcal I}\psi $, then

So it suffices to show that $\varphi \sim _{\mathcal I}\psi $. It only remains to argue that $\psi \preceq _{\mathcal{I}} \varphi $. For this purpose, take $\lambda >0$. We need to show that

By the strong openness theorem 5.16.21, we may take $\lambda '>\lambda $ such that $\mathcal{I}(\lambda \psi )=\mathcal{I}(\lambda ' \psi )$. It follows from the definition of the quasi-equisingular approximation that

for large enough $j$, where the first inclusion uses $\psi \leq P_\theta [\varphi _ j]$ and $P_\theta [\varphi _ j]\sim _{\mathcal{I}}\varphi _ j$. Our assertion follows.

It follows from the proof that we may take $\varphi _ j$ so that $\theta _{\varphi _ j}$ is a Kähler current for all $j\geq 1$.

Step 2. We handle the general case.

Assume 3 holds. By 13.13.47, we can find $\psi \in \mathrm{PSH}(X,\theta )$ so that $\theta _{\psi }$ is a Kähler current and $\psi \leq \varphi $. We let

for each $j\in \mathbb {Z}_{>1}$. Then $(\psi _ j)_ j$ is an increasing sequence converging almost everywhere to $\varphi $. Then

by proposition 16.6.30, corollary 16.9.12. From Step 1, we know that each $P_{\theta }[\psi _ j]_{\mathcal{I}}$ lies in the closure of analytic singularities, hence so is $P_{\theta }[\varphi ]\sim _ P \varphi $. Therefore, 1 follows. $\square$

Proof. 1 $\implies $ 2. By theorem 18.1.2, we can find a sequence $(\varphi _ j)_ j$ in $\mathrm{PSH}(X,\theta )$ with analytic singularities such that $\varphi _ j\xrightarrow {d_{S,\theta }}\varphi $. By corollary 16.9.36, we have $\varphi _ j\xrightarrow {d_{S,\theta +\omega }}\varphi $. Therefore, by theorem 18.1.2 again, 2 holds.

2 $\implies $ 1. Suppose that 1 fails, so that

Since $\varphi \preceq P_{\theta }[\varphi ]_{\mathcal I}$, the monotonicity theorem theorem 13.13.11 gives

for all $i=0,\ldots ,n$, and the inequality is strict for $i=n$. It follows that

So 2 fails as well. $\square$

Proof. By adding a Kähler form to $\theta $, and using 18.1.8, corollary 16.9.36, we may assume that $\varphi _ j,\varphi \in \mathrm{PSH}(X,\theta )_{>0}$ for all $j\in I$. It follows from theorem 18.1.2 that

for all $j\in I$. Taking limit with respect to $j$ with the help of theorem 16.9.57, we conclude that

Therefore, by theorem 18.1.2 again, we find that $\varphi $ is $\mathcal{I}$-good. $\square$

Proof. By corollary 16.9.36 and example 18.1.11, we may replace $\theta $ by $\theta +C\omega $ for some large constant $C>0$ and assume that $\varphi ,\varphi _ j\in \mathrm{PSH}(X,\theta -\omega )$ for all $j\geq 1$.

Here the limit in 1 is unchanged by this replacement. Indeed, $P_{\theta }[\varphi ]_{\mathcal I}$ is $\mathcal I$-model, hence $\mathcal I$-good by example 18.1.11. This implies

Thus the old and new targets are $P$-equivalent in the enlarged class, and proposition 16.9.6, corollary 16.9.36 shows that condition 1 is unchanged.

2 $\implies $ 1. This is already proved in the proof of theorem 18.1.2.

1 $\implies $ 2. This follows from theorem 16.9.59. $\square$

Proof. Let $(\epsilon _ j)_ j$ and $\omega $ be as in definition 6.17.5, so that $\epsilon _ j\searrow 0$ and $\varphi _ j\in \mathrm{PSH}(X,\theta +\epsilon _ j\omega )$ for all $j$. Choose a Kähler form $\tilde\omega $ on $Y$ and a constant $C>0$ such that $\pi ^*\omega \leq C\tilde\omega $. Then

Moreover, the sequence $(\pi ^*\varphi _ j)_ j$ is decreasing, has analytic singularities, and decreases to $\pi ^*\varphi $.

It remains to verify the quasi-equisingular condition definition 6.17.5 4. Fix a constant $A>0$. Then $(\varphi _ j)_ j$ is also a quasi-equisingular approximation of $\varphi $ with respect to $\theta +A\omega $. Since $\theta _{\varphi }+A\omega $ is a Kähler current, corollary 18.1.14 gives

Viewing the convergence in a fixed enlarged background class, pulling back and using corollary 16.9.11, proposition 16.6.10, we get

The domination $\pi ^*\omega \leq C\tilde\omega $ also gives

so the preceding convergence is in the setting of corollary 18.1.14 on $Y$. The potential $\pi ^*\varphi $ has positive mass with respect to $\pi ^*(\theta +A\omega )$ by the bimeromorphic invariance of non-pluripolar products proposition 13.13.4. Applying corollary 18.1.14 again on $Y$ shows that $(\pi ^*\varphi _ j)_ j$ satisfies the quasi-equisingular condition definition 6.17.5 4 for $\pi ^*\varphi $. Combining this with the first paragraph proves the assertion with respect to $\pi ^*\theta $. $\square$