16.9.1 The definition of the $d_ S$-pseudometricBB740CB4BE99

Proof. This follows immediately from theorem 17.10.81. $\square$

Proof. We claim that

Recall that $\lor $ is defined in definition 17.10.85. Note that this assertion implies our desired inequality by 17.10.90.

In proving this assertion, we may assume that $\varphi ,\psi \leq 0$ since

for any $C\in \mathbb {R}$.

In fact, it is clear that

so the $\leq $ direction in (16.9.5) holds.

Conversely, if $\ell '\in \mathcal{R}^1(X,\theta )$ and $\ell '\geq \ell ^{\varphi }\lor \ell ^{\psi }$, then for each $t\geq 0$,

See (17.10.96) for the first inequality. Therefore, thanks to the definition of the Perron envelopes (cf. (17.10.21)), we have

for any $0\leq s\leq t$. It follows from (17.10.96) that $\ell '_ s\geq \ell ^{\varphi \lor \psi }_ s$ for any $s\geq 0$. $\square$

Proof. By 16.8.13, we have $\varphi \sim _ P \psi $ if and only if $\varphi \sim _ P \varphi \lor \psi $ and $\psi \sim _ P \varphi \lor \psi $. By 16.9.4, $d_ S(\varphi ,\psi )=0$ if and only if $d_ S(\varphi ,\varphi \lor \psi )=0$ and $d_ S(\psi ,\varphi \lor \psi )=0$. So it suffices to prove the assertion when $\varphi \leq \psi $. Assuming this, by proposition 17.10.83, 2 holds if and only if

where $\mathbf{E}$ is introduced in definition 17.10.82. But by (17.10.94), this holds if and only if

Thanks to the monotonicity theorem theorem 13.13.11, this holds if and only if for all $j=0,\ldots ,n$,

which is equivalent to 1 by proposition 16.8.7. $\square$

Proof. Set $\chi \coloneqq \varphi \lor \psi $. By 16.8.13, we have $\chi \sim _ P\psi $. Hence proposition 16.9.6 gives $d_ S(\chi ,\psi )=0$, and the pseudometric property gives

Since $\varphi \leq \chi $, (17.10.94) and proposition 17.10.83 give

Finally, proposition 16.8.7 applied to $\psi \leq \chi $ gives

for all $j=0,\ldots ,n$. $\square$

Proof. This is an immediate consequence of 16.9.7 and the monotonicity theorem proposition 16.8.22. $\square$

Proof. The estimates (16.9.10) follow from the combination of 16.9.7 and 16.9.4.

Suppose that $\varphi _ i\xrightarrow {d_ S}\varphi $. Then $\varphi _ i\lor \varphi \xrightarrow {d_ S}\varphi $ by 16.9.4. Therefore, theorem 13.13.11 and 16.9.7 imply that

for any $j=0,\ldots ,n$. The last assertion now follows from (16.9.10) and theorem 13.13.11. $\square$

Proof. It suffices to observe that the middle term in (16.9.10) is invariant under $\pi ^*$. $\square$

Proof. Write $\varphi =\operatorname *{sup}_{j\in I}{}\! {}^{*}\varphi _ j$. Recall that by proposition 5.14.2, $\varphi \in \mathrm{PSH}(X,\theta )$. By 16.9.7, it suffices to show that for each $k=0,\ldots ,n$, we have

The latter follows from corollary 13.13.12. $\square$

Proof. We compute

where the first line is just the triangle inequality, the second line follows from corollary 16.9.9 and the third line follows from 16.9.4. $\square$

Proof. 1 $\iff $ 2. This follows immediately from 16.9.7.

Assume that $\int _ X \theta _{\varphi }^ n>0$.

2 $\implies $ 1. This is the case $k=n$ in (16.9.15).

1 $\implies $ 2. Assume (3). If for some $i\in I$, $\int _ X \theta _{\varphi _ i}^ n= \int _ X \theta _{\varphi }^ n$, then for all $j\geq i$ we have

Hence $\int _ X\theta _{\varphi _ j}^ n=\int _ X\theta _{\varphi }^ n$ by proposition 16.8.22. It follows from proposition 16.8.7 that $\varphi _ j\sim _ P\varphi $, and (2) follows from proposition 16.3.10. So in the sequel, we shall assume that for all $i\in I$, we have $\int _ X \theta _{\varphi _ i}^ n>\int _ X \theta _{\varphi }^ n$. In particular, each $\varphi _ i$ has positive mass.

Let $(b_ j)_{j\in I}$ be a net converging to $\infty $ such that

This can be achieved for example by taking $b_ j$ as the midpoint of this interval. By theorem 16.8.23, for each $j\in I$, we can find $\eta _ j\in \mathrm{PSH}(X,\theta )$ such that

It follows from the monotonicity theorem theorem 13.13.11 that for any $k=0,\ldots ,n$,

Taking the limit, we conclude the $\leq $ direction in (16.9.15). The $\geq $ direction follows from proposition 16.8.22.

Finally, we argue (16.9.16). Let $\psi _ j=P_{\theta }[\varphi _ j]$. It follows from corollary 16.3.20 that $\psi _ j$ is a model potential. Since the net $(\varphi _ j)_{j\in I}$ is decreasing with respect to the $P$-preorder, $(\psi _ j)_{j\in I}$ is decreasing pointwise. Let

By proposition 16.4.3, $\psi $ is a model potential. Since $P_{\theta }[\varphi ]\leq \psi \leq \psi _ j$ for all $j\in I$, the monotonicity theorem theorem 13.13.11 and proposition 16.3.10 give

Hence $\psi $ has the same mass as $P_{\theta }[\varphi ]$. As $\psi $ is model and $\psi \geq P_{\theta }[\varphi ]$, theorem 16.3.15 gives $\psi =P_{\theta }[\varphi ]$. $\square$

Proof. First note that $\int _ X \theta _{\varphi }^ n\geq \delta $ as well, due to the $d_ S$-continuity of non-pluripolar masses as proved in corollary 16.9.13. Replacing $\varphi $ by $P_{\theta }[\varphi ]$, we may then assume that $\varphi $ is a model potential in $\mathrm{PSH}(X,\theta )_{>0}$.

We are free to replace $(\varphi _ j)_ j$ by a subsequence. So we may assume that

where $C_ n$ is the constant in corollary 16.9.9, $D_ n$ is the constant in corollary 16.9.13.

In particular, by corollary 16.9.13,

Step 1. We handle the $\psi _ j$'s. For each $j\geq 1$ and $k\geq 1$, by 16.9.4 we have

By iteration, we find

Using corollary 16.9.12, we have

as $k\to \infty $. Hence

We conclude that $\psi _ j\xrightarrow {d_ S}\varphi $.

Moreover, we observe that

Indeed, set $\tilde{\psi }_ j=P_{\theta }[\psi _ j]$. Then $(\tilde{\psi }_ j)_ j$ is a decreasing sequence of model potentials, and $d_ S(\tilde{\psi }_ j,\psi _ j)=0$ by proposition 16.9.6. Hence $\tilde{\psi }_ j\xrightarrow {d_ S}\varphi $. If $\chi =\inf _ j\tilde{\psi }_ j$, then proposition 16.4.4 and corollary 16.9.14 give $\tilde{\psi }_ j\xrightarrow {d_ S}\chi $. Thus $d_ S(\chi ,\varphi )=0$. Since both $\chi $ and $\varphi $ are model potentials, proposition 16.9.6 gives $P_{\theta }[\chi ]=P_{\theta }[\varphi ]$, hence $\chi =\varphi $, proving (16.9.21).

Step 2. We consider the $\eta _ j$'s.

For each $j\geq 1$ and $k\geq 0$, we let

Using (16.9.18), (16.9.20) and corollary 16.9.13, we have

for all $j\geq 1$. Taking (16.9.19) into account, we have

for $j>1$.

Step 2.1. We claim that for a fixed $j\geq 1$, for any $k\in \mathbb {N}$, we have $\eta _ j^ k\in \mathrm{PSH}(X,\theta )$ and

We argue by induction on $k\geq 0$. The case $k=0$ is trivial. When $k>0$, assume that the case $k-1$ is known. Then

where the first inequality follows from the inductive hypothesis and (16.9.22).

Observe that

it follows from proposition 16.3.25 that $\eta _ j^ k\in \mathrm{PSH}(X,\theta )$. By theorem 16.3.35, we deduce that

where the second inequality follows from the inductive hypothesis and (16.9.22). Therefore, (16.9.23) follows.

Step 2.2. It follows by induction from proposition 16.3.29 that for any $j\geq 1$, $k\geq 0$,

By proposition 16.4.4, we have

for any $j\geq 1$. Letting $k\to \infty $ in (16.9.23), we find that

for $j\geq 1$. Observe that we also have

for $j\geq 1$ by theorem 13.13.11. It follows from these inequalities and (16.9.19) that

For $k\geq 1$, we have

Letting $k\to \infty $ gives $\eta _ j\leq \eta _{j+1}$. Hence $(\eta _ j)_ j$ is increasing. Hence corollary 13.13.12 gives

where $\eta =\operatorname *{sup}_{j\geq 1}{}\! {}^{*} \eta _ j$. For fixed $m$, if $j\geq m$, then $\eta _ j\leq \varphi _ j\leq \psi _ m$, whereas if $j<m$, then $\eta _ j\leq \eta _ m\leq \psi _ m$. Thus $\eta _ j$ is a candidate in the definition of $P_{\theta }[\psi _ m]$ for all $j,m$, and (16.9.21) gives $\eta \leq \varphi $. It follows from proposition 16.8.7 that $\eta \sim _ P \varphi $. By corollary 16.9.12 and proposition 16.9.6, we have $\eta _ j\xrightarrow {d_ S}\varphi $. $\square$

Proof. In a pseudometric space, sequential completeness implies completeness for Cauchy nets. So for the first assertion, we may assume that $(\varphi _ j)_ j$ is a Cauchy sequence.

We first treat the case of a decreasing sequence $(\gamma _ j)_ j$ with $\int _ X\theta _{\gamma _ j}^ n\geq \delta $ for all $j$. Put $\tilde\gamma _ j=P_\theta [\gamma _ j]$. Then $(\tilde\gamma _ j)_ j$ is a decreasing sequence of model potentials, $d_ S(\gamma _ j,\tilde\gamma _ j)=0$, and

by proposition 16.3.10. By proposition 16.4.4, the decreasing limit $\tilde\gamma =\inf _ j\tilde\gamma _ j$ is model and

Hence $\tilde\gamma _ j\to \tilde\gamma $ in $d_ S$ by corollary 16.9.14, and therefore $\gamma _ j$ also converges in $d_ S$.

Next let $(\varphi _ j)_{j>0}$ be a $d_ S$-Cauchy sequence with masses bounded from below by $\delta $. Put

By corollary 16.3.20, example 16.8.6, proposition 16.9.6, the $\tilde{\varphi }_ j$'s are model potentials, $d_ S(\varphi _ j,\tilde{\varphi }_ j)=0$, and

by proposition 16.3.10. Replacing $\varphi _ j$ by $\tilde{\varphi }_ j$, we may therefore assume that the $\varphi _ j$'s are model potentials. Passing to a subsequence, we may also assume that

where $C_ n$ is the constant in corollary 16.9.9. Define

for each $j>0$. As in the proof of proposition 16.9.17 Step 1, especially (16.9.20), we know that

The sequence $(\psi _ j)_ j$ is decreasing and satisfies $\int _ X\theta _{\psi _ j}^ n\geq \delta $ by theorem 13.13.11. By the first paragraph, $(\psi _ j)_ j$ converges in $d_ S$. Hence the chosen subsequence of $(\varphi _ j)_ j$ converges, and the original Cauchy sequence converges as well.

Finally, suppose that $(\varphi _ j)_{j\in I}$ is a decreasing net with masses bounded from below by $\delta $. For each $k=0,\ldots ,n$, the net

is decreasing and bounded from below, hence Cauchy as a real-valued net. If $j'\geq j$, then $\varphi _{j'}\leq \varphi _ j$, and 16.9.7 expresses $d_ S(\varphi _{j'},\varphi _ j)$ as a finite sum of the corresponding differences of these mixed masses. For two sufficiently large indices, we choose a common upper bound in $I$ and use the triangle inequality. It follows that $(\varphi _ j)_{j\in I}$ is $d_ S$-Cauchy. The first assertion gives its convergence. $\square$

Proof. After adding a constant, we may assume that $\varphi \leq 0$. Since $\theta _{\varphi }$ is a Kähler current, $(1-\epsilon )\varphi \in \mathrm{PSH}(X,\theta )$ for all small enough $\epsilon >0$. Fix such an $\epsilon $. Take any Kähler form $\omega $ on $X$ so that $\omega -\theta $ is still a Kähler form. Note that the choice of $\omega $ depends only on $X$ and $\theta $.

Since $\varphi \preceq (1-\epsilon )\varphi $, by 16.9.7 and the multi-linearity of the non-pluripolar product proposition 13.11.4 3, we can compute

where, for $1\leq m\leq n$,

Note that

where we used the monotonicity theorem theorem 13.13.11. Hence the whole expression is of the order of $\mathcal{O}(\epsilon )$, and our assertion follows. $\square$